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Publications (10 of 55) Show all publications
Björck, Å. & Eldén, L. (2015). Biographical-Item: Axel Ruhe 1942-2015 in BIT NUMERICAL MATHEMATICS, vol 55, issue 3, pp 621-623. SPRINGER, 55(3)
Open this publication in new window or tab >>Biographical-Item: Axel Ruhe 1942-2015 in BIT NUMERICAL MATHEMATICS, vol 55, issue 3, pp 621-623
2015 (English)Other (Other academic)
Abstract [en]

n/a

Place, publisher, year, pages
SPRINGER, 2015
National Category
Mathematics
Identifiers
urn:nbn:se:liu:diva-122115 (URN)10.1007/s10543-015-0579-4 (DOI)000361818100001 ()
Available from: 2015-10-19 Created: 2015-10-19 Last updated: 2015-10-19
Eldén, L. (2015). Computing Frechet derivatives in partial least squares regression. Linear Algebra and its Applications, 473, 316-338
Open this publication in new window or tab >>Computing Frechet derivatives in partial least squares regression
2015 (English)In: Linear Algebra and its Applications, ISSN 0024-3795, E-ISSN 1873-1856, Vol. 473, p. 316-338Article in journal (Refereed) Published
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.

Keyword
Partial Least Squares, PLS, regression, least squares, prediction, Golub-Kahan bidiagonalization, Krylov method, Frechet derivative, recursion, perturbation theory, degrees of freedom
National Category
Computational Mathematics
Identifiers
urn:nbn:se:liu:diva-117295 (URN)10.1016/j.laa.2014.09.017 (DOI)000355040400018 ()
Available from: 2015-04-23 Created: 2015-04-23 Last updated: 2017-12-04
Eldén, L. (2015). Editorial Material: Preface to BIT 55:4 in BIT NUMERICAL MATHEMATICS, vol 55, issue 4, pp 897-899. BIT Numerical Mathematics, 55(4), 897-899
Open this publication in new window or tab >>Editorial Material: Preface to BIT 55:4 in BIT NUMERICAL MATHEMATICS, vol 55, issue 4, pp 897-899
2015 (English)In: BIT Numerical Mathematics, ISSN 0006-3835, E-ISSN 1572-9125, Vol. 55, no 4, p. 897-899Article in journal, Editorial material (Other academic) Published
Abstract [en]

n/a

Place, publisher, year, edition, pages
SPRINGER, 2015
National Category
Mathematics
Identifiers
urn:nbn:se:liu:diva-124136 (URN)10.1007/s10543-015-0593-6 (DOI)000366634300001 ()
Available from: 2016-01-22 Created: 2016-01-19 Last updated: 2017-11-30
Rezghi, M., Hosseini, S. M. & Eldén, L. (2014). Best Kronecker Product Approximation of The Blurring Operator in Three Dimensional Image Restoration Problems. SIAM Journal on Matrix Analysis and Applications, 35(3), 1086-1104
Open this publication in new window or tab >>Best Kronecker Product Approximation of The Blurring Operator in Three Dimensional Image Restoration Problems
2014 (English)In: SIAM Journal on Matrix Analysis and Applications, ISSN 0895-4798, E-ISSN 1095-7162, Vol. 35, no 3, p. 1086-1104Article in journal (Refereed) Published
Abstract [en]

In this paper, we propose a method to find the best Kronecker product approximationof the blurring operator which arises in three dimensional image restoration problems. We show thatthis problem can be reduced to a well known rank-1 approximation of the scaled three dimensionalpoint spread function (PSF) array, which is much smaller. This approximation can be used as apreconditioner in solving image restoration problems with iterative methods. The comparison ofthe approximation by the new scaled PSF array and approximation by the original PSF array that is used in [J. G. Nagy and M. E. Kilmer, IEEE Trans. Image Process., 15 (2006), pp. 604–613],confirms the performance of the new proposed approximation.

Place, publisher, year, edition, pages
Philadelphia, PA: Society for Industrial and Applied Mathematics, 2014
Keyword
image restoration, Kronecker product approximation, tensor decomposition, tensor best rank-1 approximation, preconditioner
National Category
Computational Mathematics
Identifiers
urn:nbn:se:liu:diva-109945 (URN)10.1137/130917260 (DOI)000343229800013 ()
Available from: 2014-08-28 Created: 2014-08-28 Last updated: 2018-01-26Bibliographically approved
Feng, X. & Eldén, L. (2014). Solving a Cauchy problem for a 3D elliptic PDE with variable coefficients by a quasi-boundary-value method. Inverse Problems, 30(1), 015005
Open this publication in new window or tab >>Solving a Cauchy problem for a 3D elliptic PDE with variable coefficients by a quasi-boundary-value method
2014 (English)In: Inverse Problems, ISSN 0266-5611, E-ISSN 1361-6420, Vol. 30, no 1, p. 015005-Article in journal (Refereed) Published
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.

Place, publisher, year, edition, pages
Institute of Physics (IOP), 2014
Keyword
elliptic equation; ill-posed; Cauchy problem; finite difference method; quasi-boundary-value method; left-preconditioned GMRES; fast solver; variable coefficients; three dimensions; fast Fourier transform
National Category
Computational Mathematics
Identifiers
urn:nbn:se:liu:diva-103280 (URN)10.1088/0266-5611/30/1/015005 (DOI)000328861800006 ()
Available from: 2014-01-17 Created: 2014-01-16 Last updated: 2017-12-06Bibliographically approved
Ranjbar, Z. & Eldén, L. (2014). Solving an Ill-Posed Cauchy Problem for a Two-Dimensional Parabolic PDE with Variable Coefficients Using a Preconditioned GMRES Method. SIAM Journal on Scientific Computing, 36(5), B868-B886
Open this publication in new window or tab >>Solving an Ill-Posed Cauchy Problem for a Two-Dimensional Parabolic PDE with Variable Coefficients Using a Preconditioned GMRES Method
2014 (English)In: SIAM Journal on Scientific Computing, ISSN 1064-8275, E-ISSN 1095-7197, Vol. 36, no 5, p. B868-B886Article in journal (Refereed) Published
Abstract [en]

The sideways parabolic equation (SPE) is a model of the problem of determiningthe temperature on the surface of a body from the interior measurements. Mathematically it can beformulated as a noncharacteristic Cauchy problem for a parabolic partial differential equation. Thisproblem is severely ill-posed in an L2 setting. We use a preconditioned generalized minimum residualmethod (GMRES) to solve a two-dimensional SPE with variable coefficients. The preconditioner issingular and chosen in a way that allows efficient implementation using the FFT. The preconditioneris a stabilized solver for a nearby problem with constant coefficients, and it reduces the numberof iterations in the GMRES algorithm significantly. Numerical experiments are performed thatdemonstrate the performance of the proposed method.

Place, publisher, year, edition, pages
Philadelphia: Society for Industrial and Applied Mathematics, 2014
Keyword
Cauchy problem, inverse problem, ill-posed, iterative methods, GMRES precondi- tioning, singular preconditioner, parabolic PDE, FFT
National Category
Computational Mathematics
Identifiers
urn:nbn:se:liu:diva-111837 (URN)10.1137/130951166 (DOI)000346123200021 ()
Available from: 2014-11-05 Created: 2014-11-05 Last updated: 2017-12-05Bibliographically approved
Eldén, L., Merkel, M., Ahrenberg, L. & Fagerlund, M. (2013). Computing Semantic Clusters by Semantic Mirroring and Spectral Graph Partitioning. Mathematics in Computer Science, 7, 293-313
Open this publication in new window or tab >>Computing Semantic Clusters by Semantic Mirroring and Spectral Graph Partitioning
2013 (English)In: Mathematics in Computer Science, ISSN 1661-8270, Vol. 7, p. 293-313Article in journal (Refereed) Published
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.

Place, publisher, year, edition, pages
Basel: Springer, 2013
Keyword
Semantics · Cluster · Bilingual · Dictionary · Mirroring · WordNet · Graph · Spectral · Partitioning · Adjacency · Graph Laplacian · Fiedler vector · Matrix · Eigenvector · Eigenvalue
National Category
Computational Mathematics
Identifiers
urn:nbn:se:liu:diva-98175 (URN)10.1007/s11786-013-0159-4 (DOI)
Funder
Swedish Research Council
Available from: 2013-09-30 Created: 2013-09-30 Last updated: 2014-09-19
Savas, B. & Eldén, L. (2013). Krylov-type methods for tensor computations I. Linear Algebra and its Applications, 438(2), 891-918
Open this publication in new window or tab >>Krylov-type methods for tensor computations I
2013 (English)In: Linear Algebra and its Applications, ISSN 0024-3795, E-ISSN 1873-1856, Vol. 438, no 2, p. 891-918Article in journal (Refereed) Published
Abstract [en]

Several Krylov-type procedures are introduced that generalize matrix Krylov methods for tensor computations. They are denoted minimal Krylov recursion, maximal Krylov recursion, and contracted tensor product Krylov recursion. It is proved that, for a given tensor A with multilinear rank-(p; q; r), the minimal Krylov recursion extracts the correct subspaces associated to the tensor in p+q+r number of tensor-vector-vector multiplications. An optimized minimal Krylov procedure is described that, for a given multilinear rank of an approximation, produces a better approximation than the standard minimal recursion. We further generalize the matrix Krylov decomposition to a tensor Krylov decomposition. The tensor Krylov methods are intended for the computation of low multilinear rank approximations of large and sparse tensors, but they are also useful for certain dense and structured tensors for computing their higher order singular value decompositions or obtaining starting points for the best low-rank computations of tensors. A set of numerical experiments, using real-world and synthetic data sets, illustrate some of the properties of the tensor Krylov methods.

Place, publisher, year, edition, pages
Elsevier, 2013
Keyword
Tensor, Krylov-type method, tensor approximation, Tucker model, multilinear algebra, multilinear rank, sparse tensor, information science
National Category
Computational Mathematics
Identifiers
urn:nbn:se:liu:diva-73727 (URN)10.1016/j.laa.2011.12.007 (DOI)000313226900017 ()
Note

Special Issue on Tensors and Multilinear Algebra

Available from: 2012-01-12 Created: 2012-01-12 Last updated: 2017-12-08Bibliographically approved
Eldén, L. & Simoncini, V. (2012). SOLVING ILL-POSED LINEAR SYSTEMS WITH GMRES AND A SINGULAR PRECONDITIONER. SIAM Journal on Matrix Analysis and Applications, 33, 1369-1394
Open this publication in new window or tab >>SOLVING ILL-POSED LINEAR SYSTEMS WITH GMRES AND A SINGULAR PRECONDITIONER
2012 (English)In: SIAM Journal on Matrix Analysis and Applications, ISSN 0895-4798, E-ISSN 1095-7162, Vol. 33, p. 1369-1394Article in journal (Refereed) Published
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.

Keyword
ill-posed, linear system, GMRES, singular preconditioner, nearly singular
National Category
Computational Mathematics
Identifiers
urn:nbn:se:liu:diva-86985 (URN)10.1137/110832793 (DOI)
Available from: 2013-01-08 Created: 2013-01-08 Last updated: 2017-12-06
Rezgi, M. & Eldén, L. (2011). Diagonalization of Tensors with Circulant Structure. Linear Algebra and its Applications, 435(3), 422-447
Open this publication in new window or tab >>Diagonalization of Tensors with Circulant Structure
2011 (English)In: Linear Algebra and its Applications, ISSN 0024-3795, E-ISSN 1873-1856, Vol. 435, no 3, p. 422-447Article in journal (Refereed) Published
Abstract [en]

The concepts of tensors with diagonal and circulant structure are defined and aframework is developed for the analysis of such tensors. It is shown a tensor of arbitraryorder, which is circulant with respect to two particular modes, can be diagonalized inthose modes by discrete Fourier transforms. This property can be used in the efficientsolution of linear systems involving contractive products of tensors with circulantstructure. Tensors with circulant structure occur in models for image blurring withperiodic boundary conditions. It is shown that the new framework can be applied tosuch problems.

Keyword
Tensor, Diagonal tensor, Circulant structure, Discrete Fourier transform, Image blurring model, Periodic boundary conditions, Preconditioner
National Category
Computational Mathematics
Identifiers
urn:nbn:se:liu:diva-63420 (URN)10.1016/j.laa.2010.03.032 (DOI)000291142900002 ()
Note
Original Publication: Mansoor Rezgi and Lars Eldén, Diagonalization of Tensors with Circulant Structure, 2011, Linear Algebra and its Applications, (435), 3, 422-447. http://dx.doi.org/10.1016/j.laa.2010.03.032 Copyright: Elsevier Science B.V., Amsterdam. http://www.elsevier.com/ Available from: 2010-12-17 Created: 2010-12-17 Last updated: 2017-12-11
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Identifiers
ORCID iD: ORCID iD iconorcid.org/0000-0003-2281-856X

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