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  • 51.
    Elfving, Tommy
    et al.
    Linköping University, Department of Mathematics, Scientific Computing. Linköping University, The Institute of Technology.
    Nikazad, Touraj
    Dept. of Mathematics, Iran Uv of Science and Technology.
    Popa, Constantin
    Faculty of Math. and Comp. Science, Ovidius uv. Romania.
    A Class of Iterative Methods: Semi-Convergence, Stopping Rules, Inconsistency, and Constraining2010In: Biomedical Mathematics: Promising Directions in Imaging,Therapy Planning and Inverse Problems / [ed] Y. Censor, M. Jiang and G. Wang, Madison, Wi, USA: Medical Physics Publishing , 2010, p. 157-184Chapter in book (Refereed)
    Abstract [en]

    This book brings together 27 state-of-the-art research and review papers by leading experts and practitioners in mathematical methods in biomedical imaging, in intensity-modulated radiation therapy (IMRT) and in optimization and inverse problems. These papers were presented at the Huangguoshu International Interdisciplinary Conference on Biomedical Mathematics Promising Directions in Imaging, Therapy Planning, and Inverse Problems November 3 9, 2008 in China. The emphasis is on trying to discover relations and connections between these fields that will enhance progress in each of them. As this volume shows, applicable mathematical work in these fields goes hand-in-hand with real-world applications and the mutual technology transfers between them leads to further progress. The topics covered here include mathematical aspects and practical problems in current major and emerging technologies in diagnostic and therapeutic medicine and biology research. The contributed work signifies the interdisciplinary cooperation between mathematicians and scientists from medical physics, engineering, clinical medicine, and biology that leads to mathematically based better solutions of practical problems in biomedical imaging and IMRT.

  • 52.
    Elfving, Tommy
    et al.
    Linköping University, Department of Mathematics, Scientific Computing. Linköping University, The Institute of Technology.
    Skoglund, Ingegerd
    Linköping University, Department of Mathematics, Scientific Computing. Linköping University, The Institute of Technology.
    A block-preconditioner for a special regularized least-squares problem2007In: Linear Algebra with Applications, ISSN 1070-5325, Vol. 14, no 6, p. 469-484Article in journal (Refereed)
    Abstract [en]

    We consider a linear system of the form A1x1 + A2x2 + =b1. The vector consists of independent and identically distributed random variables all with mean zero. The unknowns are split into two groups x1 and x2. It is assumed that AA1 has full rank and is easy to invert. In this model, usually there are more unknowns than observations and the resulting linear system is most often consistent having an infinite number of solutions. Hence, some constraint on the parameter vector x is needed. One possibility is to avoid rapid variation in, e.g. the parameters x2. This can be accomplished by regularizing using a matrix A3, which is a discretization of some norm (e.g. a Sobolev space norm). We formulate the problem as a partially regularized least-squares problem and use the conjugate gradient method for its solution. Using the special structure of the problem we suggest and analyse block-preconditioners of Schur compliment type. We demonstrate their effectiveness in some numerical tests. The test examples are taken from an application in modelling of substance transport in rivers.

  • 53.
    Elfving, Tommy
    et al.
    Linköping University, Department of Mathematics, Scientific Computing. Linköping University, The Institute of Technology.
    Skoglund, Ingegerd
    Linköping University, Department of Mathematics, Scientific Computing. Linköping University, The Institute of Technology.
    A direct method for a regularized least-squares problem2009In: NUMERICAL LINEAR ALGEBRA WITH APPLICATIONS, ISSN 1070-5325, Vol. 16, no 8, p. 649-675Article in journal (Refereed)
    Abstract [en]

    We consider a linear system of the form A(1)x(1)+A(2)X(2)+eta=b1. The vector eta consists of identically distributed random variables all with mean zero. The unknowns are split into two groups x(1) and x(2). In the model usually there are more unknowns than observations and the resulting linear system is most often consistent having an infinite number of solutions. Hence some constraint on the parameter vector x is needed. One possibility is to avoid rapid variation in, e.g. the parameters x(2). We formulate the problem as a partially regularized least-squares problem, and propose a direct solution method based on the QR decomposition of matrix blocks. Further we consider regularizing using one and two regularization parameters, respectively. We also discuss the choice of regularization parameters, and extend Reinschs method to the case with two parameters. Also the cross-validation technique is treated. We present test examples taken from an application in modelling of the substance transport in rivers.

  • 54.
    Elfving, Tommy
    et al.
    Linköping University, The Institute of Technology. Linköping University, Department of Mathematics, Scientific Computing.
    Skoglund, Ingegerd
    Linköping University, The Institute of Technology. Linköping University, Department of Mathematics, Scientific Computing.
    A direct method for a special regularized least squares problem2007Report (Other academic)
  • 55.
    Eriksson, Sofia
    et al.
    Department of Information Technology, Scientific Computing, Uppsala University, SE-751 05 Uppsala, Sweden.
    Abbas, Qaisar
    Department of Information Technology, Scientific Computing, Uppsala University, SE-751 05 Uppsala, Sweden.
    Nordström, Jan
    Linköping University, Department of Mathematics, Scientific Computing. Linköping University, The Institute of Technology.
    A stable and conservative method for locally adapting the design order of finite difference schemes2011In: Journal of Computational Physics, ISSN 0021-9991, E-ISSN 1090-2716, Vol. 230, no 11, p. 4216-4231Article in journal (Refereed)
    Abstract [en]

    A procedure to locally change the order of accuracy of finite difference schemes is developed. The development is based on existing Summation-By-Parts operators and a weak interface treatment. The resulting scheme is proven to be accurate and stable.

     

    Numerical experiments verify the theoretical accuracy for smooth solutions. In addition, shock calculations are performed, using a scheme where the developed switching procedure is combined with the MUSCL technique.

  • 56.
    Fagerlund, Martin
    Linköping University, Department of Mathematics, Scientific Computing.
    Computing Word Senses by Semantic Mirroring and Spectral Graph Partitioning2010Independent thesis Advanced level (degree of Master (One Year)), 20 credits / 30 HE creditsStudent thesis
    Abstract [en]

    In this thesis we use the method of Semantic Mirrors to create a graph of words that are semantically related to a seed word. Spectral graph partitioning methods are then used to partition the graph into subgraphs, and thus dividing the words into different word senses. These methods are applied to a bilingual lexicon of English and Swedish adjectives. A panel of human evaluators have looked at a few examples, and evaluated consistency within the derived senses and synonymy with the seed word.

  • 57. Fagerlund, Martin
    et al.
    Merkel, Magnus
    Linköping University, Department of Computer and Information Science, NLPLAB - Natural Language Processing Laboratory. Linköping University, The Institute of Technology.
    Eldén, Lars
    Linköping University, Department of Mathematics, Scientific Computing. Linköping University, The Institute of Technology.
    Ahrenberg, Lars
    Linköping University, Department of Computer and Information Science, NLPLAB - Natural Language Processing Laboratory. Linköping University, The Institute of Technology.
    Computing Word Senses by Semantic Mirroring and Spectral Graph Partitioning2010In: Proceedings of TextGraphs-5 - 2010 Workshop on Graph-based Methods for Natural Language Processing / [ed] Carmen Banea, Alessandro Moschitti, Swapna Somasundaran and Fabio Massimo Zanzotto, Stroudsburg, PA, USA: The Association for Computational Linguistics , 2010, p. 103-107Conference paper (Refereed)
    Abstract [en]

    Using the technique of ”semantic mirroring”a graph is obtained that representswords and their translations from a parallelcorpus or a bilingual lexicon. The connectednessof the graph holds informationabout the different meanings of words thatoccur in the translations. Spectral graphtheory is used to partition the graph, whichleads to a grouping of the words accordingto different senses. We also report resultsfrom an evaluation using a small sample ofseed words from a lexicon of Swedish andEnglish adjectives.

  • 58.
    Feng, Xiao-Li
    et al.
    Lanzhou University.
    Eldén, Lars
    Linköping University, Department of Mathematics, Scientific Computing. Linköping University, The Institute of Technology.
    Fu, Chu-Li
    Lanzhou University.
    A quasi-boundary-value method for the Cauchy problem for elliptic equations with  nonhomogeneous Neumann data2010In: Journal of Inverse and Ill-Posed Problems, ISSN 0928-0219, E-ISSN 1569-3945, Vol. 18, no 6, p. 617-645Article in journal (Refereed)
    Abstract [en]

    A Cauchy problem for elliptic equations with nonhomogeneous Neumann datain a cylindrical domain is investigated in this paper. For the theoretical aspect the a-prioriand a-posteriori parameter choice rules are suggested and the corresponding error estimatesare obtained. About the numerical aspect, for a simple case results given by twomethods based on the discrete Sine transform and the finite difference method are presented;an idea of left-preconditioned GMRES (Generalized Minimum Residual) methodis proposed to deal with the high dimensional case to save the time; a view of dealingwith a general domain is suggested. Some ill-posed problems regularized by the quasiboundary-value method are listed and some rules of this method are suggested.

  • 59.
    Feng, Xiao-Li
    et al.
    Lanzhou University.
    Eldén, Lars
    Linköping University, Department of Mathematics, Scientific Computing. Linköping University, The Institute of Technology.
    Fu, Chu-Li
    Lanzhou University.
    Stability and regularization of a backward parabolic PDE with variable coefficients2010In: Journal of Inverse and Ill-Posed Problems, ISSN 0928-0219, E-ISSN 1569-3945, Vol. 18, p. 217-243Article in journal (Refereed)
    Abstract [en]

    We consider a backward parabolic partial differential equation (BPPDE) withvariable coefficient a.x; t / in time. A new modification is used on the logarithmic convexitymethod to obtain a conditional stability estimate. Based on a formal solution, wereveal the essence of the ill-posedness and propose a simple regularization method. Moreover,we apply the regularization method to two representative cases. The results of boththeoretical and numerical performance show the validity of our method.

  • 60.
    Gong, Jing
    et al.
    Department of Information Technology, Scientific Computing Division, Uppsala University, Uppsala.
    Nordström, Jan
    Linköping University, Department of Mathematics, Scientific Computing. Linköping University, The Institute of Technology.
    Interface procedures for finite difference approximations of the advection–diffusion equation2011In: Journal of Computational and Applied Mathematics, ISSN 0377-0427, E-ISSN 1879-1778, Vol. 236, no 5, p. 602-620Article in journal (Refereed)
    Abstract [en]

    We investigate several existing interface procedures for finite difference methods applied to advection–diffusion problems. The accuracy, stiffness and reflecting properties of the various interface procedures are investigated.

    The analysis and numerical experiments show that there are only minor differences between the various methods once a proper parameter choice has been made.

  • 61.
    Gyllensten, Johan
    Linköping University, Department of Mathematics, Scientific Computing.
    Numerical Aspects of Image Rendering using Spherical Harmonics2009Independent thesis Advanced level (degree of Master (Two Years)), 20 credits / 30 HE creditsStudent thesis
    Abstract [en]

    Image rendering is the process of creating realistic computer images from  geometric models and physical laws of light and reflection. This master thesis deals mainly with the numerical intricacies of implementing an image renderer using spherical harmonics. It investigates how to calculate the reflection of light in a surface using the Phong model, and employs ray tracing to create a realistic image of a geometric model. Further, it investigates different ways of calculating the spherical harmonic representation of a function defined on the sphere. The thesis also deals with the implementation of self-shadowing, and the effects of adding this component to the rendering equation.

  • 62.
    Johansson, Björn
    et al.
    Linköping University, The Institute of Technology. Linköping University, Department of Electrical Engineering, Computer Vision.
    Elfving, Tommy
    Linköping University, The Institute of Technology. Linköping University, Department of Mathematics, Scientific Computing.
    Kozlov, Vladimir
    Linköping University, The Institute of Technology. Linköping University, Department of Mathematics, Applied Mathematics.
    Censor, Y.
    Department of Mathematics, University of Haifa, Mt. Carmel, Haifa 31905, Israel.
    Forssén, Per-Erik
    Linköping University, The Institute of Technology. Linköping University, Department of Electrical Engineering, Computer Vision.
    Granlund, Gösta
    Linköping University, The Institute of Technology. Linköping University, Department of Electrical Engineering, Computer Vision.
    The application of an oblique-projected Landweber method to a model of supervised learning2006In: Mathematical and computer modelling, ISSN 0895-7177, E-ISSN 1872-9479, Vol. 43, no 7-8, p. 892-909Article in journal (Refereed)
    Abstract [en]

    This paper brings together a novel information representation model for use in signal processing and computer vision problems, with a particular algorithmic development of the Landweber iterative algorithm. The information representation model allows a representation of multiple values for a variable as well as an expression for confidence. Both properties are important for effective computation using multi-level models, where a choice between models will be implementable as part of the optimization process. It is shown that in this way the algorithm can deal with a class of high-dimensional, sparse, and constrained least-squares problems, which arise in various computer vision learning tasks, such as object recognition and object pose estimation. While the algorithm has been applied to the solution of such problems, it has so far been used heuristically. In this paper we describe the properties and some of the peculiarities of the channel representation and optimization, and put them on firm mathematical ground. We consider the optimization a convexly constrained weighted least-squares problem and propose for its solution a projected Landweber method which employs oblique projections onto the closed convex constraint set. We formulate the problem, present the algorithm and work out its convergence properties, including a rate-of-convergence result. The results are put in perspective with currently available projected Landweber methods. An application to supervised learning is described, and the method is evaluated in an experiment involving function approximation, as well as application to transient signals. © 2006 Elsevier Ltd. All rights reserved.

  • 63.
    Johansson, Björn
    et al.
    Linköping University, Department of Electrical Engineering, Computer Vision. Linköping University, The Institute of Technology.
    Elfving, Tommy
    Linköping University, Department of Mathematics, Scientific Computing. Linköping University, The Institute of Technology.
    Kozlov, Vladimir
    Linköping University, Department of Mathematics, Applied Mathematics. Linköping University, The Institute of Technology.
    Censor, Yair
    Department of Mathematics, University of Haifa, Mt. Carmel, Haifa 31905, Israel.
    Granlund, Gösta
    Linköping University, Department of Electrical Engineering. Linköping University, The Institute of Technology.
    The Application of an Oblique-Projected Landweber Method to a Model of Supervised Learning2004Report (Other academic)
    Abstract [en]

    This report brings together a novel approach to some computer vision problems and a particular algorithmic development of the Landweber iterative algorithm. The algorithm solves a class of high-dimensional, sparse, and constrained least-squares problems, which arise in various computer vision learning tasks, such as object recognition and object pose estimation. The algorithm has recently been applied to these problems, but it has been used rather heuristically. In this report we describe the method and put it on firm mathematical ground. We consider a convexly constrained weighted least-squares problem and propose for its solution a projected Landweber method which employs oblique projections onto the closed convex constraint set. We formulate the problem, present the algorithm and work out its convergence properties, including a rate-of-convergence result. The results are put in perspective of currently available projected Landweber methods. The application to supervised learning is described, and the method is evaluated in a function approximation experiment.

  • 64.
    Kozdon, Jeremy E.
    et al.
    Department of Geophysics, Stanford, CA, USA.
    Dunham, Eric M.
    Department of Geophysics, Stanford, CA, USA.
    Nordström, Jan
    Linköping University, Department of Mathematics. Linköping University, The Institute of Technology. Linköping University, Department of Mathematics, Scientific Computing.
    Interaction of waves with frictional interfaces using summation-by-parts difference operators: Weak enforcement of nonlinear boundary conditions2011Report (Refereed)
    Abstract [en]

    We present a high-order difference method for problems in elastodynamics involving the interaction of waves with highly nonlinear frictional interfaces. We restrict our attention to two-dimensional antiplane problems involving deformation in only one direction. Jump conditions that relate tractions on the interface, or fault, to the relative sliding velocity across it are of a form closely related to those used in earthquake rupture models and other frictional sliding problems. By using summation-by-parts (SBP) finite difference operators and weak enforcement of boundary and interface conditions, a strictly stable method is developed. Furthermore, it is shown that unless the nonlinear interface conditions are formulated in terms of characteristic variables, as opposed to the physical variables in terms of which they are more naturally stated, the semi-discretized system of equations can become extremely stiff, preventing efficient solution using explicit time integrators.

    The use of SBP operators also provides a rigorously defined energy balance for the discretized problem that, as the mesh is refined, approaches the exact energy balance in the continuous problem. This enables one to investigate earthquake energetics, for example the efficiency with which elastic strain energy released during rupture is converted to radiated energy carried by seismic waves, rather than dissipated by frictional sliding of the fault. These theoretical results are confirmed by several numerical tests in both one and two dimensions demonstrating the computational efficiency, the high-order convergence rate of the method, the benefits of using strictly stable numerical methods for long time integration, and the accuracy of the energy balance.

  • 65.
    Kozdon, Jeremy E.
    et al.
    Department of Geophysics, Stanford University.
    Dunham, Eric M.
    Department of Geophysics, Stanford University.
    Nordström, Jan
    Linköping University, Department of Mathematics, Scientific Computing. Linköping University, The Institute of Technology.
    Interaction of Waves with Frictional Interfaces Using Summation-by-Parts Difference Operators: Weak Enforcement of Nonlinear Boundary Conditions2012In: Journal of Scientific Computing, ISSN 0885-7474, E-ISSN 1573-7691, Vol. 50, no 2, p. 341-367Article in journal (Refereed)
    Abstract [en]

    We present a high-order difference method for problems in elastodynamics involving

    the interaction of waves with highly nonlinear frictional interfaces. We restrict our

    attention to two-dimensional antiplane problems involving deformation in only one direction.

    Jump conditions that relate tractions on the interface, or fault, to the relative sliding velocity

    across it are of a form closely related to those used in earthquake rupture models and

    other frictional sliding problems. By using summation-by-parts (SBP) finite difference operators

    and weak enforcement of boundary and interface conditions, a strictly stable method

    is developed. Furthermore, it is shown that unless the nonlinear interface conditions are formulated

    in terms of characteristic variables, as opposed to the physical variables in terms of

    which they are more naturally stated, the semi-discretized system of equations can become

    extremely stiff, preventing efficient solution using explicit time integrators.

    The use of SBP operators also provides a rigorously defined energy balance for the discretized

    problem that, as the mesh is refined, approaches the exact energy balance in the

    continuous problem. This enables one to investigate earthquake energetics, for example the

    efficiency with which elastic strain energy released during rupture is converted to radiated

    energy carried by seismic waves, rather than dissipated by frictional sliding of the fault.

    These theoretical results are confirmed by several numerical tests in both one and two dimensions

    demonstrating the computational efficiency, the high-order convergence rate of

    the method, the benefits of using strictly stable numerical methods for long time integration,

    and the accuracy of the energy balance.

  • 66.
    Kozdon, Jeremy E.
    et al.
    Department of Geophysics, Stanford University, Stanford, CA, USA.
    Dunham, Eric M.
    Department of Geophysics and Institute for Computational and Mathematical Engineering, Stanford University, Stanford, CA, USA.
    Nordström, Jan
    Linköping University, Department of Mathematics, Scientific Computing. Linköping University, The Institute of Technology.
    Simulation of Dynamic Earthquake Ruptures in Complex Geometries Using High-Order Finite Difference Methods2012Report (Other academic)
    Abstract [en]

    We develop a stable and high-order accurate finite difference method for problems in earthquake rupture dynamics capable of handling complex geometries and multiple faults. The bulk material is an isotropic elastic solid cut by preexisting fault interfaces. The fields across the interfaces are related through friction laws which depend on the sliding velocity, tractions acting on the interface, and state variables which evolve according to ordinary differential equations involving local fields.

    The method is based on summation-by-parts finite difference operators with irregular geometries handled through coordinate transforms and multi-block meshes. Boundary conditions as well as block interface conditions (whether frictional or otherwise) are enforced weakly through the simultaneous approximation term method, resulting in a provably stable discretization.

    The theoretical accuracy and stability results are confirmed with the method of manufactured solutions. The practical benefits of the new methodology are illustrated in a simulation of a subduction zone megathrust earthquake, a challenging application problem involving complex free-surface topography, nonplanar faults, and varying material properties.

  • 67.
    Lambrix, Patrick
    et al.
    Linköping University, Department of Computer and Information Science, Database and information techniques. Linköping University, The Institute of Technology.
    Ouchterlony, Ulla
    Linköping University, Department of Mathematics, Scientific Computing. Linköping University, The Institute of Technology.
    Integration of Psychology, Economics and Information Technology in an Engineering Curriculum1999In: Computer Science Education, ISSN 0899-3408, E-ISSN 1744-5175, Vol. 9, no 2, p. 162-180Article in journal (Refereed)
    Abstract [en]

    Engineers often work together with professionals from entirely different areas. Therefore, it is important for the engineers to understand enough of these other areas, where technology is used, to obtain good results. To this aim a term has been developed within the civil engineering curriculum in information technology at Linköpings universitet, where the students work and study together with students from the psychology and economics education programs. The information technology and economics students build companies together and perform a project. The psychology students act as consultants for the different companies. Subjects from six departments are integrated into the project. The students work and study together for different parts of the project, although the learning goals differ between the different programs. The cooperation in the project leads to a transfer of information and knowledge between the students. Experience also suggests that the students obta in a better motivation and an integrated view of technology and the other areas as a result of the integrated project. The learning method also promotes cooperation and understanding between different professional cultures.

  • 68.
    Lenferink, Wilhelmus
    Linköping University, The Institute of Technology. Linköping University, Department of Mathematics, Scientific Computing.
    A second order scheme for a time-dependent, singularly perturbed convection-diffusion equation2002In: Journal of Computational and Applied Mathematics, ISSN 0377-0427, E-ISSN 1879-1778, Vol. 143, no 1, p. 49-68Article in journal (Refereed)
    Abstract [en]

    We consider a numerical scheme for a one-dimensional, time-dependent, singularly perturbed convection-diffusion problem. The problem is discretized in space by a standard finite element method on a Bakhvalov-Shishkin type mesh. The space error is measured in an L2 norm. For the time integration, the implicit midpoint rule is used. The fully discrete scheme is shown to be convergent of order 2 in space and time, uniformly in the singular perturbation parameter. © 2002 Elsevier Science B.V. All rights reserved.

  • 69.
    Lenferink, Wilhelmus
    Linköping University, The Institute of Technology. Linköping University, Department of Mathematics, Scientific Computing.
    Pointwise convergence of approximations to a convection-diffusion equation on a Shishkin mesh2000In: Applied Numerical Mathematics, ISSN 0168-9274, E-ISSN 1873-5460, Vol. 32, no 1, p. 69-86Article in journal (Refereed)
    Abstract [en]

    A centered difference or finite element discretization is applied to a singularly perturbed, one-dimensional boundary value problem. The discretization uses a piecewise equidistant mesh. It is proved that the pointwise error is (almost) of second order with respect to the number of nodes, uniformly in the perturbation parameter. The proof is based on a monotonicity argument.

  • 70.
    Lindgren, David
    et al.
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, The Institute of Technology.
    Savas, Berkant
    Linköping University, Department of Mathematics, Scientific Computing. Linköping University, The Institute of Technology.
    Rank Reduction and Volume Minimization Approach to State-Space Subspace System Identification2006In: Signal Processing, ISSN 0165-1684, E-ISSN 1872-7557, Vol. 86, no 11, p. 3275-3285Article in journal (Refereed)
    Abstract [en]

    In this paper we consider the reduced rank regression problem

    solved by maximum-likelihood-inspired state-space subspace system identification algorithms. We conclude that the determinant criterion is, due to potential rank-deficiencies, not general enough to handle all problem instances. The main part of the paper analyzes the structure of the reduced rank minimization problem and identifies signal properties in terms of geometrical concepts. A more general minimization criterion is considered, rank reduction followed by volume minimization. A numerically sound algorithm for minimizing this criterion is presented and validated on both simulated and experimental data.

  • 71.
    Lindström Berg, Jens
    et al.
    Uppsala University, Department of Information Technology, SE-751 05, Uppsala, Sweden.
    Nordström, Jan
    Linköping University, Department of Mathematics, Scientific Computing. Linköping University, The Institute of Technology.
    Stable Robin boundary conditions for the Navier-Stokes equations2011Report (Other academic)
    Abstract [en]

    In this paper we prove stability of Robin solid wall boundary conditions for the compressible Navier-Stokes equations. Applications include the no-slip boundary conditions with prescribed temperature or temperature gradient and the rst order slip-ow boundary conditions. The formulation is uni-form and the transitions between dierent boundary conditions are done by a change of parameters. We give dierent sharp energy estimates depending on the choice of parameters.

    The discretization is done using nite dierences on Summation-By-Parts form with weak boundary conditions using the Simultaneous Approximation Term. We verify convergence by the method of manufactured solutions and show computations of ows ranging from no-slip to substantial slip.

  • 72. Lindström, Jens
    et al.
    Nordström, Jan
    Linköping University, Department of Mathematics, Scientific Computing. Linköping University, The Institute of Technology.
    Spectral analysis of the continuous and discretized heat and advection equation on single and multiple domains2010Report (Other academic)
    Abstract [en]

    In this paper we study the heat and advection equation in single and multiple domains. We discretize using a second order accurate finite difference method on Summation-By-Parts form with weak boundary and interface conditions. We derive analytic expressions for the spectrum of the continuous problem and for their corresponding discretization matrices. We show how the spectrum of the single domain operator is contained in the multi domain operator spectrum when artificial interfaces are introduced. We study the impact on the spectrum and discretization errors depending on the interface treatment and verify that the results are carried over to higher order accurate schemes.

  • 73.
    Lundquist, Tomas
    Linköping University, Department of Mathematics, Scientific Computing.
    Stability of SBP schemes on overlapping domains2011Independent thesis Advanced level (degree of Master (Two Years)), 20 credits / 30 HE creditsStudent thesis
    Abstract [en]

    Using fnite difference methods for partial differential equations, this thesis focuses on the problem of connecting overlapping solution domains in the context of a frst order hyperbolic problem. Especially the stability properties of such constructions is studied, and a stable general implementation of the the test problem is proposed. However, no energy estimate could be found, and indeed proven not to exist in the natural norm. Finally, an example is also put forward where the interface conditions derived are, for stability considerations, incompatible with the boundary conditions in a coupled system of hyperbolic equations.

  • 74. Lundstrom, E.
    et al.
    Elden, Lars
    Linköping University, The Institute of Technology. Linköping University, Department of Mathematics, Scientific Computing.
    Adaptive eigenvalue computations using Newton's method on the Grassmann manifold2002In: SIAM Journal on Matrix Analysis and Applications, ISSN 0895-4798, E-ISSN 1095-7162, Vol. 23, no 3, p. 819-839Article in journal (Refereed)
    Abstract [en]

    We consider the problem of updating an invariant subspace of a large and structured Hermitian matrix when the matrix is modified slightly. The problem can be formulated as that of computing stationary values of a certain function with orthogonality constraints. The constraint is formulated as the requirement that the solution must be on the Grassmann manifold, and Newton's method on the manifold is used. In each Newton iteration a Sylvester equation is to be solved. We discuss the properties of the Sylvester equation and conclude that for large problems preconditioned iterative methods can be used. Preconditioning techniques are discussed. Numerical examples from signal subspace computations are given in which the matrix is Toeplitz and we compute a partial singular value decomposition corresponding to the largest singular values. Further we solve numerically the problem of computing the smallest eigenvalues and corresponding eigenvectors of a large sparse matrix that has been slightly modified.

  • 75.
    Löw, Joakim
    et al.
    Linköping University, Department of Science and Technology, Visual Information Technology and Applications (VITA). Linköping University, The Institute of Technology.
    Ynnerman, Anders
    Linköping University, Department of Science and Technology, Visual Information Technology and Applications (VITA). Linköping University, The Institute of Technology.
    Eldén, Lars
    Linköping University, Department of Mathematics, Scientific Computing. Linköping University, The Institute of Technology.
    Numerical Analysis of BRDFs for Inverse Rendering2009Report (Other academic)
    Abstract [en]

    The properties of materials which are present in a scene determine how geometry reflects and distributes light in the scene. This text presents work-in-progress on numerical analysis of bidirectional reflection distribution functions (BRDF) corresponding to various materials, with a focus on inverse rendering. An analysis of these functions is vital for the understanding of the behaviour of reflected light under different lighting conditions, and in the application of inverse rendering, it is important in order to determine what quality one can expect from recovered data. We discuss the singular value decompositions of a few materials, their effect on the ill-posedness of the inverse problem related to the reflectance equation and how regularization affects the solution of the problem.

  • 76.
    Nikazad, Touraj
    Linköping University, Department of Mathematics, Scientific Computing. Linköping University, The Institute of Technology.
    Algebraic Reconstruction Methods2008Doctoral thesis, comprehensive summary (Other academic)
    Abstract [en]

    Ill-posed sets of linear equations typically arise when discretizing certain types of integral transforms. A well known example is image reconstruction, which can be modeled using the Radon transform. After expanding the solution into a finite series of basis functions a large, sparse and ill-conditioned linear system occurs. We consider the solution of such systems. In particular we study a new class of iteration methods named DROP (for Diagonal Relaxed Orthogonal Projections) constructed for solving both linear equations and linear inequalities. This class can also be viewed, when applied to linear equations, as a generalized Landweber iteration. The method is compared with other iteration methods using test data from a medical application and from electron microscopy. Our theoretical analysis include convergence proofs of the fully-simultaneous DROP algorithm for linear equations without consistency assumptions, and of block-iterative algorithms both for linear equations and linear inequalities, for the consistent case.

    When applying an iterative solver to an ill-posed set of linear equations the error usually initially decreases but after some iterations, depending on the amount of noise in the data, and the degree of ill-posedness, it starts to increase. This phenomenon is called semi-convergence. We study the semi-convergence performance of Landweber-type iteration, and propose new ways to specify the relaxation parameters. These are computed so as to control the propagated error.

    We also describe a class of stopping rules for Landweber-type iteration for solving linear inverse problems. The class includes the well known discrepancy principle, and the monotone error rule. We unify the error analysis of these two methods. The stopping rules depend critically on a certain parameter whose value needs to be specified. A training procedure is therefore introduced for securing robustness. The advantages of using trained rules are demonstrated on examples taken from image reconstruction from projections.

    Kaczmarz's method, also called ART (Algebraic Reconstruction Technique) is often used for solving the linear system which appears in image reconstruction. This is a fully sequential method. We examine and compare ART and its symmetric version. It is shown that the cycles of symmetric ART, unlike ART, converge to a weighted least squares solution if and only if the relaxation parameter lies between zero and two. Further we show that ART has faster asymptotic rate of convergence than symmetric ART. Also a stopping criterion is proposed and evaluated for symmetric ART.

    We further investigate a class of block-iterative methods used in image reconstruction. The cycles of the iterative sequences are characterized in terms of the original linear system. We define symmetric block-iteration and compare the behavior of symmetric and non-symmetric block-iteration. The results are illustrated using some well-known methods. A stopping criterion is offered and assessed for symmetric block-iteration.

    List of papers
    1. On Diagonally Relaxed Orthogonal Projection Methods
    Open this publication in new window or tab >>On Diagonally Relaxed Orthogonal Projection Methods
    2008 (English)In: SIAM Journal on Scientific Computing, ISSN 1064-8275, E-ISSN 1095-7197, Vol. 30, no 1, p. 473-504Article in journal (Refereed) Published
    Abstract [en]

    We propose and studya block-iterative projection method for solving linear equations and/or inequalities.The method allows diagonal componentwise relaxation in conjunction with orthogonalprojections onto the individual hyperplanes of the system, and isthus called diagonally relaxed orthogonal projections (DROP). Diagonal relaxation hasproven useful in accelerating the initial convergence of simultaneous andblock-iterative projection algorithms, but until now it was available onlyin conjunction with generalized oblique projections in which there isa special relation between the weighting and the oblique projections.DROP has been used by practitioners, and in this papera contribution to its convergence theory is provided. The mathematicalanalysis is complemented by some experiments in image reconstruction fromprojections which illustrate the performance of DROP.

    Place, publisher, year, edition, pages
    Philadelphia, PA, United States: Society for Industrial and Applied Mathematics, 2008
    Keywords
    block iteration, convex feasibility, diagonal relaxation, projection methods, simultaneous algorithms
    National Category
    Mathematics
    Identifiers
    urn:nbn:se:liu:diva-13235 (URN)10.1137/050639399 (DOI)000208048600006 ()
    Available from: 2008-05-21 Created: 2008-05-21 Last updated: 2017-12-13Bibliographically approved
    2. Stopping Rules for Landweber-type Iteration
    Open this publication in new window or tab >>Stopping Rules for Landweber-type Iteration
    2007 (English)In: Inverse Problems, ISSN 0266-5611, Vol. 23, no 4, p. 1417-1432Article in journal (Refereed) Published
    Abstract [en]

    We describe a class of stopping rules for Landweber-type iterations for solving linear inverse problems. The class includes both the discrepancy principle (DP rule) and the monotone error rule (ME rule). We also unify the error analysis of the two methods. The stopping rules depend critically on a certain parameter whose value needs to be specified. A training procedure is therefore introduced for securing robustness. The advantages of using a trained rule are demonstrated on examples taken from image reconstruction from projections. After training the stopping rules became quite robust and only small differences were observed between, e.g. the DP rule and ME rule.

    National Category
    Mathematics
    Identifiers
    urn:nbn:se:liu:diva-13236 (URN)10.1088/0266-5611/23/4/004 (DOI)
    Available from: 2008-05-21 Created: 2008-05-21
    3. Some Properties of ART-type Reconstruction Algorithms
    Open this publication in new window or tab >>Some Properties of ART-type Reconstruction Algorithms
    2008 (English)In: Mathematical Methods in Biomedical Imaging and Intensity-Modulated Radiation Therapy (IMRT), / [ed] Yair Censor, Ming Jiang, Alfred K. Louis, 2008, 1, p. 526-Chapter in book (Other academic)
    Abstract [en]

    This book contains papers presented by leading experts at the "Interdisciplinary Workshop on Mathematical Methods in Biomedical Imaging and Intensity-Modulated Radiation Therapy (IMRT)" held at the Centro di Ricerca Matematica (CRM) Ennio De Giorgi at Pisa, Italy, from October 15 to 19, 2007. The interdisciplinary book consists of research and review papers by leading experts and practitioners in biomedical imaging and intensity-modulated radiation therapy (IMRT).

    National Category
    Engineering and Technology
    Identifiers
    urn:nbn:se:liu:diva-13237 (URN)8876423141 (ISBN)9788876423147 (ISBN)
    Available from: 2008-05-21 Created: 2008-05-21 Last updated: 2013-05-24Bibliographically approved
    4. Some Block-Iterative Methods used in Image Reconstruction
    Open this publication in new window or tab >>Some Block-Iterative Methods used in Image Reconstruction
    2008 (English)Article in journal (Refereed) Submitted
    National Category
    Mathematics
    Identifiers
    urn:nbn:se:liu:diva-13238 (URN)
    Available from: 2008-05-21 Created: 2008-05-21
    5. Semi-Convergence and Choice of Relaxation Parameters in Landweber-type Algorithms
    Open this publication in new window or tab >>Semi-Convergence and Choice of Relaxation Parameters in Landweber-type Algorithms
    Manuscript (Other academic)
    Identifiers
    urn:nbn:se:liu:diva-13239 (URN)
    Available from: 2008-05-21 Created: 2008-05-21 Last updated: 2010-01-13
  • 77.
    Nikazad, Touraj
    Linköping University, Department of Mathematics, Scientific Computing. Linköping University, The Institute of Technology.
    The Use of Landweber Algorithm in Image Reconstruction2007Licentiate thesis, comprehensive summary (Other academic)
    Abstract [en]

    Ill-posed sets of linear equations typically arise when discretizing certain types of integral transforms. A well known example is image reconstruction, which can be modelled using the Radon transform. After expanding the solution into a finite series of basis functions a large, sparse and ill-conditioned linear system arises. We consider the solution of such systems. In particular we study a new class of iteration methods named DROP (for Diagonal Relaxed Orthogonal Projections) constructed for solving both linear equations and linear inequalities. This class can also be viewed, when applied to linear equations, as a generalized Landweber iteration. The method is compared with other iteration methods using test data from a medical application and from electron microscopy. Our theoretical analysis include convergence proofs of the fully-simultaneous DROP algorithm for linear equations without consistency assumptions, and of block-iterative algorithms both for linear equations and linear inequalities, for the consistent case.

    When applying an iterative solver to an ill-posed set of linear equations the error typically initially decreases but after some iterations (depending on the amount of noise in the data, and the degree of ill-posedness) it starts to increase. This phenomena is called semi-convergence. It is therefore vital to find good stopping rules for the iteration.

    We describe a class of stopping rules for Landweber type iterations for solving linear inverse problems. The class includes, e.g., the well known discrepancy principle, and also the monotone error rule. We also unify the error analysis of these two methods. The stopping rules depend critically on a certain parameter whose value needs to be specified. A training procedure is therefore introduced for securing robustness. The advantages of using trained rules are demonstrated on examples taken from image reconstruction from projections.

    List of papers
    1. On Diagonally Relaxed Orthogonal Projection Methods
    Open this publication in new window or tab >>On Diagonally Relaxed Orthogonal Projection Methods
    2008 (English)In: SIAM Journal on Scientific Computing, ISSN 1064-8275, E-ISSN 1095-7197, Vol. 30, no 1, p. 473-504Article in journal (Refereed) Published
    Abstract [en]

    We propose and studya block-iterative projection method for solving linear equations and/or inequalities.The method allows diagonal componentwise relaxation in conjunction with orthogonalprojections onto the individual hyperplanes of the system, and isthus called diagonally relaxed orthogonal projections (DROP). Diagonal relaxation hasproven useful in accelerating the initial convergence of simultaneous andblock-iterative projection algorithms, but until now it was available onlyin conjunction with generalized oblique projections in which there isa special relation between the weighting and the oblique projections.DROP has been used by practitioners, and in this papera contribution to its convergence theory is provided. The mathematicalanalysis is complemented by some experiments in image reconstruction fromprojections which illustrate the performance of DROP.

    Place, publisher, year, edition, pages
    Philadelphia, PA, United States: Society for Industrial and Applied Mathematics, 2008
    Keywords
    block iteration, convex feasibility, diagonal relaxation, projection methods, simultaneous algorithms
    National Category
    Mathematics
    Identifiers
    urn:nbn:se:liu:diva-13235 (URN)10.1137/050639399 (DOI)000208048600006 ()
    Available from: 2008-05-21 Created: 2008-05-21 Last updated: 2017-12-13Bibliographically approved
    2. Stopping Rules for Landweber-type Iteration
    Open this publication in new window or tab >>Stopping Rules for Landweber-type Iteration
    2007 (English)In: Inverse Problems, ISSN 0266-5611, Vol. 23, no 4, p. 1417-1432Article in journal (Refereed) Published
    Abstract [en]

    We describe a class of stopping rules for Landweber-type iterations for solving linear inverse problems. The class includes both the discrepancy principle (DP rule) and the monotone error rule (ME rule). We also unify the error analysis of the two methods. The stopping rules depend critically on a certain parameter whose value needs to be specified. A training procedure is therefore introduced for securing robustness. The advantages of using a trained rule are demonstrated on examples taken from image reconstruction from projections. After training the stopping rules became quite robust and only small differences were observed between, e.g. the DP rule and ME rule.

    National Category
    Mathematics
    Identifiers
    urn:nbn:se:liu:diva-13236 (URN)10.1088/0266-5611/23/4/004 (DOI)
    Available from: 2008-05-21 Created: 2008-05-21
  • 78.
    Niu, Steve S.
    et al.
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, The Institute of Technology.
    Ljung, Lennart
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, The Institute of Technology.
    Björk, Åke
    Linköping University, Department of Mathematics, Scientific Computing. Linköping University, The Institute of Technology.
    Decomposition Methods for Solving Least-Squares Parameter Estimation1995Report (Other academic)
    Abstract [en]

    In this paper leastsquares method with matrix decomposition is revisited and a multiple model formulation is proposed The proposed formulation takes advantage of the wellestablished decomposition methods but possesses a multiple model structure which leads to simpler and more exible implementations and produces more infor mation than the original least squares methods Several application examples in signal processing and system identication are included As a basic numerical analysis tool the proposed methods can be used in many dierent application areas

  • 79.
    Niu, Steve S.
    et al.
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, The Institute of Technology.
    Ljung, Lennart
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, The Institute of Technology.
    Björk, Åke
    Linköping University, Department of Mathematics, Scientific Computing. Linköping University, The Institute of Technology.
    Decomposition Methods for Solving Least-Squares Parameter Estimation1996In: IEEE Transactions on Signal Processing, ISSN 1053-587X, E-ISSN 1941-0476, Vol. 44, no 11, p. 2847-2852Article in journal (Refereed)
    Abstract [en]

    A multiple model least-squares method based on matrix decomposition is proposed. Compared with the conventional implementation of the least-squares method, the proposed method is simpler and more flexible in implementation and produces more information. An application example in parameter estimation is included. As a basic numerical tool, the proposed method can be used in many different application areas.

  • 80.
    Nordström, Jan
    et al.
    Linköping University, Department of Mathematics, Scientific Computing. Linköping University, The Institute of Technology.
    Berg, Jens
    Department of Information Technology, Uppsala University, SE-751 05, Uppsala, Sweden.
    Conjugate heat transfer using modified interface conditions for the Navier-Stokes equations2011Report (Other academic)
    Abstract [en]

    This paper evaluates the use of the compressible Navier-Stokes equations, with prescribed zero velocities, as a model for heat transfer in solids. In particular in connection with conjugate heat transfer problems.

    We derive estimates, and show how to choose and scale the coefficients of the energy part in the Navier-Stokes equations, such that the difference between the energy equation and the heat equation is minimal.

    A rigorous analysis of the physical interface conditions for the conjugate heat transfer problem is performed and energy estimates are derived in non-standard L2-equivalent norms. The numerical schemes are proven energy stable with the physical interface conditions and the stability of the schemes are independent of the order of accuracy.

    We have performed computations of a conjugate heat transfer problem in two different ways. One where, as traditionally, the heat transfer in the solid is governed by the heat equation. The other where the heat transfer in the solid is governed by the Navier-Stokes equations. The simulations are compared and the numerical results corroborate the theoretical results.

  • 81.
    Nordström, Jan
    et al.
    Linköping University, Department of Mathematics, Scientific Computing. Linköping University, The Institute of Technology.
    Eriksson, Sofia
    Department of Information Technology, Scientific Computing, Uppsala University.
    Eliasson, Peter
    Department of Aeronautics and Systems Integration, FOI, The Swedish Defence Research Agency, SE-164 90 Stockholm.
    Weak and Strong Wall Boundary Procedures and Convergence to Steady-State of the Navier-Stokes Equations2011Report (Refereed)
    Abstract [en]

    We study the influence of different implementations of no-slip solid wall boundary conditions on the convergence to steady-state of the Navier-Stokes equations. The various approaches are investigated using the energy method and an eigenvalue analysis. It is shown that the weak implementation is superior and enhances the convergence to steady-state for coarse meshes. It is also demonstrated that all the stable approaches produce the same convergence rate as the mesh size goes to zero. The numerical results obtained by using a fully nonlinear finite volume solver support the theoretical findings from the linear analysis.

  • 82.
    Nordström, Jan
    et al.
    Linköping University, Department of Mathematics, Scientific Computing. Linköping University, The Institute of Technology.
    Eriksson, Sofia
    Uppsala University.
    Eliasson, Peter
    Swedish Def Research Agency.
    Weak and strong wall boundary procedures and convergence to steady-state of the Navier-Stokes equations2012In: Journal of Computational Physics, ISSN 0021-9991, E-ISSN 1090-2716, Vol. 231, no 14, p. 4867-4884Article in journal (Refereed)
    Abstract [en]

    We study the influence of different implementations of no-slip solid wall boundary conditions on the convergence to steady-state of the Navier-Stokes equations. The various approaches are investigated using the energy method and an eigenvalue analysis. It is shown that the weak implementation is superior and enhances the convergence to steady-state for coarse meshes. It is also demonstrated that all the stable approaches produce the same convergence rate as the mesh size goes to zero. The numerical results obtained by using a fully nonlinear finite volume solver support the theoretical findings from the linear analysis. 

  • 83.
    O´Reilly, Ossian
    et al.
    Department of Information Technology, Uppsala University.
    Kozdon, Jeremy E.
    Department of Geophysics, Stanford University.
    Dunham, Eric M.
    Department of Geophysics, Stanford University.
    Nordström, Jan
    Linköping University. Linköping University, Department of Mathematics, Scientific Computing. Linköping University, The Institute of Technology.
    Coupled High-Order Finite Difference and Unstructured Finite Volume Methods for Earthquake Rupture Dynamics in Complex Geometries2011Conference paper (Other academic)
    Abstract [en]

    Unstructured grid methods are well suited for earthquake problems in complex geometries, such as non-planar and branching faults. Unfortunately they are indefficient in comparsion wiht high-order finite differences. With the use of summation-by-parts (SBP) operators and the SAT penalty method (simultaneous approximation term) it is possible to couple unstructured finite volume methods with high-order finite difference methods in an accurate and stable way. The couple method is more efficient than the unstructured method alone. Another advantage of the SBP and SAT method is that it is possible to prove strict stability, meaning that the semi-discrete solution dissipates energy at a slightly faster rate than the continuous solution so that the error remains bounded in time, which is particulary useful for long time computations.

  • 84.
    O´Reilly, Ossian
    et al.
    Department of Information Technology, Uppsala University.
    Kozdon, Jeremy E.
    Department of Geophysics, Stanford University.
    Dunham, Eric M.
    Department of Geophysics, Stanford University.
    Nordström, Jan
    Linköping University, Department of Mathematics, Scientific Computing. Linköping University, The Institute of Technology.
    High-order finite difference methods for eartquake rupture dynamics in complex geometries2010Conference paper (Other academic)
    Abstract [en]

    Unstructured grid methods are well suited for earthquake problems in complex geometries, such as non-planar and branching faults. Unfortunately they are inefficient in comparisioin with high-order finite difference. With the use of summertion-by-parts (SBP) operators and the SAT penalty method (simultaneous approximation term) it is possible to couple unstructuren finite volume methods wiht high order finite difference methods in an accurate and stable way. The coupled methods is more efficient than the unstructured method alone. Another advantage of the SBP and SAT method is that it is possible to prove strict stability, meaning that the semi-discrete solution dissipates energy at a slightly faster rate than the continuous solution so that the error remains bounded in time, which is particulary useful for long time computations.

  • 85.
    Ouchterlony, Ulla
    Linköping University, Department of Mathematics, Scientific Computing. Linköping University, The Institute of Technology.
    IT-programmets projekttermin, termin 5 1997-20082010Report (Other academic)
    Abstract [sv]

    Det är en unik kurs som sträcker sig över en hel termin, där samarbete sker mellan lärare från sju institutioner och studenter från tre program, nämligen IT-programmet, Psykologprogrammet och Ekonomprogrammet.

    De kunskaper som krävdes för projektets genomförande inhämtades under terminens gång i form av delkurser. Dessa examinerades individuellt och flera olika examinationsformer användes. Projektarbetet examinerades gruppvis i form av muntliga redovisningar och en skriftlig rapport.

    Jag var ansvarig för terminen under tolv år, 1997-2008.  När jag gick i pension ändrade  projektterminen inriktning. Avsikten med denna rapport är att beskriva hur terminen fungerade under de  år som jag var ansvarig.

  • 86.
    Park, H
    et al.
    Univ Minnesota, Dept Comp Sci & Engn, Minneapolis, MN 55455 USA Linkoping Univ, Dept Math, S-58183 Linkoping, Sweden.
    Elden, Lars
    Linköping University, The Institute of Technology. Linköping University, Department of Mathematics, Scientific Computing.
    Schur-type methods for solving least squares problems with Toeplitz structure2000In: SIAM Journal on Scientific Computing, ISSN 1064-8275, E-ISSN 1095-7197, Vol. 22, no 2, p. 406-430Article in journal (Refereed)
    Abstract [en]

    We give an overview of fast algorithms for solving least squares problems with Toeplitz structure, based on generalization of the classical Schur algorithm, and discuss their stability properties. In order to obtain more accurate triangular factors of a Toeplitz matrix as well as accurate solutions for the least squares problems, methods based on corrected seminormal equations (CSNE) can be used. We show that the applicability of the generalized Schur algorithm is considerably enhanced when the algorithm is used in conjunction with CSNE. Several numerical tests are reported, where different variants of the generalized Schur algorithm and CSNE are compared for their accuracy and speed.

  • 87.
    Ranjbar, Zohreh
    Linköping University, The Institute of Technology. Linköping University, Department of Mathematics, Scientific Computing.
    Analysis of an ill-posed Cauchy problem for a convection-diffusion equation2004In: Workshop i tillämpad matematik,2004, 2004Conference paper (Other academic)
  • 88.
    Ranjbar, Zohreh
    Linköping University, Department of Mathematics, Scientific Computing. Linköping University, The Institute of Technology.
    Numerical Solution of a Cauchy Problem for a Parabolic Equation in Two or more Space Dimensions by the Arnoldi Method2010Report (Other academic)
    Abstract [en]

    We consider the numerical solution of a Cauchy problem for a parabolic equation in multi-dimensional space with cylindrical domain in one spatial space direction. It is desired to find the lower boundary values from the Cauchy data on the upper boundary. This problem is severely ill-posed. The formal solution is written as a hyperbolic cosine function in terms of a multidimensional parabolic (unbounded) operator. We compute an approximate solution by projecting onto a smaller subspace generated via the Arnoldi algorithm applied on the discretized inverse of the operator. Further we regularize the projected problem. The hyperbolic cosine is evaluated explicitly on a low-dimensional subspace. In each iteration step of the Arnoldi method a well-posed parabolic problem is solved. Numerical examples are given to illustrate the performance of the method.

  • 89.
    Ranjbar, Zohreh
    Linköping University, Department of Mathematics, Scientific Computing. Linköping University, The Institute of Technology.
    Numerical Solution of Ill-posed Cauchy Problems for Parabolic Equations2010Doctoral thesis, comprehensive summary (Other academic)
    Abstract [en]

    Ill-posed mathematical problem occur in many interesting scientific and engineering applications. The solution of such a problem, if it exists, may not depend continuously on the observed data. For computing a stable approximate solution it is necessary to apply a regularization method. The purpose of this thesis is to investigate regularization approaches and develop numerical methods for solving certain ill-posed problems for parabolic partial differential equations. In thermal engineering applications one wants to determine the surface temperature of a body when the surface itself is inaccessible to measurements. This problem can be modelled by a sideways heat equation. The mathematical and numerical properties of the sideways heat equation with constant convection and diffusion coefficients is first studied. The problem is reformulated as a Volterra integral equation of the first kind with smooth kernel. The influence of the coefficients on the degree of ill-posedness are also studied. The rate of decay of the singular values of the Volterra integral operator determines the degree of ill-posedness. It is shown that the sign of the coefficient in the convection term influences the rate of decay of the singular values.

    Further a sideways heat equation in cylindrical geometry is studied. The equation is a mathematical model of the temperature changes inside a thermocouple, which is used to approximate the gas temperature in a combustion chamber. The heat transfer coefficient at the surface of thermocouple is also unknown. This coefficient is approximated via a calibration experiment. Then the gas temperature in the combustion chamber is computed using the convection boundary condition. In both steps the surface temperature and heat flux are approximated using Tikhonov regularization and the method of lines.

    Many existing methods for solving sideways parabolic equations are inadequate for solving multi-dimensional problems with variable coefficients. A new iterative regularization technique for solving a two-dimensional sideways parabolic equation with variable coefficients is proposed. A preconditioned Generalized Minimum Residuals Method (GMRS) is used to regularize the problem. The preconditioner is based on a semi-analytic solution formula for the corresponding problem with constant coefficients. Regularization is used in the preconditioner as well as truncating the GMRES algorithm. The computed examples indicate that the proposed PGMRES method is well suited for this problem.

    In this thesis also a numerical method is presented for the solution of a Cauchy problem for a parabolic equation in multi-dimensional space, where the domain is cylindrical in one spatial direction. The formal solution is written as a hyperbolic cosine function in terms of a parabolic unbounded operator. The ill-posedness is dealt with by truncating the large eigenvalues of the operator. The approximate solution is computed by projecting onto a smaller subspace generated by the Arnoldi algorithm applied on the inverse of the operator. A well-posed parabolic problem is solved in each iteration step. Further the hyperbolic cosine is evaluated explicitly only for a small triangular matrix. Numerical examples are given to illustrate the performance of the method.

    List of papers
    1. Numerical analysis of an ill-posed Cauchy problem for a convection - Diffusion equation
    Open this publication in new window or tab >>Numerical analysis of an ill-posed Cauchy problem for a convection - Diffusion equation
    2007 (English)In: Inverse Problems in Science and Engineering, ISSN 1741-5977, E-ISSN 1741-5985, Vol. 15, no 3, p. 191-211Article in journal (Refereed) Published
    Abstract [en]

    The mathematical and numerical properties of an ill-posed Cauchy problem for a convection - diffusion equation are investigated in this study. The problem is reformulated as a Volterra integral equation of the first kind with a smooth kernel. The rate of decay of the singular values of the integral operator determines the degree of ill-posedness. The purpose of this article is to study how the convection term influences the degree of ill-posedness by computing numerically the singular values. It is also shown that the sign of the coefficient in the convection term determines the rate of decay of the singular values. Some numerical examples are also given to illustrate the theory.

    Keywords
    Cauchy problem, Convection - diffusion equation, Ill-posed, Inverse problem, Singular value decomposition, Volterra integral operator
    National Category
    Engineering and Technology
    Identifiers
    urn:nbn:se:liu:diva-50032 (URN)10.1080/17415970600557299 (DOI)
    Available from: 2009-10-11 Created: 2009-10-11 Last updated: 2017-12-12
    2. A Sideways Heat Equation Applied to the Measurement of the Gas Temperature in a Combustion Chamber
    Open this publication in new window or tab >>A Sideways Heat Equation Applied to the Measurement of the Gas Temperature in a Combustion Chamber
    2010 (English)Report (Other academic)
    Abstract [en]

    We consider a Cauchy problem for a parabolic equation as a mathematical model of the temperature development inside a suction pyrometer. Such devices are often used to calibrate the temperature sensor in a combustion chamber. Mathematically the problem is severely ill-posed and needs to be regularized. The model is simplified to make it one-dimensional in space. The temperature measurements are done in two steps. First, the heat transfer coefficient is approximated via a calibration experiment. Then the gas temperature in the combustion chamber is computed using a convection boundary condition. In both steps one computes the surface temperature and heat flux based on interior measurements in the thermocouple. Numerical examples are presented to test the proposed approach.

    Place, publisher, year, edition, pages
    Linköping: Linköping University Electronic Press, 2010. p. 27
    Series
    LiTH-MAT-R, ISSN 0348-2960 ; 2010:2
    Keywords
    Conduction, convective boundary condition, heat transfer coefficient, ill-posed, Volterra integral operator
    National Category
    Mathematics
    Identifiers
    urn:nbn:se:liu:diva-54294 (URN)
    Available from: 2010-03-08 Created: 2010-03-08 Last updated: 2013-08-30Bibliographically approved
    3. A Preconditioned GMRES Method for Solving a Sideways Parabolic Equation in Two Space Dimensions
    Open this publication in new window or tab >>A Preconditioned GMRES Method for Solving a Sideways Parabolic Equation in Two Space Dimensions
    2010 (English)Report (Other academic)
    Abstract [en]

    The sideways parabolic equation (SPE) is a model of the problem of determining the temperature on the surface of a body from the interior measurements. Mathematically it can be formulated as a non-characteristic Cauchy problem for a parabolic partial differential equation. This problem is severely ill-posed: the solution does not depend continuously on the data. We consider both one and two-dimensional SPE with both constant and variable coefficients. We apply the preconditioned Generalized Minimum Residuals Method (GMRES) on these problems. Preconditioners are chosen in ways that allow efficient implementation using the Fast Fourier Transform (FFT). Regularization is used in the preconditioner as well as truncating the GMRES algorithm. Numerical experiments demonstrate that the proposed method works well.

    Place, publisher, year, edition, pages
    Linköping: Linköping University Electronic Press, 2010. p. 36
    Series
    LiTH-MAT-R, ISSN 0348-2960 ; 2010:3
    Keywords
    Cauchy problem, inverse problem, ill-posed, iterative methods, GMRES preconditioning, FFT, parabolic PDE
    National Category
    Mathematics
    Identifiers
    urn:nbn:se:liu:diva-54298 (URN)
    Available from: 2010-03-08 Created: 2010-03-08 Last updated: 2013-08-30Bibliographically approved
    4. Numerical Solution of a Cauchy Problem for a Parabolic Equation in Two or more Space Dimensions by the Arnoldi Method
    Open this publication in new window or tab >>Numerical Solution of a Cauchy Problem for a Parabolic Equation in Two or more Space Dimensions by the Arnoldi Method
    2010 (English)Report (Other academic)
    Abstract [en]

    We consider the numerical solution of a Cauchy problem for a parabolic equation in multi-dimensional space with cylindrical domain in one spatial space direction. It is desired to find the lower boundary values from the Cauchy data on the upper boundary. This problem is severely ill-posed. The formal solution is written as a hyperbolic cosine function in terms of a multidimensional parabolic (unbounded) operator. We compute an approximate solution by projecting onto a smaller subspace generated via the Arnoldi algorithm applied on the discretized inverse of the operator. Further we regularize the projected problem. The hyperbolic cosine is evaluated explicitly on a low-dimensional subspace. In each iteration step of the Arnoldi method a well-posed parabolic problem is solved. Numerical examples are given to illustrate the performance of the method.

    Place, publisher, year, edition, pages
    Linköping: Linköping University Electronic Press, 2010. p. 23
    Series
    LiTH-MAT-R, ISSN 0348-2960 ; 2010:4
    Keywords
    Cauchy problem, inverse problem, ill-posed, iterative method, Arnoldi method, Schur decomposition, parabolic PDE
    National Category
    Mathematics
    Identifiers
    urn:nbn:se:liu:diva-54299 (URN)
    Available from: 2010-03-08 Created: 2010-03-08 Last updated: 2011-03-09Bibliographically approved
  • 90.
    Ranjbar, Zohreh
    et al.
    Linköping University, The Institute of Technology. Linköping University, Department of Mathematics, Scientific Computing.
    Elden, Lars
    Linköping University, The Institute of Technology. Linköping University, Department of Mathematics, Scientific Computing.
    Numerical analysis of an ill-posed Cauchy problem for a convection - Diffusion equation2007In: Inverse Problems in Science and Engineering, ISSN 1741-5977, E-ISSN 1741-5985, Vol. 15, no 3, p. 191-211Article in journal (Refereed)
    Abstract [en]

    The mathematical and numerical properties of an ill-posed Cauchy problem for a convection - diffusion equation are investigated in this study. The problem is reformulated as a Volterra integral equation of the first kind with a smooth kernel. The rate of decay of the singular values of the integral operator determines the degree of ill-posedness. The purpose of this article is to study how the convection term influences the degree of ill-posedness by computing numerically the singular values. It is also shown that the sign of the coefficient in the convection term determines the rate of decay of the singular values. Some numerical examples are also given to illustrate the theory.

  • 91.
    Ranjbar, Zohreh
    et al.
    Linköping University, Department of Mathematics, Scientific Computing. Linköping University, The Institute of Technology.
    Eldén, Lars
    Linköping University, Department of Mathematics, Scientific Computing. Linköping University, The Institute of Technology.
    A Preconditioned GMRES Method for Solving a 1D Sideways Heat  Equation2010Report (Other academic)
    Abstract [en]

    The sideways Heat equation (SHE) is a model of the problem of determining the temperature on the surface of a body from the interior measurements. Mathematically it can be formulated as a non-characteristic Cauchy problem for a  parabolic partial differential equation. This problem is severely ill-posed: the solution does not depend continuously on the data. We use a preconditioned Generalized Minimum Residuals Method (GMRES) to solve a 1D SHE. Regularization is used in the preconditioner as well as truncating the GMRES algorithm. Numerical experiments demonstrate that the proposed method works well.

  • 92.
    Ranjbar, Zohreh
    et al.
    Linköping University, Department of Mathematics, Scientific Computing. Linköping University, The Institute of Technology.
    Eldén, Lars
    Linköping University, Department of Mathematics, Scientific Computing. Linköping University, The Institute of Technology.
    A Preconditioned GMRES Method for Solving a Sideways Parabolic Equation in Two Space Dimensions2010Report (Other academic)
    Abstract [en]

    The sideways parabolic equation (SPE) is a model of the problem of determining the temperature on the surface of a body from the interior measurements. Mathematically it can be formulated as a non-characteristic Cauchy problem for a parabolic partial differential equation. This problem is severely ill-posed: the solution does not depend continuously on the data. We consider both one and two-dimensional SPE with both constant and variable coefficients. We apply the preconditioned Generalized Minimum Residuals Method (GMRES) on these problems. Preconditioners are chosen in ways that allow efficient implementation using the Fast Fourier Transform (FFT). Regularization is used in the preconditioner as well as truncating the GMRES algorithm. Numerical experiments demonstrate that the proposed method works well.

  • 93.
    Ranjbar, Zohreh
    et al.
    Linköping University, Department of Mathematics, Scientific Computing. Linköping University, The Institute of Technology.
    Eldén, Lars
    Linköping University, Department of Mathematics, Scientific Computing. Linköping University, The Institute of Technology.
    A Sideways Heat Equation Applied to the Measurement of the Gas Temperature in a Combustion Chamber2010Report (Other academic)
    Abstract [en]

    We consider a Cauchy problem for a parabolic equation as a mathematical model of the temperature development inside a suction pyrometer. Such devices are often used to calibrate the temperature sensor in a combustion chamber. Mathematically the problem is severely ill-posed and needs to be regularized. The model is simplified to make it one-dimensional in space. The temperature measurements are done in two steps. First, the heat transfer coefficient is approximated via a calibration experiment. Then the gas temperature in the combustion chamber is computed using a convection boundary condition. In both steps one computes the surface temperature and heat flux based on interior measurements in the thermocouple. Numerical examples are presented to test the proposed approach.

  • 94.
    Rezghi, Mansoor
    et al.
    Department of Mathematics Tarbiat Modares University.
    Elden, Lars
    Linköping University, The Institute of Technology. Linköping University, Department of Mathematics, Scientific Computing.
    Diagonalization of circulant tensors with application in image deblurring2008Report (Other academic)
    Abstract [en]

      

  • 95.
    Savas, Berkant
    Linköping University, Department of Mathematics, Scientific Computing. Linköping University, The Institute of Technology.
    Algorithms in data mining: reduced rank regression and classification by tensor methods2005Licentiate thesis, comprehensive summary (Other academic)
    Abstract [en]

    In many fields of science, engineering, and economics large amounts of data are stored and there is a need to analyze these data in order to extract information for various purposes. Data mining is a general concept involving different tools for performing this kind of analysis. The development of mathematical models and efficient algorithms is of key importance. In this thesis, which consists of three appended manuscripts, we discuss algorithms for reduced rank regression and for classification in the context of tensor theory.

    The first two manuscripts deal with the reduced rank regression problem, which is encountered in the field of state-space subspace system identification. More specifically the problem is

    where A and B are given matrices and we want to find X under a certain rank condition that minimizes the determinant. This problem is not properly stated since it involves implicit assumptions on A and B so that (B - XA)(B - XA)T is never singular. This deficiency of the determinant criterion is fixed by generalizing the minimization criterion to rank reduction and volume minimization of the objective matrix. The volume of a matrix is defined as the product of its nonzero singular values. We give an algorithm that solves the generalized problem and identify properties of the input and output signals causing singularity on the objective matrix.

    Classification problems occur in many applications. The task is to determine the label or class of an unknown object. The third appended manuscript concerns with classification of hand written digits in the context of tensors or multidimensional data arrays. Tensor theory is also an area that attracts more and more attention because of the multidimensional structure of the collected data in a various applications. Two classification algorithms are given based on the higher order singular value decomposition (HOSVD). The main algorithm makes a data reduction using HOSVD of 98%- 99% prior the construction of the class models. The models are computed as a set of orthonormal bases spanning the dominant subspaces for the different classes. An unknown digit is expressed as a linear combination of the basis vectors. The amount of computations is fairly low and the performance reasonably good, 5% in error rate.

    List of papers
    1. Dimensionality reduction and volume minimization - generalization of the determinant minimization criterion for reduced rank regression problems
    Open this publication in new window or tab >>Dimensionality reduction and volume minimization - generalization of the determinant minimization criterion for reduced rank regression problems
    2006 (English)In: Linear Algebra and its Applications, ISSN 0024-3795, Vol. 418, no 1, p. 201-214Article in journal (Refereed) Published
    Abstract [en]

    In this article we propose a generalization of the determinant minimization criterion. The problem of minimizing the determinant of a matrix expression has implicit assumptions that the objective matrix is always nonsingular. In case of singular objective matrix the determinant would be zero and the minimization problem would be meaningless. To be able to handle all possible cases we generalize the determinant criterion to rank reduction and volume minimization of the objective matrix. The generalized minimization criterion is used to solve the following ordinary reduced rank regression problem:

    minrank(X)=kdet(B-XA)(B-XA)T,

    where A and B are known and X is to be determined. This problem is often encountered in the system identification context.

    Keywords
    Volume; Minimization criterion; Determinant; Rank deficient matrix
    National Category
    Mathematics
    Identifiers
    urn:nbn:se:liu:diva-13190 (URN)10.1016/j.laa.2006.01.032 (DOI)
    Available from: 2008-04-29 Created: 2008-04-29 Last updated: 2013-11-06
    2. Rank Reduction and Volume Minimization Approach to State-Space Subspace System Identification
    Open this publication in new window or tab >>Rank Reduction and Volume Minimization Approach to State-Space Subspace System Identification
    2006 (English)In: Signal Processing, ISSN 0165-1684, E-ISSN 1872-7557, Vol. 86, no 11, p. 3275-3285Article in journal (Refereed) Published
    Abstract [en]

    In this paper we consider the reduced rank regression problem

    solved by maximum-likelihood-inspired state-space subspace system identification algorithms. We conclude that the determinant criterion is, due to potential rank-deficiencies, not general enough to handle all problem instances. The main part of the paper analyzes the structure of the reduced rank minimization problem and identifies signal properties in terms of geometrical concepts. A more general minimization criterion is considered, rank reduction followed by volume minimization. A numerically sound algorithm for minimizing this criterion is presented and validated on both simulated and experimental data.

    Place, publisher, year, edition, pages
    Elsevier, 2006
    Keywords
    Reduced rank regression, System identification, General algorithm, Determinant minimization criterion, Rank reduction, Volume minimization
    National Category
    Mathematics Control Engineering
    Identifiers
    urn:nbn:se:liu:diva-13191 (URN)10.1016/j.sigpro.2006.01.008 (DOI)
    Available from: 2008-04-29 Created: 2008-04-29 Last updated: 2017-12-13
    3. Handwritten digit classification using higher order singular value decomposition
    Open this publication in new window or tab >>Handwritten digit classification using higher order singular value decomposition
    2007 (English)In: Pattern Recognition, ISSN 0031-3203, E-ISSN 1873-5142, Vol. 40, no 3, p. 993-1003Article in journal (Refereed) Published
    Abstract [en]

    In this paper we present two algorithms for handwritten digit classification based on the higher order singular value decomposition (HOSVD). The first algorithm uses HOSVD for construction of the class models and achieves classification results with error rate lower than 6%. The second algorithm uses the HOSVD for tensor approximation simultaneously in two modes. Classification results for the second algorithm are almost down at 5% even though the approximation reduces the original training data with more than 98% before the construction of the class models. The actual classification in the test phase for both algorithms is conducted by solving a series least squares problems. Considering computational amount for the test presented the second algorithm is twice as efficient as the first one.

    Keywords
    Handwritten digit classification, Tensors, Higher order singular value decomposition, Tensor approximation, Least squares
    National Category
    Mathematics
    Identifiers
    urn:nbn:se:liu:diva-13192 (URN)10.1016/j.patcog.2006.08.004 (DOI)
    Available from: 2008-04-29 Created: 2008-04-29 Last updated: 2017-12-13
  • 96.
    Savas, Berkant
    Linköping University, Department of Mathematics, Scientific Computing. Linköping University, The Institute of Technology.
    Algorithms in data mining using matrix and tensor methods2008Doctoral thesis, comprehensive summary (Other academic)
    Abstract [en]

    In many fields of science, engineering, and economics large amounts of data are stored and there is a need to analyze these data in order to extract information for various purposes. Data mining is a general concept involving different tools for performing this kind of analysis. The development of mathematical models and efficient algorithms is of key importance. In this thesis we discuss algorithms for the reduced rank regression problem and algorithms for the computation of the best multilinear rank approximation of tensors.

    The first two papers deal with the reduced rank regression problem, which is encountered in the field of state-space subspace system identification. More specifically the problem is

    \[

    \min_{\rank(X) = k} \det (B - X A)(B - X A)\tp,

    \]

    where $A$ and $B$ are given matrices and we want to find $X$ under a certain rank condition that minimizes the determinant. This problem is not properly stated since it involves implicit assumptions on $A$ and $B$ so that $(B - X A)(B - X A)\tp$ is never singular. This deficiency of the determinant criterion is fixed by generalizing the minimization criterion to rank reduction and volume minimization of the objective matrix. The volume of a matrix is defined as the product of its nonzero singular values. We give an algorithm that solves the generalized problem and identify properties of the input and output signals causing a singular objective matrix.

    Classification problems occur in many applications. The task is to determine the label or class of an unknown object. The third paper concerns with classification of handwritten digits in the context of tensors or multidimensional data arrays. Tensor and multilinear algebra is an area that attracts more and more attention because of the multidimensional structure of the collected data in various applications. Two classification algorithms are given based on the higher order singular value decomposition (HOSVD). The main algorithm makes a data reduction using HOSVD of 98--99 \% prior the construction of the class models. The models are computed as a set of orthonormal bases spanning the dominant subspaces for the different classes. An unknown digit is expressed as a linear combination of the basis vectors. The resulting algorithm achieves 5\% in classification error with fairly low amount of computations.

    The remaining two papers discuss computational methods for the best multilinear

    rank approximation problem

    \[

    \min_{\cB} \| \cA - \cB\|

    \]

    where $\cA$ is a given tensor and we seek the best low multilinear rank approximation tensor $\cB$. This is a generalization of the best low rank matrix approximation problem. It is well known that for matrices the solution is given by truncating the singular values in the singular value decomposition (SVD) of the matrix. But for tensors in general the truncated HOSVD does not give an optimal approximation. For example, a third order tensor $\cB \in \RR^{I \x J \x K}$ with rank$(\cB) = (r_1,r_2,r_3)$ can be written as the product

    \[

    \cB = \tml{X,Y,Z}{\cC}, \qquad b_{ijk}=\sum_{\lambda,\mu,\nu}

    x_{i\lambda} y_{j\mu} z_{k\nu} c_{\lambda\mu\nu},

    \]

    where $\cC \in \RR^{r_1 \x r_2 \x r_3}$ and $X \in \RR^{I \times r_1}$, $Y \in \RR^{J \times r_2}$, and $Z \in \RR^{K \times r_3}$ are matrices of full column rank. Since it is no restriction to assume that $X$, $Y$, and $Z$ have orthonormal columns and due to these constraints, the approximation problem can be considered as a nonlinear optimization problem defined on a product of Grassmann manifolds. We introduce novel techniques for multilinear algebraic manipulations enabling means for theoretical analysis and algorithmic implementation. These techniques are used to solve the approximation problem using Newton and Quasi-Newton methods specifically adapted to operate on products of Grassmann manifolds. The presented algorithms are suited for small, large and sparse problems and, when applied on difficult problems, they clearly outperform alternating least squares methods, which are standard in the field.

    List of papers
    1. Dimensionality reduction and volume minimization - generalization of the determinant minimization criterion for reduced rank regression problems
    Open this publication in new window or tab >>Dimensionality reduction and volume minimization - generalization of the determinant minimization criterion for reduced rank regression problems
    2006 (English)In: Linear Algebra and its Applications, ISSN 0024-3795, Vol. 418, no 1, p. 201-214Article in journal (Refereed) Published
    Abstract [en]

    In this article we propose a generalization of the determinant minimization criterion. The problem of minimizing the determinant of a matrix expression has implicit assumptions that the objective matrix is always nonsingular. In case of singular objective matrix the determinant would be zero and the minimization problem would be meaningless. To be able to handle all possible cases we generalize the determinant criterion to rank reduction and volume minimization of the objective matrix. The generalized minimization criterion is used to solve the following ordinary reduced rank regression problem:

    minrank(X)=kdet(B-XA)(B-XA)T,

    where A and B are known and X is to be determined. This problem is often encountered in the system identification context.

    Keywords
    Volume; Minimization criterion; Determinant; Rank deficient matrix
    National Category
    Mathematics
    Identifiers
    urn:nbn:se:liu:diva-13190 (URN)10.1016/j.laa.2006.01.032 (DOI)
    Available from: 2008-04-29 Created: 2008-04-29 Last updated: 2013-11-06
    2. Rank Reduction and Volume Minimization Approach to State-Space Subspace System Identification
    Open this publication in new window or tab >>Rank Reduction and Volume Minimization Approach to State-Space Subspace System Identification
    2006 (English)In: Signal Processing, ISSN 0165-1684, E-ISSN 1872-7557, Vol. 86, no 11, p. 3275-3285Article in journal (Refereed) Published
    Abstract [en]

    In this paper we consider the reduced rank regression problem

    solved by maximum-likelihood-inspired state-space subspace system identification algorithms. We conclude that the determinant criterion is, due to potential rank-deficiencies, not general enough to handle all problem instances. The main part of the paper analyzes the structure of the reduced rank minimization problem and identifies signal properties in terms of geometrical concepts. A more general minimization criterion is considered, rank reduction followed by volume minimization. A numerically sound algorithm for minimizing this criterion is presented and validated on both simulated and experimental data.

    Place, publisher, year, edition, pages
    Elsevier, 2006
    Keywords
    Reduced rank regression, System identification, General algorithm, Determinant minimization criterion, Rank reduction, Volume minimization
    National Category
    Mathematics Control Engineering
    Identifiers
    urn:nbn:se:liu:diva-13191 (URN)10.1016/j.sigpro.2006.01.008 (DOI)
    Available from: 2008-04-29 Created: 2008-04-29 Last updated: 2017-12-13
    3. Handwritten digit classification using higher order singular value decomposition
    Open this publication in new window or tab >>Handwritten digit classification using higher order singular value decomposition
    2007 (English)In: Pattern Recognition, ISSN 0031-3203, E-ISSN 1873-5142, Vol. 40, no 3, p. 993-1003Article in journal (Refereed) Published
    Abstract [en]

    In this paper we present two algorithms for handwritten digit classification based on the higher order singular value decomposition (HOSVD). The first algorithm uses HOSVD for construction of the class models and achieves classification results with error rate lower than 6%. The second algorithm uses the HOSVD for tensor approximation simultaneously in two modes. Classification results for the second algorithm are almost down at 5% even though the approximation reduces the original training data with more than 98% before the construction of the class models. The actual classification in the test phase for both algorithms is conducted by solving a series least squares problems. Considering computational amount for the test presented the second algorithm is twice as efficient as the first one.

    Keywords
    Handwritten digit classification, Tensors, Higher order singular value decomposition, Tensor approximation, Least squares
    National Category
    Mathematics
    Identifiers
    urn:nbn:se:liu:diva-13192 (URN)10.1016/j.patcog.2006.08.004 (DOI)
    Available from: 2008-04-29 Created: 2008-04-29 Last updated: 2017-12-13
    4. A Newton-Grassmann method for computing the best multilinear rank-(r1,r2,r3) approximation of a tensor
    Open this publication in new window or tab >>A Newton-Grassmann method for computing the best multilinear rank-(r1,r2,r3) approximation of a tensor
    2009 (English)In: SIAM Journal on Matrix Analysis and Applications, ISSN 0895-4798, Vol. 32, no 2, p. 248-271Article in journal (Refereed) Published
    Abstract [en]

    We derive a Newton method for computing the best rank-$(r_1,r_2,r_3)$ approximation of a given $J\times K\times L$ tensor $\mathcal{A}$. The problem is formulated as an approximation problem on a product of Grassmann manifolds. Incorporating the manifold structure into Newton's method ensures that all iterates generated by the algorithm are points on the Grassmann manifolds. We also introduce a consistent notation for matricizing a tensor, for contracted tensor products and some tensor-algebraic manipulations, which simplify the derivation of the Newton equations and enable straightforward algorithmic implementation. Experiments show a quadratic convergence rate for the Newton–Grassmann algorithm.

    Keywords
    tensor, multilinear, rank, approximation, Grassmann manifold, Newton
    National Category
    Mathematics
    Identifiers
    urn:nbn:se:liu:diva-13193 (URN)10.1137/070688316 (DOI)
    Available from: 2008-04-29 Created: 2008-04-29 Last updated: 2013-10-11
    5. Best multilinear rank approximation of tensors with quasi-Newton methods on Grassmannians
    Open this publication in new window or tab >>Best multilinear rank approximation of tensors with quasi-Newton methods on Grassmannians
    Manuscript (Other academic)
    Identifiers
    urn:nbn:se:liu:diva-13194 (URN)
    Available from: 2008-04-29 Created: 2008-04-29 Last updated: 2010-01-13
  • 97.
    Savas, Berkant
    Linköping University, The Institute of Technology. Linköping University, Department of Mathematics, Scientific Computing.
    Analyses and Tests of Handwritten Digit Recognition Algorithms2004In: Workshop i tillämpad matematik,2004, 2004Conference paper (Other academic)
  • 98.
    Savas, Berkant
    Linköping University, The Institute of Technology. Linköping University, Department of Mathematics, Scientific Computing.
    Dimensionality reduction and volume minimization - generalization of the determinant minimization criterion for reduced rank regression problems2005In: Working Group on Matrix Computations and Statistics,2005, 2005Conference paper (Other academic)
  • 99.
    Savas, Berkant
    Linköping University, Department of Mathematics, Scientific Computing. Linköping University, The Institute of Technology.
    Dimensionality reduction and volume minimization - generalization of the determinant minimization criterion for reduced rank regression problems2006In: Linear Algebra and its Applications, ISSN 0024-3795, Vol. 418, no 1, p. 201-214Article in journal (Refereed)
    Abstract [en]

    In this article we propose a generalization of the determinant minimization criterion. The problem of minimizing the determinant of a matrix expression has implicit assumptions that the objective matrix is always nonsingular. In case of singular objective matrix the determinant would be zero and the minimization problem would be meaningless. To be able to handle all possible cases we generalize the determinant criterion to rank reduction and volume minimization of the objective matrix. The generalized minimization criterion is used to solve the following ordinary reduced rank regression problem:

    minrank(X)=kdet(B-XA)(B-XA)T,

    where A and B are known and X is to be determined. This problem is often encountered in the system identification context.

  • 100.
    Savas, Berkant
    Linköping University, The Institute of Technology. Linköping University, Department of Mathematics, Scientific Computing.
    Dimensionality Reduction and Volume Minimization: Generalization of the Determinant Minimization Criterion for Reduced Rank Regression Problems2005Report (Other academic)
    Abstract [en]

    In this article we propose a generalization of the determinant minimization criterion. The problem of minimizing the determinant of a matrix expression has implicit assumptions that the objective matrix is always nonsingular. In case of singular objective matrix the determinant would be zero and the minimization problem would be meaningless. To be able to handle all possible cases we generalize the determinant criterion to rank reduction and volume minimization of the objective matrix. The generalized minimization criterion is used to solve the following ordinary reduced rank regression problem:minrank(X)=kdet(B-XA)(B-XA)T,where A and B are known and X is to be determined. This problem is often encountered in the system identification context.

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