2023 | Tikhonov regularization in general form with Chebfun | A. Alqahtani, T. Mach, L. ReichelZeitschrift: PAMMSeiten: e202200041, 6 pagesBand: 22(1)Link zur Publikation
Tikhonov regularization in general form with Chebfun
Autoren: A. Alqahtani, T. Mach, L. Reichel
Linear ill-posed problems often are analyzed in function spaces using tools from functional analysis, while their numerical solution typically is computed by first discretizing the problem and then applying tools from finite-dimensional linear algebra. The Chebfun package makes it possible to solve linear ill-posed problems without explicit discretization. This work discusses the use of Tikhonov regularization with a fairly general linear regularization operator within the Chebfun framework.
e202200041, 6 pages
2023 | Solution of Ill-posed Problems with Chebfun | A. Alqahtani, T. Mach, L. ReichelZeitschrift: Numerical AlgorithmsVerlag: SpringerSeiten: 2341‒2364Band: 92Link zur Publikation
Link zum Preprint
Solution of Ill-posed Problems with Chebfun
Autoren: A. Alqahtani, T. Mach, L. Reichel
The analysis of linear ill-posed problems often is carried out in function spaces using tools from functional analysis. However, the numerical solution of these problems typically is computed by first discretizing the problem and then applying tools from finite-dimensional linear algebra. The present paper explores the feasibility of applying the Chebfun package to solve ill-posed problems with a regularize-first approach numerically. This allows a user to work with functions instead of vectors and with integral operators instead of matrices. The solution process therefore is much closer to the analysis of ill-posed problems than standard linear algebra-based solution methods. Furthermore, the difficult process of explicitly choosing a suitable discretization is not required.
2023 | Solving the Parametric Eigenvalue Problem by Taylor Series and Chebyshev Expansion | T. Mach, M.A. FreitagZeitschrift: arXivBand: 2302.03661Link zum Preprint
Solving the Parametric Eigenvalue Problem by Taylor Series and Chebyshev Expansion
Autoren: T. Mach, M.A. Freitag
We discuss two approaches to solving the parametric (or stochastic) eigenvalue problem A(μ)λ(μ)=A(μ)v(μ). One of them uses a Taylor expansion and the other a Chebyshev expansion. The parametric eigenvalue problem assumes that the matrix A depends on a parameter μ, where μ might be a random variable. Consequently, the eigenvalues and eigenvectors are also functions of μ. We compute a Taylor approximation of these functions about μ₀ by iteratively computing the Taylor coefficients. The complexity of this approach is O(n³) for all eigenpairs, if the derivatives of A(μ) at μ₀ are given. The Chebyshev expansion works similarly. We first find an initial approximation iteratively which we then refine with Newton's method. This second method is more expensive but provides a good approximation over the whole interval of the expansion instead around a single point.
We present numerical experiments confirming the complexity and demonstrating that the approaches are capable of tracking eigenvalues at intersection points. Further experiments shed light on the limitations of the Taylor expansion approach with respect to the distance from the expansion point μ₀.
2023 | Adaptive Cross Approximation for Tikhonov Regularization in General Form | T. Mach, L. Reichel, M. Van BarelZeitschrift: Numerical AlgorithmsVerlag: SpringerSeiten: 815‒830Band: 92Link zur Publikation
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Adaptive Cross Approximation for Tikhonov Regularization in General Form
Autoren: T. Mach, L. Reichel, M. Van Barel
Many problems in Science and Engineering give rise to linear integral equations of the first kind with a smooth kernel. Discretization of the integral operator yields a matrix, whose singular values cluster at the origin. We describe the approximation of such matrices by adaptive cross approximation, which avoids forming the entire matrix. The choice of the number of steps of adaptive cross approximation is discussed. The discretized right-hand side represents data that commonly are contaminated by measurement error. Solution of the linear system of equations so obtained is not meaningful because the matrix determined by adaptive cross approximation is rank-deficient. We remedy this difficulty by using Tikhonov regularization and discuss how a fairly general regularization matrix can be used. Computed examples illustrate that the use of a regularization matrix different from the identity can improve the quality of the computed approximate solutions significantly.
2022 | Hyperbolic Embedding for Finding Syntax in BERT (short paper) | T. Auyespek, T. Mach, Zh. AssylbekovVerlag: CEUR Workshop ProceedingsBuchtitel: AIxIA 2021 Discussion PapersSeiten: 58‒64Link zur Publikation
Hyperbolic Embedding for Finding Syntax in BERT (short paper)
Autoren: T. Auyespek, T. Mach, Zh. Assylbekov
Recent advances in natural language processing have improved our understanding of what kind of linguistic knowledge is encoded in modern word representations. For example, methods for testing the ability to extract syntax trees from a language model architecture were developed by Hewitt and Manning (2019)—they project word vectors into Euclidean subspace in such a way that the corresponding squared Euclidean distance approximates the tree distance between words in the syntax tree. This work proposes a method for assessing whether embedding word representations in hyperbolic space can better reflect the graph structure of syntax trees. We show that the tree distance between words in a syntax tree can be approximated well by the hyperbolic distance between corresponding word vectors.
CEUR Workshop Proceedings
AIxIA 2021 Discussion Papers
2021 | New matrix function approximations and quadratue rules based on the Arnoldi process | N. Eshghi, T. Mach, L. ReichelZeitschrift: Journal of Computational and Applied MathematicsVerlag: ElsevierSeiten: 113442Band: 391Link zur Publikation
New matrix function approximations and quadratue rules based on the Arnoldi process
Autoren: N. Eshghi, T. Mach, L. Reichel
The Arnoldi process can be applied to inexpensively approximate matrix functions of the form f(A)v and matrix functionals of the form v∗(f(A))∗g(A)v, where A is a large square non-Hermitian matrix, v is a vector, and the superscript ∗ denotes transposition and complex conjugation. Here f and g are analytic functions that are defined in suitable regions in the complex plane. This paper reviews available approximation methods and describes new ones that provide higher accuracy for essentially the same computational effort by exploiting available, but generally not used, moment information.
Numerical experiments show that in some cases the modifications of the Arnoldi decompositions proposed can improve the accuracy of v∗(f(A))∗g(A)v about as much as performing an additional step of the Arnoldi process.
Journal of Computational and Applied Mathematics