Multipreconditioned Gmres for Shifted Systems

Bakhos, Tania; Kitanidis, Peter K.; Ladenheim, Scott; Saibaba, Arvind K.; Szyld, Daniel B.

doi:10.1137/16m1068694

Cited by 23 publications

(35 citation statements)

References 44 publications

(49 reference statements)

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“…On the other hand, the selective version remains viable for small t, especially when using a parallel implementation. We note, however, that in some practical applications, the complete version of MPGMRES has shown only linear growth of the storage needs, i.e., comparable to that of selective MPGMRES; see [2] and Section 2.5. Table 3.1 Number of matrix-vector products, inner products and preconditioner solves at the k th iteration when using t preconditioners, for complete and selective versions of MPGMRES, and FGMRES with cycling multiple preconditioners.…”

Section: Derivation Of Mpgmresmentioning

confidence: 99%

“…In our Arnoldi-type block procedure to obtain an orthonormal basis of the search space, we start by orthogonalizing every column of W := AZ (1) with respect to V (1) := r (0) / r (0) 2 , and among themselves (using a reduced QR factorization), and storing the coefficients in the matrices H (j,1) , j = 1, 2, which are part of the upper Hessenberg matrix H k (see diagram below); thus obtaining V (2) . Then we increase the space by applying the multiple preconditioners, i.e., at step k, compute 6) and repeat the process.…”

Section: Derivation Of Mpgmresmentioning

confidence: 99%

“…Observe that Z (1) and V (2) have t columns, while Z (2) and V (3) have t 2 columns, and in general Z (i) and V (i+1) have t i columns. Therefore V k+1 has…”

Section: Derivation Of Mpgmresmentioning

confidence: 99%

“…Thus, we consider also a heuristic variant where a selective set of directions from this rich space are used. We mention that there are cases where the selective and the complete versions coincide, i.e., the problems are such that the growth in storage in only linear; see, e.g., Section 2.5 and [2].…”

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confidence: 99%

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GMRES with multiple preconditioners

Greif

Rees

Szyld

2016

SeMA

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Abstract. We propose a variant of GMRES, where multiple (two or more) preconditioners are applied simultaneously, while maintaining minimal residual optimality properties. To accomplish this, a block version of Flexible GMRES is used, but instead of considering blocks associated with multiple right hand sides, we consider a single right-hand side and grow the space by applying each of the preconditioners to all current search directions, minimizing the residual norm over the resulting larger subspace. To alleviate the difficulty of rapidly increasing storage requirements, we present a heuristic limited-memory selective algorithm, and demonstrate the effectiveness of this approach. Numerical results for problems in PDE-constrained optimization and fluid flow problems are presented, illustrating the viability and the potential of the proposed method.

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Section: Derivation Of Mpgmresmentioning

confidence: 99%

Section: Derivation Of Mpgmresmentioning

confidence: 99%

“…Observe that Z (1) and V (2) have t columns, while Z (2) and V (3) have t 2 columns, and in general Z (i) and V (i+1) have t i columns. Therefore V k+1 has…”

Section: Derivation Of Mpgmresmentioning

confidence: 99%

mentioning

confidence: 99%

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GMRES with multiple preconditioners

Greif

Rees

Szyld

2016

SeMA

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“…Computing explicit factorizations may be prohibitively expensive for large-scale problems in three-dimensional physical domains. In this case, the action of M 1/2 (and its inverse) on vectors can be computed using polynomial approaches [14,15] or rational approaches [8,23].…”

Section: The Operatormentioning

confidence: 99%

Efficient D-Optimal Design of Experiments for Infinite-Dimensional Bayesian Linear Inverse Problems

Alexanderian¹,

Saibaba²

2018

SIAM J. Sci. Comput.

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We develop a computational framework for D-optimal experimental design for PDEbased Bayesian linear inverse problems with infinite-dimensional parameters. We follow a formulation of the experimental design problem that remains valid in the infinite-dimensional limit. The optimal design is obtained by solving an optimization problem that involves repeated evaluation of the logdeterminant of high-dimensional operators along with their derivatives. Forming and manipulating these operators is computationally prohibitive for large-scale problems. Our methods exploit the lowrank structure in the inverse problem in three different ways, yielding efficient algorithms. Our main approach is to use randomized estimators for computing the D-optimal criterion, its derivative, as well as the Kullback-Leibler divergence from posterior to prior. Two other alternatives are proposed based on low-rank approximation of the prior-preconditioned data misfit Hessian, and a fixed low-rank approximation of the prior-preconditioned forward operator. Detailed error analysis is provided for each of the methods, and their effectiveness is demonstrated on a model sensor placement problem for initial state reconstruction in a time-dependent advection-diffusion equation in two space dimensions.we seek sensor placements that maximize the expected information gain in parameter inversion. Our approach, however, is general in that it is applicable to D-optimal experimental design for a broad class of linear inverse problems. ]. In particular, A-optimal experimental design for large-scale applications has been addressed in [3, 4, 17, 20-22, 24, 43]. Computing Aoptimal designs for large-scale applications faces similar challenges to the D-optimal designs. The use of Monte-Carlo trace estimators [7] has been instrumental in making A-optimal design computationally feasible for such applications.Fast algorithms for computing expected information gain, i.e., the D-optimal criterion, for nonlinear inverse problems, via Laplace approximations, were proposed in [31,32]. The focus of the aforementioned works was efficient computation of the D-optimal criterion. The optimal design was then computed by an exhaustive search over prespecified sets of experimental scenarios. We also mention [26-28] that use polynomial chaos (PC) expansions to build easy-to-evaluate surrogates of the forward model. The expected information gain is then evaluated using an appropriate Monte Carlo procedure. In [27,28], the authors use a gradient based approach to obtain the optimal design. In [44], the authors propose an alternate design criterion given by a lower bound of the expected information gain, and use PC representation of the forward model to accelerate objective function evaluations. However, the approaches based on PC representations remain limited in scope to problems with low to moderate parameter dimensions (e.g., parameter dimensions in order of tens).Efficient estimators for the evaluation of D-optimal criterion were developed in [38,39]; however, these works do not...

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Preconditioners for Krylov subspace methods: An overview

Pearson

Pestana

2020

GAMM-Mitteilungen

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When simulating a mechanism from science or engineering, or an industrial process, one is frequently required to construct a mathematical model, and then resolve this model numerically. If accurate numerical solutions are necessary or desirable, this can involve solving large-scale systems of equations. One major class of solution methods is that of preconditioned iterative methods, involving preconditioners which are computationally cheap to apply while also capturing information contained in the linear system. In this article, we give a short survey of the field of preconditioning. We introduce a range of preconditioners for partial differential equations, followed by optimization problems, before discussing preconditioners constructed with less standard objectives in mind.

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Multipreconditioned Gmres for Shifted Systems

Cited by 23 publications

References 44 publications

GMRES with multiple preconditioners

GMRES with multiple preconditioners

Efficient D-Optimal Design of Experiments for Infinite-Dimensional Bayesian Linear Inverse Problems

Preconditioners for Krylov subspace methods: An overview

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