Solvability of the Core Problem with Multiple Right-Hand Sides in the TLS Sense

Hnětýnková, Iveta; Plešinger, Martin; Sima, Diana M.

doi:10.1137/15m1028339

Cited by 15 publications

(18 citation statements)

References 8 publications

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“…The core problem theory has been introduced in [7] and then extended in [2,4,5] for the vector and matrix right-hand sides problems, respectively. It states that there exist orthogonal matrices P ∈ R m×m , P −1 = P T , Q ∈ R n×n , Q −1 = Q T , and in the second case also R ∈ R d×d , R −1 = R T , such that…”

Section: Classical Tls Formulationsmentioning

confidence: 99%

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Towards tensor generalizations of TLS & core problem theory

Hnětýnková

Plešinger

Žáková

2018

Proc Appl Math and Mech

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Despite the wide attention, there are still several not well understood parts in the theory of total least squares (TLS) formulation of linear algebraic approximation problems. In particular, in the problems with multiple (matrix) right-hand sides one can ask about the meaning of the nongeneric solution in the context of the original data, the nonexistence of the TLS solution for the so-called irreducible core problems, etc. In the single (vector) right-hand side TLS, these problems can be easily explained through the core problem theory or simply do not appear. Here we summarize, how the existing TLS and core problem theory can be generalized to problems with tensor right-hand sides. Such generalization also gives a natural and wider context for further analysis of the above mentioned questions.

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Section: Classical Tls Formulationsmentioning

confidence: 99%

“…The core problem theory through this mechanism explains the concept of the nongeneric solution in terms of original data from a novel point of view. However, this mechanism cannot be directly generalized to the matrix right-hand side case; see [2].…”

Section: Classical Tls Formulationsmentioning

confidence: 99%

Towards tensor generalizations of TLS & core problem theory

Hnětýnková

Plešinger

Žáková

2018

Proc Appl Math and Mech

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“…The TLS problem is significantly more complicated than the ordinary LS problem. It has been studied for a long time, see in particular [6], [18], [21], [20], [14], and recently also [13]. The analysis is based on the SVD of the system matrix A (2.2) and of the extended matrix [B, A].…”

Section: Total Least Squares Based Regularization By Filteringmentioning

confidence: 99%

Filter factors of truncated TLS regularization with multiple observations

2017

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“…The core concept to the case with multiple right-hand sides AX ≈ B is considered in [17,19,20] and realized by the Golub-Kahan bidiagonalization [21]. Recently Hnětynková et al extend the core reduction to tensor for problems with structured right-hand sides [22].…”

Section: Introductionmentioning

confidence: 99%

Randomized core reduction for discrete ill-posed problem

Zhang

Wei

2020

Journal of Computational and Applied Mathematics

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In this paper, we apply randomized algorithms to approximate the total least squares (TLS) solution of the problem Ax ≈ b in the large-scale discrete ill-posed problems. A regularization technique, based on the multiplicative randomization and the subspace iteration, is proposed to obtain the approximate core problem. In the error analysis, we provide upper bounds for the errors of the solution and the residual of the randomized core reduction.Illustrative numerical examples and comparisons are presented.

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Solvability of the Core Problem with Multiple Right-Hand Sides in the TLS Sense

Cited by 15 publications

References 8 publications

Towards tensor generalizations of TLS & core problem theory

Towards tensor generalizations of TLS & core problem theory

Filter factors of truncated TLS regularization with multiple observations

Randomized core reduction for discrete ill-posed problem

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