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• 1.
Linköping University, Department of Mathematics, Scientific Computing.
Energy estimates and variance estimation for hyperbolic stochastic partial differentialequations2011Independent thesis Advanced level (professional degree), 30 credits / 45 HE creditsStudent thesis

In this thesis the connections between the boundary conditions and the vari- ance of the solution to a stochastic partial differential equation (PDE) are investigated. In particular a hyperbolical system of PDE’s with stochastic initial and boundary data are considered. The problem is shown to be well- posed on a class of boundary conditions through the energy method. Stability is shown by using summation-by-part operators coupled with simultaneous- approximation-terms. By using the energy estimates, the relative variance of the solutions for different boundary conditions are analyzed. It is concluded that some types of boundary conditions yields a lower variance than others. This is verified by numerical computations.

• 2.
Department of Computer Science and Engineering, The Pennsylvania State University, USA.
Linköping University, Department of Mathematics, Scientific Computing. Linköping University, The Institute of Technology. Department of Statistics, University of Bologna, Italy.
Editorial Material: 3rd Special issue on matrix computations and statistics2010In: Computational Statistics & Data Analysis, ISSN 0167-9473, E-ISSN 1872-7352, Vol. 54, no 12, p. 3379-3380Article in journal (Other academic)

n/a

• 3.
Uppsala University, Department of Information Technology.
Linköping University, The Institute of Technology. Linköping University, Department of Mathematics, Scientific Computing.
Stable Robin solid wall boundary conditions for the Navier-Stokes equations2011In: Journal of Computational Physics, ISSN 0021-9991, E-ISSN 1090-2716, Vol. 230, no 19, p. 7519-7532Article in journal (Refereed)

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 first order slip-flow boundary conditions. The formulation is uniform and the transitions between different boundary conditions are done by a change of parameters. We give different sharp energy estimates depending on the choice of parameters.

The discretization is done using finite differences 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 flows ranging from no-slip to almost full slip.

• 4.
Linköping University, Department of Mathematics, Scientific Computing. Linköping University, The Institute of Technology.
A survey of methods for determinig surface temperatures using interior measurements2001In: Trends in Heat, Mass & Momentum Transfer, ISSN 0973-2446, Vol. 7, no pp, p. 105-128Article in journal (Refereed)
• 5.
Linköping University, The Institute of Technology. Linköping University, Department of Mathematics, Scientific Computing.
Boundary identification for an elliptic equation2002In: Inverse Problems, ISSN 0266-5611, E-ISSN 1361-6420, Vol. 18, no 6, p. 1579-1592Article in journal (Refereed)

We consider an inverse problem for the two-dimensional steady-state heat equation. More precisely, the heat equation is valid in a domain O, that is a subset of the unit square. Temperature and heat-flux measurements are available on the line y = 0, and the sides x = 0 and 1 are assumed to be insulated. From these we wish to determine the temperature in the domain O. Furthermore, a part of the boundary ?O is considered to be unknown, and must also be determined. The problem is ill-posed in the sense that the solution does not depend continuously on the data. We stabilize the computations by replacing the x-derivative in the heat equation by an operator, representing differentiation of least-squares cubic splines. We discretize in the x-coordinate, and obtain an initial value problem for a system of ordinary differential equations, which can be solved using standard numerical methods. The inverse problem that we consider in this paper arises in iron production, where the walls of a melting furnace are subject to physical and chemical wear. In order to avoid a situation where molten metal breaks out the remaining thickness of the walls should constantly be monitored. This is done by recording the temperature at several locations inside the walls. The shape of the interface boundary between the molten iron and the walls of the furnace can then be determined by solving an inverse heat conduction problem.

• 6.
Linköping University, Department of Mathematics, Scientific Computing. Linköping University, The Institute of Technology.
Numerical Methods for an Inverse Heat Conduction Problem1998Conference paper (Other academic)
• 7.
Linköping University, The Institute of Technology. Linköping University, Department of Mathematics, Scientific Computing.
Sequential solution of the sideways heat equation by windowing of the data2003In: Inverse Problems in Engineering, ISSN 1068-2767, E-ISSN 1029-0281, Vol. 11, no 2, p. 91-103Article in journal (Refereed)

The sideways heat equation is a one-dimensional model of a problem, where one wants to determine the temperature on the surface of a body using interior measurements. More precisely, we consider a heat conduction problem, where temperature and heat-flux data are available along the line x = 1 and the solution is sought in the interval 0 = x < 1. The problem is ill-posed in the sense that the solution does not depend continuously on the data. Stability can be restored by replacing the time derivative in the heat equation with a bounded spectral approximation. The cut off level in the spectral approximation acts as a regularization parameter, that controls the degree of smoothness in the solution. In certain applications one wants to solve the sideways heat equation in real time, i.e. to constantly update the solution as new measurements are recorded. For this case sequential solution methods are required.

• 8.
Linköping University, Department of Mathematics, Scientific Computing. Linköping University, The Institute of Technology.
Simulation Tools for Injection Moulding1997Conference paper (Other academic)
• 9.
Linköping University, The Institute of Technology. Linköping University, Department of Mathematics, Scientific Computing.
Linköping University, The Institute of Technology. Linköping University, Department of Mathematics, Scientific Computing.
Numerical Solution of Cauchy Problems for Elliptic Equations in "Rectangle-like" Geometries2005In: FEMLAB Conference,2005, Stockholm: Comsol AB , 2005Conference paper (Other academic)

We consider two dimensional inverse steady state heat conduction problems in complex geometries. The coefficients of the elliptic equation are assumed to be non-constant. Cauchy data are given on one part of the boundary and we want to find the solution in the whole domain. The problem is ill--posed in the sense that the solution does not depend continuously on the data. Using an orthogonal coordinate transformation the domain is mapped onto a rectangle. The Cauchy problem can then be solved by replacing one derivative by a bounded approximation. The resulting well--posed problem can then be solved by a method of lines. A bounded approximation of the derivative can be obtained by differentiating a cubic spline, that approximate the function in the least squares sense. This particular approximation of the derivative is computationally efficient and flexible in the sense that its easy to handle different kinds of boundary conditions. This inverse problem arises in iron production, where the walls of a melting furnace are subject to physical and chemical wear. Temperature and heat--flux data are collected by several thermocouples located inside the walls. The shape of the interface between the molten iron and the walls can then be determined by solving an inverse heat conduction problem. In our work we make extensive use of Femlab for creating test problems. By using Femlab we solve relatively complex model problems for the purpose of creating numerical test data used for validating our methods. For the types of problems we are intressted in numerical artefacts appear, near corners in the domain, in the gradients that Femlab calculates. We demonstrate why this happen and also how we deal with the problem.

• 10.
Linköping University, Department of Mathematics, Scientific Computing. Linköping University, The Institute of Technology.
Linköping University, Department of Mathematics, Scientific Computing. Linköping University, The Institute of Technology.
Numerical Solution of an Inverse Steady State Heat Conduction Problem2000Conference paper (Other academic)
• 11.
Linköping University, Department of Mathematics, Scientific Computing. Linköping University, The Institute of Technology.
Linköping University, Department of Mathematics, Scientific Computing. Linköping University, The Institute of Technology.
Spectral and Wavelet Methods for Solving an Inverse Heat Conduction Problem1998In: Inverse Problems in Engineering Mechanics: International Symposium on Inverse Problems in Engineering Mechanics, 1998 (ISIP 98) / [ed] M. Tanaka and G.S. Dulikravich, Oxford: Elsevier Science , 1998, p. 3-10Conference paper (Other academic)
• 12.
Linköping University, Department of Mathematics, Scientific Computing. Linköping University, The Institute of Technology.
Linköping University, Department of Mathematics, Scientific Computing. Linköping University, The Institute of Technology. Linköping University, Department of Mechanical Engineering, Applied Thermodynamics and Fluid Mechanics. Linköping University, The Institute of Technology. Linköping University, Department of Mechanical Engineering. Linköping University, The Institute of Technology.
A Comparison of Three Numerical Methods for an Inverse Heat Conduction Problem and an Industrial Application1997Conference paper (Other academic)
• 13.
Linköping University, Department of Mathematics, Scientific Computing. Linköping University, The Institute of Technology.
Linköping University, Department of Mathematics, Applied Mathematics. Linköping University, The Institute of Technology. Linköping University, Department of Mathematics, Applied Mathematics. Linköping University, The Institute of Technology. Linköping University, Department of Mathematics, Applied Mathematics. Linköping University, The Institute of Technology.
An accelerated alternating procedure for the Cauchy problem for the Helmholtz equation2014In: Computers and Mathematics with Applications, ISSN 0898-1221, E-ISSN 1873-7668, Vol. 68, no 1-2, p. 44-60Article in journal (Refereed)

In this paper we study the Cauchy problem for the Helmholtz equation. This problem appears in various applications and is severely ill–posed. The modified alternating procedure has been proposed by the authors for solving this problem but the convergence has been rather slow. We demonstrate how to instead use conjugate gradient methods for accelerating the convergence. The main idea is to introduce an artificial boundary in the interior of the domain. This addition of the interior boundary allows us to derive an inner product that is natural for the application and that gives us a proper framework for implementing the steps of the conjugate gradient methods. The numerical results performed using the finite difference method show that the conjugate gradient based methods converge considerably faster than the modified alternating iterative procedure studied previously.

• 14.
Linköping University, Department of Mathematics, Scientific Computing. Linköping University, The Institute of Technology.
Linköping University, Department of Mathematics, Applied Mathematics. Linköping University, The Institute of Technology. Linköping University, Department of Mathematics, Applied Mathematics. Linköping University, The Institute of Technology. Linköping University, Department of Mathematics, Applied Mathematics. Linköping University, The Institute of Technology.
Robin–Dirichlet algorithms for the Cauchy problem for the Helmholtz equation2014Manuscript (preprint) (Other academic)

The Cauchy problem for the Helmholtz equation is considered. It was demonstrated in a previous paper by the authors that the alternating algorithm suggested by V.A. Kozlov and V.G. Maz’ya does not converge for large wavenumbers in the Helmholtz equation. We prove here that if we alternate Robin and Dirichlet boundary conditions instead of Neumann and Dirichlet boundary conditions, then the algorithm will converge. We present also another algorithm based on the same idea, which converges for large wavenumbers. Numerical implementations obtained using the finite difference method are presented. Numerical results illustrate that the algorithms suggested in this paper, produce a convergent iterative sequences.

• 15.
Linköping University, Department of Mathematics, Scientific Computing. Linköping University, The Institute of Technology.
Department of Informatics, University of Bergen, NO-5020 Bergen, Norway.
Methods for large scale total least squares problems2001In: SIAM Journal on Matrix Analysis and Applications, ISSN 0895-4798, E-ISSN 1095-7162, Vol. 22, no 2, p. 413-429Article in journal (Refereed)

The solution of the total least squares (TLS) problems, minE,f ?(E,f)?F subject to (A + E)x = b + f, can in the generic case be obtained from the right singular vector corresponding to the smallest singular value sn+1 of (A, b). When A is large and sparse (or structured) a method based on Rayleigh quotient iteration (RQI) has been suggested by Björck. In this method the problem is reduced to the solution of a sequence of symmetric, positive definite linear systems of the form (ATA - s¯2I)z = g, where s¯ is an approximation to sn+1. These linear systems are then solved by a preconditioned conjugate gradient method (PCGTLS). For TLS problems where A is large and sparse a (possibly incomplete) Cholesky factor of AT A can usually be computed, and this provides a very efficient preconditioner. The resulting method can be used to solve a much wider range of problems than it is possible to solve by using Lanczos-type algorithms directly for the singular value problem. In this paper the RQI-PCGTLS method is further developed, and the choice of initial approximation and termination criteria are discussed. Numerical results confirm that the given algorithm achieves rapid convergence and good accuracy.

• 16.
University of Copenhagen.
Linköping University, Department of Mathematics, Scientific Computing. Linköping University, The Institute of Technology.
PLS works2009In: Journal of Chemometrics, ISSN 0886-9383, E-ISSN 1099-128X, Vol. 23, no 1-2, p. 69-71Article in journal (Refereed)

In a recent paper, claims were made that most current implementations of PLS provide wrong and misleading residuals [1]. In this paper the relation between PLS and Lanczos bidiagonalization is described and it is shown that there is a good rationale behind current implementations of PLS. Most importantly, the residuals determined in current implementations of PLS are independent of the scores used for predicting the dependent variable(s). Oppositely, in the newly suggested approach, the residuals are correlated to the scores and hence may be high due to variation that is actually used for predicting. It is concluded that the current practice of calculating residuals be maintained.

• 17.
Univ Haifa, Dept Math, IL-31905 Haifa, Israel Linkoping Univ, Dept Math, SE-58183 Linkoping, Sweden.
Linköping University, The Institute of Technology. Linköping University, Department of Mathematics, Scientific Computing.
Block-iterative algorithms with diagonally scaled oblique projections for the linear feasibility problem2002In: SIAM Journal on Matrix Analysis and Applications, ISSN 0895-4798, E-ISSN 1095-7162, Vol. 24, no 1, p. 40-58Article in journal (Refereed)

We formulate a block- iterative algorithmic scheme for the solution of systems of linear inequalities and/ or equations and analyze its convergence. This study provides as special cases proofs of convergence of ( i) the recently proposed component averaging ( CAV) method of Censor, Gordon, and Gordon [ Parallel Comput., 27 ( 2001), pp. 777 808], ( ii) the recently proposed block- iterative CAV ( BICAV) method of the same authors [ IEEE Trans. Medical Imaging, 20 ( 2001), pp. 1050 1060], and ( iii) the simultaneous algebraic reconstruction technique ( SART) of Andersen and Kak [ Ultrasonic Imaging, 6 ( 1984), pp. 81 94] and generalizes them to linear inequalities. The first two algorithms are projection algorithms which use certain generalized oblique projections and diagonal weighting matrices which reflect the sparsity of the underlying matrix of the linear system. The previously reported experimental acceleration of the initial behavior of CAV and BICAV is thus complemented here by a mathematical study of the convergence of the algorithms.

• 18. Censor, Yair
Linköping University, The Institute of Technology. Linköping University, Department of Mathematics, Scientific Computing.
Iterative algorithms with seminorm-induced oblique projections2003Article in journal (Refereed)

A definition of oblique projections onto closed convex sets that use seminorms induced by diagonal matrices which may have zeros on the diagonal is introduced. Existence and uniqueness of such projections are secured via directional affinity of the sets with respect to the diagonal matrices involved. A block-iterative algorithmic scheme for solving the convex feasibility problem, employing seminorm-induced oblique projections, is constructed and its convergence for the consistent case is established. The fully simultaneous algorithm converges also in the inconsistent case to the minimum of a certain proximity function.

• 19.
Department of Mathematics, University of Haifa, Mt. Carmel, Haifa, Israel.
Linköping University, Department of Mathematics, Scientific Computing. Linköping University, The Institute of Technology. Department of Computer Science, The Graduate Center, City University of New York, New York, USA. Linköping University, Department of Mathematics, Scientific Computing. Linköping University, The Institute of Technology.
On Diagonally Relaxed Orthogonal Projection Methods2008In: SIAM Journal on Scientific Computing, ISSN 1064-8275, E-ISSN 1095-7197, Vol. 30, no 1, p. 473-504Article in journal (Refereed)

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.

• 20.
University of Haifa Israel.
Linköping University, The Institute of Technology. Linköping University, Department of Mathematics, Scientific Computing. University of Haifa Israel. Harvard Medical School USA.
The multiple-sets split feasibility problem and its applications for inverse problems2005In: Inverse Problems, ISSN 0266-5611, E-ISSN 1361-6420, Vol. 21, p. 2071-2084Article in journal (Refereed)
• 21.
Inst Computat Technol, Novosibirsk 630090, Russia Linkoping Univ, Dept Math, S-58138 Linkoping, Sweden.
Inst Computat Technol, Novosibirsk 630090, Russia Linkoping Univ, Dept Math, S-58138 Linkoping, Sweden. Inst Computat Technol, Novosibirsk 630090, Russia Linkoping Univ, Dept Math, S-58138 Linkoping, Sweden. Linköping University, The Institute of Technology. Linköping University, Department of Mathematics, Scientific Computing.
Comparative analysis of nonlinear dispersive shallow water models2000In: International journal of computational fluid dynamics (Print), ISSN 1061-8562, E-ISSN 1029-0257, Vol. 14, no 1, p. 55-73Article in journal (Refereed)

The results of comparative analysis of some nonlinear dispersive models of shallow water are presented. The aim is to find their individual properties relevant for the numerical solution of some model problems of long wave transformation over submerged obstacles The study considers basic properties of the listed models and their numerical implementation. Computations are obtained compared with the analytical solution and experimental data. Attention is primarily focused on the models suggested by Peregrine (1967), Zheleznyak and Pelinovsky (1985), Kim, Reid, Whitaker (1988), Fedotova and Pashkova (1997). Also classical equations of shallow water are considered in both linear and nonlinear approximations.

• 22. Domeij Bäckryd, Rebecka
Linköping University, The Institute of Technology. Linköping University, Department of Mathematics, Scientific Computing.
Simulation of heat transfer on a gas sensor component2005In: FEMLAB Conference,2005, Stockholm: Comsol AB , 2005, p. 177-181Conference paper (Other academic)

Gas sensors are used in many different application areas. As many gas sensor components are battery heated, one major limit of the operation time is the power dissipated as heat. The aim of this work has been to simulate the heat transfer on a hydrogen gas sensor component. Modelling and simulations have been performed in FEMLAB. The partial differential equation with boundary conditions was solved and the solution was validated against experimental data. Convection increases with the increase of hydrogen concentration. A great effort was made to find a model for the convection. When the simulations were compared to experiments, it turned out that the theoretical convection model was insufficient to describe this small system involving hydrogen, which was an unexpected but interesting result.

• 23.
Linköping University, The Institute of Technology. Linköping University, Department of Mathematics, Scientific Computing.
Accuracy and Reliability in Scientific Computing2005Collection (editor) (Other academic)

Numerical software is used to test scientific theories, design airplanes and bridges, operate manufacturing lines, control power plants and refineries, analyze financial derivatives, identify genomes, and provide the understanding necessary to derive and analyze cancer treatments. Because of the high stakes involved, it is essential that results computed using software be accurate, reliable, and robust. Unfortunately, developing accurate and reliable scientific software is notoriously difficult. This book investigates some of the difficulties related to scientific computing and provides insight into how to overcome them and obtain dependable results. The tools to assess existing scientific applications are described, and a variety of techniques that can improve the accuracy and reliability of newly developed applications is discussed. Accuracy and Reliability in Scientific Computing can be considered a handbook for improving the quality of scientific computing.

It will help computer scientists address the problems that affect software in general as well as the particular challenges of numerical computation: approximations occurring at all levels, continuous functions replaced by discretized versions, infinite processes replaced by finite ones, and real numbers replaced by finite precision numbers. Divided into three parts, it starts by illustrating some of the difficulties in producing robust and reliable scientific software. Well-known cases of failure are reviewed and the what and why of numerical computations are considered. The second section describes diagnostic tools that can be used to assess the accuracy and reliability of existing scientific applications.

In the last section, the authors describe a variety of techniques that can be employed to improve the accuracy and reliability of newly developed scientific applications. The authors of the individual chapters are international experts, many of them members of the IFIP Working Group on Numerical Software. Accuracy and Reliability in Scientific Computing contains condensed information on the main features of six major programming languages - Ada, C, C++, Fortran, Java, and Python - and the INTLAB toolbox of the MATLAB software and the PRECISE toolbox of Fortran are discussed in detail. This book has an accompanying website, www.nsc.liu.se/wg25/book/, with codes, links, color versions of some illustrations, and additional material.

• 24.
Linköping University, Department of Mathematics, Scientific Computing. Linköping University, The Institute of Technology.
Visual Numerics Inc., Albuquerque, NM. University of Kent, Canterbury, England.
Standardized mixed language programming for Fortran and C2009In: ACM SIGPLAN Fortran Forum, ISSN 1061-7264, Vol. 28, no 3, p. 8-22Article in journal (Refereed)

Programmers have long practiced the matter of mixed language procedure calls. This is particularly true for the programming languages C and Fortran. The use of the alternate language often results in efficient running time or the effective use of human or other resources.

Prior to the Fortran 2003 standard there was silence about how the two languages interoperated. Before this release there existed a set of differing ad hoc methods for making the inter-language calls. These typically depended on the Fortran and C compilers. The newer Fortran standard provides an intrinsic module, iso_c_binding, that permits the languages to interoperate. There remain restrictions regarding interoperable data types.

This paper illustrates several programs that contain core exercises likely to be encountered by programmers. The source code is available from the first author's web site. Included is an illustration of a "trap" based on use of the ad hoc methods: A call from a C to a Fortran 2003 routine that passes a character in C to a character variable in Fortran results in a run-time error.

• 25.
Linköping University, The Institute of Technology. Linköping University, Department of Mathematics, Scientific Computing.
Matrix methods in data mining and pattern recognition2007Other (Other (popular science, discussion, etc.))
• 26.
Linköping University, The Institute of Technology. Linköping University, Department of Mathematics, Scientific Computing.
Multi-linear mappings, SVD, HOSVD, and the numerical solution of ill-conditioned tensor least squares problems2005In: Second Workshop on Tensor Decompositions and Applications TDA05,2005, 2005Conference paper (Other academic)
• 27.
Linköping University, The Institute of Technology. Linköping University, Department of Mathematics, Scientific Computing.
Numerical Linear Algebra and Applications in Data Mining and IT2003Other (Other (popular science, discussion, etc.))
• 28.
Linköping University, The Institute of Technology. Linköping University, Department of Mathematics, Scientific Computing.
Numerical linear algebra in data mining2006In: Acta Numerica, ISSN 0962-4929, E-ISSN 1474-0508, Vol. 15, p. 327-384Article, review/survey (Refereed)

Ideas and algorithms from numerical linear algebra are important in several areas of data mining. We give an overview of linear algebra methods in text mining (information retrieval), pattern recognition (classification of handwritten digits), and Page Rank computations for web search engines. The emphasis is on rank reduction as a method of extracting information from a data matrix, low-rank approximation of matrices using the singular value decomposition and clustering, and on eigenvalue methods for network analysis. © Cambridge University Press, 2006.

• 29.
Linköping University, The Institute of Technology. Linköping University, Department of Mathematics, Scientific Computing.
Numerical Solution of Cauchy Problems for Elliptic PDE's in ComplexGeometries2003In: Applied Inverse Problems,2003, 2003Conference paper (Other academic)
• 30.
Linköping University, The Institute of Technology. Linköping University, Department of Mathematics, Scientific Computing.
Numerical Solution of Cauchy Problems for Elliptic PDE's in ComplexGeometries2003In: Conference on Numerical Analysis,2003, 2003Conference paper (Other academic)
• 31.
Linköping University, The Institute of Technology. Linköping University, Department of Mathematics, Scientific Computing.
Solving quadratically constrained least squares problems using a differential-geometric approach2002In: BIT Numerical Mathematics, ISSN 0006-3835, E-ISSN 1572-9125, Vol. 42, no 2, p. 323-335Article in journal (Refereed)

A quadratically constrained linea least squares problem is usually solved using a Lagrange multiplier for the constraint and then solving iteratively a nonlinear secular equation for the optimal Lagrange multiplier. It is well-known that, due to the closeness to a pole for the secular equation, standard methods for solving the secular equation can be slow, and sometimes it is not easy to select a good starting value for the iteration. The problem can be reformulated as that of minimizing the residual of the least squares problem on the unit sphere. Using a differential-geometric approach we formulate Newton's method on the sphere, and thereby avoid the difficulties associated with the Lagrange multiplier formulation. This Newton method on the sphere can be implemented efficiently, and since it is easy to find a good starting value for the iteration, and the convergence is often quite fast, it has a clear advantage over the Lagrange multiplier method. A numerical example is given.

• 32.
Linköping University, The Institute of Technology. Linköping University, Department of Mathematics, Scientific Computing.
• 33.
Linköping University, The Institute of Technology. Linköping University, Department of Mathematics, Scientific Computing.
The Maximum Likelihood Estimate in Reduced-Rank Regression2003In: Numerical Linear Algebra and its Applications,2003, 2003Conference paper (Other academic)
• 34.
Linköping University, The Institute of Technology. Linköping University, Department of Mathematics, Scientific Computing.
Linköping University, The Institute of Technology. Linköping University, Department of Mathematics, Scientific Computing.
A stability estimate for a Cauchy problem for an elliptic partial differential equation2005In: Inverse Problems, ISSN 0266-5611, E-ISSN 1361-6420, Vol. 21, no 5, p. 1643-1653Article in journal (Refereed)

A two-dimensional inverse steady state heat conduction problem in the unit square is considered. Cauchy data are given for y ≤ 0, and boundary data are for x ≤ 0 and x ≤ 1. The elliptic operator is self-adjoint with non-constant, smooth coefficients. The solution for y ≤ 1 is sought. This Cauchy problem is ill-posed in an L2-setting. A stability functional is defined, for which a differential inequality is derived. Using this inequality a stability result of Hölder type is proved. It is demonstrated explicitly how the stability depends on the smoothness of the coefficients. The results can also be used for rectangle-like regions that can be mapped conformally onto a rectangle. © 2005 IOP Publishing Ltd.

• 35.
Linköping University, The Institute of Technology. Linköping University, Department of Mathematics, Scientific Computing.
Minimization of linear functionals defined on solutions of large-scale discrete Ill-posed problems2005In: BIT Numerical Mathematics, ISSN 0006-3835, E-ISSN 1572-9125, Vol. 45, no 2, p. 329-340Article in journal (Refereed)

The minimization of linear functionals defined on the solutions of discrete ill-posed problems arises, e.g., in the computation of confidence intervals for these solutions. In 1990, Eldén proposed an algorithm for this minimization problem based on a parametric programming reformulation involving the solution of a sequence of trust-region problems, and using matrix factorizations. In this paper, we describe MLFIP, a large-scale version of this algorithm where a limited-memory trust-region solver is used on the subproblems. We illustrate the use of our algorithm in connection with an inverse heat conduction problem.

• 36.
Linköping University, The Institute of Technology. Linköping University, Department of Mathematics, Scientific Computing.
Linköping University, The Institute of Technology. Linköping University, Department of Mathematics, Scientific Computing.
Introduction to Numerical Computation2004Book (Other (popular science, discussion, etc.))
• 37.
Linköping University, The Institute of Technology. Linköping University, Department of Mathematics, Scientific Computing.
Matrix rank reduction for data analysis and feature extraction2006In: Handbook Parallel Computing and Statistics / [ed] Haesun Park and Lars Eldén, Boca Raton: CRC Press , 2006, p. 415-447Chapter in book (Other academic)

Numerical techniques for data analysis and feature extraction are discussed using the framework of matrix rank reduction. The singular value decomposition (SVD) and its properties are reviewed, and the relation to Latent Semantic Indexing (LSI) and Principal Component Analysis (PCA) is described. Methods that approximate the SVD are reviewed. A few basic methods for linear regression, in particular the Partial Least Squares (PLS) method, arepresented, and analyzed as rank reduction methods. Methods for feature extraction, based on centroids and the classical Linear Discriminant Analysis (LDA), as well as an improved LDA based on the generalized singular value decomposition (LDA/GSVD) are described. The effectiveness of these methods are illustrated using examples from information retrieval, and 2 dimensional representation of clustered data.

• 38.
Linköping University, The Institute of Technology. Linköping University, Department of Mathematics, Scientific Computing.
Linköping University, The Institute of Technology. Linköping University, Department of Mathematics, Scientific Computing.
Pattern Recognition using Higher Order SVD2005In: 3rd IASC world conference onComputational Statistics and Data Analysis,2005, 2005Conference paper (Other academic)
• 39.
Linköping University, The Institute of Technology. Linköping University, Department of Mathematics, Scientific Computing.
Linköping University, The Institute of Technology. Linköping University, Department of Mathematics, Scientific Computing.
The Maximum likelihood estimate in reduced-rank regression2003Report (Other academic)
• 40.
Linköping University, The Institute of Technology. Linköping University, Department of Mathematics, Scientific Computing.
Linköping University, The Institute of Technology. Linköping University, Department of Mathematics, Scientific Computing.
The maximum likelihood estimate in reduced-rank regression2005In: Numerical Linear Algebra with Applications, ISSN 1070-5325, E-ISSN 1099-1506, Vol. 12, no 8, p. 731-741Article in journal (Refereed)

In previous work by Stoica and Viberg the reduced-rank regression problem is solved in a maximum likelihood sense. The present paper proposes an alternative numerical procedure. The solution is written in terms of the principal angles between subspaces spanned by the data matrices. It is demonstrated that the solution is meaningful also in the case when the maximum likelihood criterion is not valid. A numerical example is given. Copyright (c) 2005 John Wiley & Sons, Ltd.

• 41.
Linköping University, Department of Mathematics, Scientific Computing. Linköping University, The Institute of Technology.
Partial least-squares vs. Lanczos bidiagonalization—I: analysis of a projection method for multiple regression2004In: Computational Statistics & Data Analysis, ISSN 0167-9473, E-ISSN 1872-7352, Vol. 46, no 1, p. 11-31Article in journal (Refereed)

Multiple linear regression is considered and the partial least-squares method (PLS) for computing a projection onto a lower-dimensional subspace is analyzed. The equivalence of PLS to Lanczos bidiagonalization is a basic part of the analysis. Singular value analysis, Krylov subspaces, and shrinkage factors are used to explain why, in many cases, PLS gives a faster reduction of the residual than standard principal components regression. It is also shown why in some cases the dimension of the subspace, given by PLS, is not as small as desired.

• 42.
Linköping University, Department of Mathematics, Scientific Computing. Linköping University, The Institute of Technology.
Linköping University, Department of Mathematics, Scientific Computing. Linköping University, The Institute of Technology.
A Newton-Grassmann method for computing the best multilinear rank-(r1,r2,r3) approximation of a tensor2009In: SIAM Journal on Matrix Analysis and Applications, ISSN 0895-4798, Vol. 32, no 2, p. 248-271Article in journal (Refereed)

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.

• 43.
Linköping University, Department of Mathematics, Scientific Computing. Linköping University, The Institute of Technology.
Linköping University, Department of Mathematics, Scientific Computing. Linköping University, The Institute of Technology.
Perturbation Theory and Optimality Conditions for the Best Multilinear Rank Approximation of a Tensor2011In: SIAM Journal on Matrix Analysis and Applications, ISSN 0895-4798, E-ISSN 1095-7162, Vol. 32, no 4, p. 1422-1450Article in journal (Refereed)

The problem of computing the best rank-(p,q,r) approximation of a third order tensor is considered. First the problem is reformulated as a maximization problem on a product of three Grassmann manifolds. Then expressions for the gradient and the Hessian are derived in a local coordinate system at a stationary point, and conditions for a local maximum are given. A first order perturbation analysis is performed using the Grassmann manifold framework. The analysis is illustrated in a few examples, and it is shown that the perturbation theory for the singular value decomposition is a special case of the tensor theory.

• 44.
Linköping University, Department of Mathematics, Scientific Computing. Linköping University, The Institute of Technology.
University of Bologna.
A numerical solution of a Cauchy problem for an elliptic equation by Krylov subspaces2009In: INVERSE PROBLEMS, ISSN 0266-5611 , Vol. 25, no 6, p. 065002-Article in journal (Refereed)

We study the numerical solution of a Cauchy problem for a self-adjoint elliptic partial differential equation u(zz) - L-u = 0 in three space dimensions (x, y, z), where the domain is cylindrical in z. Cauchy data are given on the lower boundary and the boundary values on the upper boundary are sought. The problem is severely ill-posed. The formal solution is written as a hyperbolic cosine function in terms of the two-dimensional elliptic operator L (via its eigenfunction expansion), and it is shown that the solution is stabilized (regularized) if the large eigenvalues are cut off. We suggest a numerical procedure based on the rational Krylov method, where the solution is projected onto a subspace generated using the operator L-1. This means that in each Krylov step, a well-posed two-dimensional elliptic problem involving L is solved. Furthermore, the hyperbolic cosine is evaluated explicitly only for a small symmetric matrix. A stopping criterion for the Krylov recursion is suggested based on the relative change of an approximate residual, which can be computed very cheaply. Two numerical examples are given that demonstrate the accuracy of the method and the efficiency of the stopping criterion.

• 45.
Linköping University, The Institute of Technology. Linköping University, Department of Mathematics, Scientific Computing.
A projection method for semidefinite linear systems and its applications2004Article in journal (Refereed)

We study the solution of consistent, semidefinite and symmetric linear systems by iterative techniques. Given a finite sequence of subspaces a block-iterative projection type algorithm is considered. For two specific choices of iteration parameters we show convergence. We apply our results to over and under determined linear equations. These methods are based on decomposing the system matrix into blocks of rows or blocks of columns. Thereby several algorithms, many used in image reconstruction, are presented in a unified way.

• 46.
Linköping University, Department of Mathematics, Scientific Computing. Linköping University, The Institute of Technology.
Iran University of Science & Technology.
Properties of a class of block-iterative methods2009In: INVERSE PROBLEMS, ISSN 0266-5611, Vol. 25, no 11Article in journal (Refereed)

We study a class of block-iterative (BI) methods proposed in image reconstruction for solving linear systems. A subclass, symmetric block-iteration (SBI), is derived such that for this subclass both semi-convergence analysis and stopping-rules developed for fully simultaneous iteration apply. Also results on asymptotic convergence are given, e. g., BI exhibit cyclic convergence irrespective of the consistency of the linear system. Further it is shown that the limit points of SBI satisfy a weighted least-squares problem. We also present numerical results obtained using a trained stopping rule on SBI.

• 47.
Linköping University, Department of Mathematics, Scientific Computing. Linköping University, The Institute of Technology.
Linköping University, Department of Mathematics, Scientific Computing. Linköping University, The Institute of Technology.
Some Block-Iterative Methods used in Image Reconstruction2008Article in journal (Refereed)
• 48.
Linköping University, Department of Mathematics, Scientific Computing. Linköping University, The Institute of Technology.
Linköping University, Department of Mathematics, Scientific Computing. Linköping University, The Institute of Technology.
Some Properties of ART-type Reconstruction Algorithms2008In: 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)

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).

• 49.
Linköping University, Department of Mathematics, Scientific Computing. Linköping University, The Institute of Technology.
Linköping University, Department of Mathematics, Scientific Computing. Linköping University, The Institute of Technology.
Stopping Rules for Landweber-type Iteration2007In: Inverse Problems, ISSN 0266-5611, Vol. 23, no 4, p. 1417-1432Article in journal (Refereed)

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.

• 50.
Linköping University, Department of Mathematics, Scientific Computing. Linköping University, The Institute of Technology.
Iran University of Science and Technology. Technical University of Denmark.
Semi-convergence and relaxation parameters for a class of SIRT algorithms2010In: Electronic Transactions on Numerical Analysis, ISSN 1068-9613, E-ISSN 1068-9613, Vol. 37, p. 321-336Article in journal (Refereed)

This paper is concerned with the Simultaneous Iterative Reconstruction Technique (SIRT) class of iterative methods for solving inverse problems. Based on a careful analysis of the semi-convergence behavior of these methods, we propose two new techniques to specify the relaxation parameters adaptively during the iterations, so as to control the propagated noise component of the error. The advantage of using this strategy for the choice of relaxation parameters on noisy and ill-conditioned problems is demonstrated with an example from tomography

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