An Error Analysis of Galerkin Projection Methods for Linear Systems with Tensor Product Structure

Beckermann, Bernhard; Kreßner, Daniel; Tobler, Christine

doi:10.1137/120900204

Cited by 10 publications

(7 citation statements)

References 24 publications

(42 reference statements)

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“…Then the following representation holds for the semi-residualR k . The result was first proved for the Sylvester equation in [4] and then generalized to the multi-term linear case in [6]. We prove the result for C * having a single column, the generalization to multiple columns can be obtained by working with each column of C, since the whole matrix C is used to build the approximation space.…”

Section: A New Expression For the Residual And The Choice Of Shiftsmentioning

confidence: 79%

Analysis of the Rational Krylov Subspace Projection Method for Large-Scale Algebraic Riccati Equations

Simoncini¹

2016

SIAM J. Matrix Anal. & Appl.

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Abstract. In the numerical solution of the algebraic Riccati equation A * X + XA − XBB * X + C * C = 0, where A is large, sparse and stable, and B, C have low rank, projection methods have recently emerged as a possible alternative to the more established Newton-Kleinman iteration. In spite of convincing numerical experiments, a systematic matrix analysis of this class of methods is still lacking. We derive new relations for the approximate solution, the residual and the error matrices, giving new insights into the role of the matrix A − BB * X and of its approximations in the numerical procedure. The new results provide theoretical ground for recently proposed modifications of projection methods onto rational Krylov subspaces.

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Section: A New Expression For the Residual And The Choice Of Shiftsmentioning

confidence: 79%

Analysis of the Rational Krylov Subspace Projection Method for Large-Scale Algebraic Riccati Equations

Simoncini¹

2016

SIAM J. Matrix Anal. & Appl.

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“…6 GMRES was chosen because biconjugate gradient stabilized method (Bi-CG-STAB) 7 loses orthogonality quickly, though both methods are known to require good preconditioners for fast convergence. For d = 3, we also have the more restricted methods by Chen et al 1 and Beik et al 8 See also the error analysis for the related Galerkin method by Beckermann et al, 9 the preliminary results in the report by Kressner et al, 10 the Alternating Least Squares (ALS) method by Beylkin et al, 11,12 and the optimization approach by Espig et al 13 We may also include the alternating direction iterative (ADI) method by Mach et al, 14,15 although the reports have stated that the approach is not competitive against the density matrix renormalization group (DMRG) solver for tensor-train matrices by Oseledets et al 16 (which has not been proven to be convergent).…”

Section: Existing Methodsmentioning

confidence: 99%

Numerical solution to a linear equation with tensor product structure

Fan

Zhang

Chu

et al. 2017

Numerical Linear Algebra App

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We consider the numerical solution of a c-stable linear equation in the tensor product space R n 1 ×· · ·×n d , arising from a discretized elliptic partial differential equation in R d . Utilizing the stability, we produce an equivalent d-stable generalized Stein-like equation, which can be solved iteratively. For large-scale problems defined by sparse and structured matrices, the methods can be modified for further efficiency, producing algorithms of O( ∑ i n i ) + O(n s ) computational complexity, under appropriate assumptions (with n s being the flop count for solving a linear system associated with A i − I n i ). Illustrative numerical examples will be presented. KEYWORDS Cayley transform, elliptic partial differential equation, Kronecker product, large-scale problem, linear equation, Stein equation, Sylvester equationfor a constant c ∈ R d . In many similar applications, A i can be formulated to be c-stable (with eigenvalues in the open left-half plane). We make use of the stability of A i and transform the corresponding LET_d (also described as c-stable) to an equivalent d-stable Stein-like equation with tensor product structure (SET), which are defined by matrices with spectra inside the unit Numer Linear Algebra Appl. 2017;24:e2106.wileyonlinelibrary.com/journal/nla

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“…See [36,Proposition 3.1] for more details. The bound can be generalized to the use of other spaces, such as rational Krylov subspaces, see, e.g., [5,6,12,18].…”

Section: The Galerkin Conditionmentioning

confidence: 99%

Optimality Properties of Galerkin and Petrov–Galerkin Methods for Linear Matrix Equations

Palitta

Simoncini

2020

Vietnam J. Math.

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Galerkin and Petrov-Galerkin methods are some of the most successful solution procedures in numerical analysis. Their popularity is mainly due to the optimality properties of their approximate solution. We show that these features carry over to the (Petrov-)Galerkin methods applied for the solution of linear matrix equations.Some novel considerations about the use of Galerkin and Petrov-Galerkin schemes in the numerical treatment of general linear matrix equations are expounded and the use of constrained minimization techniques in the Petrov-Galerkin framework is proposed.

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An Error Analysis of Galerkin Projection Methods for Linear Systems with Tensor Product Structure

Cited by 10 publications

References 24 publications

Analysis of the Rational Krylov Subspace Projection Method for Large-Scale Algebraic Riccati Equations

Analysis of the Rational Krylov Subspace Projection Method for Large-Scale Algebraic Riccati Equations

Numerical solution to a linear equation with tensor product structure

Optimality Properties of Galerkin and Petrov–Galerkin Methods for Linear Matrix Equations

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