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  • 1.
    Barlow, Jesse
    et al.
    Department of Computer Science and Engineering, The Pennsylvania State University, USA.
    Elden, Lars
    Linköping University, Department of Mathematics, Scientific Computing. Linköping University, The Institute of Technology.
    Foschi, Paolo
    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)
    Abstract [en]

    n/a

  • 2.
    Berntsson, Fredrik
    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 Solution of Cauchy Problems for Elliptic Equations in "Rectangle-like" Geometries2005In: FEMLAB Conference,2005, Stockholm: Comsol AB , 2005Conference paper (Other academic)
    Abstract [en]

    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.

  • 3.
    Berntsson, Fredrik
    et al.
    Linköping University, Department of Mathematics. Linköping University, The Institute of Technology.
    Eldén, Lars
    Linköping University, Department of Mathematics. Linköping University, The Institute of Technology.
    An inverse heat conduction problem and an application to heat treatment of aluminium2000In: Inverse Problems in Engineering Mechanics II / [ed] Masataka Tanaka, G.S. Dulikravich, 2000, p. 99-106Conference paper (Refereed)
    Abstract [en]

    We consider an inverse heat conduction problem, the sideways heat equation, which is a model of a problem where one wants to determine the temperature on the surface of a body using internal measurements. The problem is ill-posed in the sense that the solution does not depend continuously on the data. We discuss the nature of the ill-posedness as well as methods for restoring stability with respect to measurement errors.

    Successful heat treatment requires good control of the temperature and cooling rates during the process. In an experiment a aluminium block, of the alloy AA7010, was cooled rapidly by spraying water on one surface. Thermocouples inside the block recorded the temperature, and we demonstrate that it is possible to find the temperature distribution in the region between the thermocouple and the surface, by solving numerically the sideways heat equation.

  • 4.
    Berntsson, Fredrik
    et al.
    Linköping University, Department of Mathematics. Linköping University, The Institute of Technology.
    Eldén, Lars
    Linköping University, Department of Mathematics. Linköping University, The Institute of Technology.
    Numerical solution of a Cauchy problem for the Laplace equation2001In: Inverse Problems, ISSN 0266-5611, E-ISSN 1361-6420, Vol. 17, no 4, p. 839-853Article in journal (Refereed)
    Abstract [en]

    We consider a two-dimensional steady state heat conduction problem. The Laplace equation is valid in a domain with a hole. Temperature and heat-flux data are specified on the outer boundary, and we wish to compute the temperature on the inner boundary. This Cauchy problem is ill-posed, i.e. the solution does not depend continuously on the boundary data, and small errors in the data can destroy the numerical solution. We consider two numerical methods for solving this problem. A standard approach is to discretize the differential equation by finite differences, and use Tikhonov regularization on the discrete problem, which leads to a large sparse least squares problem. We propose to use a conformal mapping that maps the region onto an annulus, where the equivalent problem is solved using a technique based on the fast Fourier transform. The ill-posedness is dealt with by filtering away high frequencies in the solution. Numerical results using both methods are given.

  • 5.
    Berntsson, Fredrik
    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.
    Numerical Solution of an Inverse Steady State Heat Conduction Problem2000Conference paper (Other academic)
  • 6.
    Berntsson, Fredrik
    et al.
    Linköping University, Department of Mathematics, Computational Mathematics. Linköping University, Faculty of Science & Engineering.
    Eldén, Lars
    Linköping University, Department of Mathematics, Computational Mathematics. Linköping University, Faculty of Science & Engineering.
    Numerical Solution of Cauchy Problems for Elliptic Equations in ``Rectangle-like'' Geometries2005In: Proceedings for the FEMLAB Conference 2005, 2005Conference paper (Other academic)
    Abstract [en]

    We consider two dimensional inverse steady state heat conductionproblems in complex geometries. The coefficients of the elliptic equation are assumed to be non-constant. Cauchy data are given on onepart of the boundary and we want to find the solution in the wholedomain. The problem is ill--posed in the sense that the solution doesnot depend continuously on the data.

    Using an orthogonal coordinate transformation the domain is mappedonto 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 theleast squares sense. This particular approximation of the derivativeis 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 amelting furnace are subject to physical and chemical wear. Temperature and heat--flux data are collected by several thermocouples locatedinside the walls. The shape of the interface between the molten ironand the walls can then be determined by solving an inverse heatconduction problem.  In our work we make extensive use of Femlab for creating testproblems. 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.

    Download full text (pdf)
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  • 7.
    Berntsson, Fredrik
    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.
    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)
  • 8.
    Berntsson, Fredrik
    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.
    Loyd, Dan
    Linköping University, Department of Mechanical Engineering, Applied Thermodynamics and Fluid Mechanics. Linköping University, The Institute of Technology.
    Garcia-Padrón, Ricardo
    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)
  • 9.
    Björck, Åke
    et al.
    Linköping University, Department of Mathematics, Computational Mathematics. Linköping University, Faculty of Science & Engineering.
    Eldén, Lars
    Linköping University, Department of Mathematics, Computational Mathematics. Linköping University, Faculty of Science & Engineering.
    Axel Ruhe 1942-20152015In: BIT Numerical Mathematics, ISSN 0006-3835, E-ISSN 1572-9125, Vol. 55, no 3, p. 621-623Article in journal (Other academic)
    Abstract [en]

    Axel Ruhe passed away April 4, 2015. He was cross-country-skiing with friends in the Swedish mountains when after 21 km he suddenly died. He is survived by his wife Gunlaug and three children from his first marriage....

  • 10.
    Bro, Rasmus
    et al.
    University of Copenhagen.
    Elden , Lars
    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)
    Abstract [en]

    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.

  • 11. Domeij Bäckryd, Rebecka
    et al.
    Elden, Lars
    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)
    Abstract [en]

    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.

  • 12.
    Elden, Lars
    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.))
  • 13.
    Elden, Lars
    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)
  • 14.
    Elden, Lars
    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.))
  • 15.
    Elden, Lars
    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)
    Abstract [en]

    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.

  • 16.
    Elden, Lars
    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)
  • 17.
    Elden, Lars
    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)
  • 18.
    Elden, Lars
    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)
    Abstract [en]

    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.

  • 19.
    Elden, Lars
    Linköping University, The Institute of Technology. Linköping University, Department of Mathematics, Scientific Computing.
    The Eigenvalues of the Google Matrix2004Report (Other academic)
  • 20.
    Elden, Lars
    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)
  • 21.
    Elden, Lars
    et al.
    Linköping University, The Institute of Technology. Linköping University, Department of Mathematics, Scientific Computing.
    Berntsson, Fredrik
    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)
    Abstract [en]

    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.

  • 22.
    Elden, Lars
    et al.
    Linköping University, Department of Mathematics. Linköping University, The Institute of Technology.
    Berntsson, Fredrik
    Linköping University, Department of Mathematics. Linköping University, The Institute of Technology.
    Reginska, Teresa
    Institute of Mathematics, Polish Academy of Sciences, Warsaw, Poland.
    Wavelet and Fourier methods for solving the sideways heat equation2000In: SIAM Journal on Scientific Computing, ISSN 1064-8275, E-ISSN 1095-7197, Vol. 21, no 6, p. 2187-2205Article in journal (Refereed)
    Abstract [en]

    We consider an inverse heat conduction problem, the sideways heat equation, which is a model of a problem, where one wants to determine the temperature on both sides of a thick wall, but where one side is inaccessible to measurements. Mathematically it is formulated as a Cauchy problem for the heat equation in a quarter plane, with data given along the line x = 1, where the solution is wanted for 0 ≤ x < 1.

    The problem is ill-posed, in the sense that the solution (if it exists) does not depend continuously on the data. We consider stabilizations based on replacing the time derivative in the heat equation by wavelet-based approximations or a Fourier-based approximation. The resulting problem is an initial value problem for an ordinary differential equation, which can be solved by standard numerical methods, e.g., a Runge–Kutta method.

    We discuss the numerical implementation of Fourier and wavelet methods for solving the sideways heat equation. Theory predicts that the Fourier method and a method based on Meyer wavelets will give equally good results. Our numerical experiments indicate that also a method based on Daubechies wavelets gives comparable accuracy. As test problems we take model equations with constant and variable coefficients. We also solve a problem from an industrial application with actual measured data.

  • 23.
    Elden, Lars
    et al.
    Linköping University, The Institute of Technology. Linköping University, Department of Mathematics, Scientific Computing.
    Hansen, Per Christian
    Rojas, Marielba
    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)
    Abstract [en]

    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.

  • 24.
    Elden, Lars
    et al.
    Linköping University, The Institute of Technology. Linköping University, Department of Mathematics, Scientific Computing.
    Koch, Linde
    Linköping University, The Institute of Technology. Linköping University, Department of Mathematics, Scientific Computing.
    Bruun Nielsen, Hans
    Introduction to Numerical Computation2004Book (Other (popular science, discussion, etc.))
  • 25.
    Elden, Lars
    et al.
    Linköping University, The Institute of Technology. Linköping University, Department of Mathematics, Scientific Computing.
    Park, Haesun
    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)
    Abstract [en]

    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.

  • 26.
    Elden, Lars
    et al.
    Linköping University, The Institute of Technology. Linköping University, Department of Mathematics, Scientific Computing.
    Savas, Berkant
    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)
  • 27.
    Elden, Lars
    et al.
    Linköping University, The Institute of Technology. Linköping University, Department of Mathematics, Scientific Computing.
    Savas, Berkant
    Linköping University, The Institute of Technology. Linköping University, Department of Mathematics, Scientific Computing.
    The Maximum likelihood estimate in reduced-rank regression2003Report (Other academic)
  • 28.
    Elden, Lars
    et al.
    Linköping University, The Institute of Technology. Linköping University, Department of Mathematics, Scientific Computing.
    Savas, Berkant
    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)
    Abstract [en]

    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.

  • 29.
    Eldén, Lars
    Linköping University, Department of Mathematics, Computational Mathematics. Linköping University, The Institute of Technology.
    Computing Frechet derivatives in partial least squares regression2015In: Linear Algebra and its Applications, ISSN 0024-3795, E-ISSN 1873-1856, Vol. 473, p. 316-338Article in journal (Refereed)
    Abstract [en]

    Partial least squares is a common technique for multivariate regression. The pro- cedure is recursive and in each step basis vectors are computed for the explaining variables and the solution vectors. A linear model is fitted by projection onto the span of the basis vectors. The procedure is mathematically equivalent to Golub-Kahan bidiagonalization, which is a Krylov method, and which is equiv- alent to a pair of matrix factorizations. The vectors of regression coefficients and prediction are non-linear functions of the right hand side. An algorithm for computing the Frechet derivatives of these functions is derived, based on perturbation theory for the matrix factorizations. From the Frechet derivative of the prediction vector one can compute the number of degrees of freedom, which can be used as a stopping criterion for the recursion. A few numerical examples are given.

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  • 30.
    Eldén, Lars
    Linköping University, Department of Mathematics, Computational Mathematics. Linköping University, Faculty of Science & Engineering.
    Editorial Material: Preface in BIT NUMERICAL MATHEMATICS, vol 57, issue 1, pp2017In: BIT Numerical Mathematics, ISSN 0006-3835, E-ISSN 1572-9125, Vol. 57, no 1Article in journal (Other academic)
    Abstract [en]

    n/a

  • 31.
    Eldén, Lars
    Linköping University, Department of Mathematics, Computational Mathematics. Linköping University, Faculty of Science & Engineering.
    Editorial Material: Preface to BIT 55:4 in BIT NUMERICAL MATHEMATICS, vol 55, issue 4, pp 897-8992015In: BIT Numerical Mathematics, ISSN 0006-3835, E-ISSN 1572-9125, Vol. 55, no 4, p. 897-899Article in journal (Other academic)
    Abstract [en]

    n/a

  • 32.
    Eldén, Lars
    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)
    Abstract [en]

    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.

  • 33.
    Eldén, Lars
    Linköping University, Department of Mathematics, Computational Mathematics. Linköping University, Faculty of Science & Engineering.
    Preface to BIT 56:4 in BIT NUMERICAL MATHEMATICS2016In: BIT Numerical Mathematics, ISSN 0006-3835, E-ISSN 1572-9125, Vol. 56, no 4, p. 1163-1164Article in journal (Other academic)
    Abstract [en]

    n/a

  • 34.
    Eldén, Lars
    et al.
    Linköping University, Department of Mathematics, Computational Mathematics. Linköping University, Faculty of Science & Engineering.
    Ahmadi-Asl, Salman
    Skolkovo Inst Sci and Technol Skoltech, Russia.
    Solving bilinear tensor least squares problems and application to Hammerstein identification2019In: Numerical Linear Algebra with Applications, ISSN 1070-5325, E-ISSN 1099-1506, Vol. 26, no 2, article id e2226Article in journal (Refereed)
    Abstract [en]

    Bilinear tensor least squares problems occur in applications such as Hammerstein system identification and social network analysis. A linearly constrained problem of medium size is considered, and nonlinear least squares solvers of Gauss-Newton-type are applied to numerically solve it. The problem is separable, and the variable projection method can be used. Perturbation theory is presented and used to motivate the choice of constraint. Numerical experiments with Hammerstein models and random tensors are performed, comparing the different methods and showing that a variable projection method performs best.

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  • 35.
    Eldén, Lars
    et al.
    Linköping University, Department of Mathematics, Applied Mathematics. Linköping University, Faculty of Science & Engineering.
    Dehghan, Maryam
    Persian Gulf Univ, Iran.
    A Krylov-Schur-like method for computing the best rank-(r1, r2, r3) approximation of large and sparse tensors2022In: Numerical Algorithms, ISSN 1017-1398, E-ISSN 1572-9265, Vol. 91, p. 1315-1347Article in journal (Refereed)
    Abstract [en]

    The paper is concerned with methods for computing the best low multilinear rank approximation of large and sparse tensors. Krylov-type methods have been used for this problem; here block versions are introduced. For the computation of partial eigenvalue and singular value decompositions of matrices the Krylov-Schur (restarted Arnoldi) method is used. A generalization of this method to tensors is described, for computing the best low multilinear rank approximation of large and sparse tensors. In analogy to the matrix case, the large tensor is only accessed in multiplications between the tensor and blocks of vectors, thus avoiding excessive memory usage. It is proved that if the starting approximation is good enough, then the tensor Krylov-Schur method is convergent. Numerical examples are given for synthetic tensors and sparse tensors from applications, which demonstrate that for most large problems the Krylov-Schur method converges faster and more robustly than higher order orthogonal iteration.

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  • 36.
    Eldén, Lars
    et al.
    Linköping University, Department of Mathematics, Applied Mathematics. Linköping University, Faculty of Science & Engineering.
    Dehghan, Maryam
    Persian Gulf Univ, Iran.
    Spectral partitioning of large and sparse 3-tensors using low-rank tensor approximation2022In: Numerical Linear Algebra with Applications, ISSN 1070-5325, E-ISSN 1099-1506, Vol. 29, no 5, article id e2435Article in journal (Refereed)
    Abstract [en]

    The problem of partitioning a large and sparse tensor is considered, where the tensor consists of a sequence of adjacency matrices. Theory is developed that is a generalization of spectral graph partitioning. A best rank-(2,2,lambda) approximation is computed for lambda=1,2,3, and the partitioning is computed from the orthogonal matrices and the core tensor of the approximation. It is shown that if the tensor has a certain reducibility structure, then the solution of the best approximation problem exhibits the reducibility structure of the tensor. Further, if the tensor is close to being reducible, then still the solution of the exhibits the structure of the tensor. Numerical examples with synthetic data corroborate the theoretical results. Experiments with tensors from applications show that the method can be used to extract relevant information from large, sparse, and noisy data.

  • 37.
    Eldén, Lars
    et al.
    Linköping University, Department of Mathematics. Linköping University, The Institute of Technology.
    Kilmer, Misha
    Tufts University.
    O'Leary, Dianne
    University of Maryland.
    Updating and Downdating Matrix Decompositions2010In: Selected Works of G. W. Stewart with Commentaries / [ed] M. Kilmer and D. O'Öeary, Boston: Birkhäuser , 2010, p. 45-58Chapter in book (Other academic)
    Abstract [en]

    Published in honor of his 70th birthday, this volume explores and celebrates the work of G.W. (Pete) Stewart, a world-renowned expert in computational linear algebra. This volume includes: forty-four of Stewart's most influential research papers in two subject areas: matrix algorithms, and rounding and perturbation theory; a biography of Stewart; a complete list of his publications, students, and honors; selected photographs; and commentaries on his works in collaboration with leading experts in the field. G.W. Stewart: Selected Works with Commentaries will appeal to graduate students, practitioners, and researchers in computational linear algebra and the history of mathematics.

  • 38.
    Eldén, Lars
    et al.
    Linköping University, Department of Mathematics, Computational Mathematics. Linköping University, The Institute of Technology.
    Merkel, Magnus
    Linköping University, Department of Computer and Information Science, Human-Centered systems. Linköping University, The Institute of Technology.
    Ahrenberg, Lars
    Linköping University, Department of Computer and Information Science, Human-Centered systems. Linköping University, The Institute of Technology.
    Fagerlund, Martin
    Linköping University, Department of Mathematics, Computational Mathematics. Linköping University, The Institute of Technology.
    Computing Semantic Clusters by Semantic Mirroring and Spectral Graph Partitioning2013In: Mathematics in Computer Science, ISSN 1661-8270, Vol. 7, p. 293-313Article in journal (Refereed)
    Abstract [en]

    Using the technique of semantic mirroring a graph is obtained that represents words and their translationsfrom a parallel corpus or a bilingual lexicon. The connectedness of the graph holds information about the semanticrelations of words that occur in the translations. Spectral graph theory is used to partition the graph, which leadsto a grouping of the words in different clusters. We illustrate the method using a small sample of seed words froma lexicon of Swedish and English adjectives and discuss its application to computational lexical semantics andlexicography.

  • 39.
    Eldén, Lars
    et al.
    Linköping University, Department of Mathematics, Scientific Computing. Linköping University, The Institute of Technology.
    Savas, Berkant
    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)
    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.

  • 40.
    Eldén, Lars
    et al.
    Linköping University, Department of Mathematics, Scientific Computing. Linköping University, The Institute of Technology.
    Savas, Berkant
    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)
    Abstract [en]

    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.

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  • 41.
    Eldén, Lars
    et al.
    Linköping University, Department of Mathematics, Scientific Computing. Linköping University, The Institute of Technology.
    Simoncini , Valeria
    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)
    Abstract [en]

    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.

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  • 42.
    Eldén, Lars
    et al.
    Linköping University, Department of Mathematics. Linköping University, The Institute of Technology.
    Simoncini, Valeria
    Department of Mathematics, University of Bologna.
    SOLVING ILL-POSED LINEAR SYSTEMS WITH GMRES AND A SINGULAR PRECONDITIONER2012In: SIAM Journal on Matrix Analysis and Applications, ISSN 0895-4798, E-ISSN 1095-7162, Vol. 33, p. 1369-1394Article in journal (Refereed)
    Abstract [en]

    Almost singular linear systems arise in discrete ill-posed problems. Either because ofthe intrinsic structure of the problem or because of preconditioning, the spectrum of the coefficientmatrix is often characterized by a sizable gap between a large group of numerically zero eigenvaluesand the rest of the spectrum. Correspondingly, the right-hand side has leading eigencomponentsassociated with the eigenvalues away from zero. In this paper the effect of this setting in theconvergence of the generalized minimal residual (GMRES) method is considered. It is shown thatin the initial phase of the iterative algorithm, the residual components corresponding to the largeeigenvalues are reduced in norm, and these can be monitored without extra computation. Theanalysis is supported by numerical experiments. In particular, ill-posed Cauchy problems for partialdifferential equations with variable coefficients are considered, where the preconditioner is a fast,low-rank solver for the corresponding problem with constant coefficients.

  • 43.
    Eldén, Lars
    et al.
    Linköping University, Department of Mathematics, Computational Mathematics. Linköping University, Faculty of Science & Engineering.
    Trendafilov, Nickolay
    Open Univ, England.
    Semi-sparse PCA2019In: Psychometrika, ISSN 0033-3123, E-ISSN 1860-0980, Vol. 84, no 1, p. 164-185Article in journal (Refereed)
    Abstract [en]

    It is well known that the classical exploratory factor analysis (EFA) of data with more observations than variables has several types of indeterminacy. We study the factor indeterminacy and show some new aspects of this problem by considering EFA as a specific data matrix decomposition. We adopt a new approach to the EFA estimation and achieve a new characterization of the factor indeterminacy problem. A new alternative model is proposed, which gives determinate factors and can be seen as a semi-sparse principal component analysis (PCA). An alternating algorithm is developed, where in each step a Procrustes problem is solved. It is demonstrated that the new model/algorithm can act as a specific sparse PCA and as a low-rank-plus-sparse matrix decomposition. Numerical examples with several large data sets illustrate the versatility of the new model, and the performance and behaviour of its algorithmic implementation.

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  • 44. 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.

  • 45.
    Feng, Xiaoli
    et al.
    Xidian University, Xi'an, China .
    Eldén, Lars
    Linköping University, Department of Mathematics, Computational Mathematics. Linköping University, The Institute of Technology.
    Solving a Cauchy problem for a 3D elliptic PDE with variable coefficients by a quasi-boundary-value method2014In: Inverse Problems, ISSN 0266-5611, E-ISSN 1361-6420, Vol. 30, no 1, p. 015005-Article in journal (Refereed)
    Abstract [en]

    An ill-posed Cauchy problem for a 3D elliptic partial differential equation with variable coefficients is considered. A well-posed quasi-boundary-value (QBV) problem is given to approximate it. Some stability estimates are given. For the numerical implementation, a large sparse system is obtained from discretizing the QBV problem using the finite difference method. A left-preconditioned generalized minimum residual method is used to solve the large system effectively. For the preconditioned system, a fast solver using the fast Fourier transform is given. Numerical results show that the method works well.

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  • 46.
    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.

  • 47.
    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.

  • 48.
    Kreiss, Gunilla
    et al.
    Uppsala Univ, Sweden.
    Eldén, Lars
    Linköping University, Department of Mathematics, Computational Mathematics. Linköping University, Faculty of Science & Engineering.
    Preface2018In: BIT Numerical Mathematics, ISSN 0006-3835, E-ISSN 1572-9125, Vol. 58, no 1Article in journal (Other academic)
    Abstract [en]

    n/a

  • 49. 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.

  • 50.
    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.

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