Greedy algorithms for high-dimensional non-symmetric linear problems

Cancès, Éric; Ehrlacher, Virginie; Lelièvre, Tony

doi:10.1051/proc/201341005

Cited by 16 publications

(21 citation statements)

References 16 publications

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“…The strong coercivity of the least-squares formulation has the added benefit that it bypasses the stability conditions associated with weakly coercive problems. This is similar to the notion of the minimal residual PGD (e.g., [16,28]). However, we use the terminology "least-squares PGD" to highlight the fact that we construct PGD algorithms based on rigorously defined least-squares principles.…”

mentioning

confidence: 77%

Least-Squares Proper Generalized Decompositions for Weakly Coercive Elliptic Problems

Croft¹,

Phillips²

2017

SIAM J. Sci. Comput.

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Abstract. Proper generalized decompositions (PGDs) are a family of methods for efficiently solving high-dimensional PDEs, which seek to find a low-rank approximation to the solution of the PDE a priori. Convergence of PGD algorithms can only be proven for problems which are continuous, symmetric, and strongly coercive. In the particular case of problems which are only weakly coercive we have the additional issue that weak coercivity estimates are not guaranteed to be inherited by the low-rank PGD approximation. This can cause stability issues when employing a Galerkin PGD approximation of weakly coercive problems. In this paper we propose the use of PGD algorithms based on least-squares formulations which always lead to symmetric and strongly coercive problems and hence provide stable and provably convergent algorithms. Taking the Stokes problem as a prototypical example of a weakly coercive problem, we develop and compare rigorous leastsquares PGD algorithms based on continuous least-squares estimates for two different reformulations of the problem. We show that these least-squares PGDs provide a much stabler algorithm than an equivalent Galerkin PGDs, and we provide proofs of convergence of the algorithms. 1. Introduction. The systems of partial differential equations (PDEs) arising from many models in science and engineering are defined in high-dimensional spaces. Examples include the kinetic theory description of polymer dynamics, option pricing in financial mathematics, and quantum chemistry. The numerical solution of these systems represents a tremendous computational challenge since they exhibit the so-called curse of dimensionality when standard methods of discretization are used. This issue arises since many algorithms do not scale well with increasing dimensions, typically requiring computational effort (time or memory) that is exponential in the number of dimensions. Therefore, new algorithms are required to circumvent the curse of dimensionality to make the problems tractable.One possibility lies in the use of sparse grids [12]. However, Achdou and Pironneau [1] argue that the use of sparse grids is restricted to models that possess moderate dimensionality. An alternative approach uses low-rank tensor methods to search for an approximation of the solution in a low-dimensional subset of the solution space. Classical low-rank subsets include canonical tensors and Tucker tensors and their variants [25].Proper generalized decompositions (PGDs) are a relatively new family of methods which were introduced by Ammar et al. [4] for the efficient approximation of the solution to PDEs defined in high-dimensional spaces. The main concept underpinning all PGD algorithms is the approximation of the solution, u, to a d-dimensional PDE

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confidence: 77%

Least-Squares Proper Generalized Decompositions for Weakly Coercive Elliptic Problems

Croft¹,

Phillips²

2017

SIAM J. Sci. Comput.

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“…This technique can be used in particular to solve high-dimensional partial differential equations. This section is related to a series of recent works [4][5][6]16]. We also refer to the contribution of Virginie Ehrlacher to this volume for an application of this technique to eigenvalue problems.…”

Section: Greedy Algorithms and Model Reductionmentioning

confidence: 99%

“…In this case however, we have no convergence rates. In [6], we investigated various techniques to generalize the approach to linear but non-symmetric problems, which are thus not simply associated to an energy minimization problem: there is up to now no satisfactory technique to treat non-symmetric problems. We have also on-going works on parametric eigenvalue problems.…”

Section: Extensions and Open Questionsmentioning

confidence: 99%

Statistical inference for partial differential equations

et al. 2014

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Abstract. Many physical phenomena are modeled by parametrized PDEs. The poor knowledge on the involved parameters is often one of the numerous sources of uncertainties on these models. Some of these parameters can be estimated, with the use of real world data. The aim of this mini-symposium is to introduce some of the various tools from both statistical and numerical communities to deal with this issue. Parametric and non-parametric approaches are developed in this paper. Some of the estimation procedures require many evaluations of the initial model. Some interpolation tools and some greedy algorithms for model reduction are therefore also presented, in order to reduce time needed for running the model.

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“…The idea is to introduce an ideal residual norm, so that minimizing this residual norm is equivalent to minimizing the error in the target norm. Notice that other variants have been also proposed in [36,12].…”

Section: Introductionmentioning

confidence: 99%

Ideal minimal residual-based proper generalized decomposition for non-symmetric multi-field models – Application to transient elastodynamics in space-time domain

Boucinha

Ammar

Gravouil

et al. 2014

Computer Methods in Applied Mechanics and Engineering

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is an open access repository that collects the work of Arts et Métiers ParisTech researchers and makes it freely available over the web where possible. b s t r a c tIt is now well established that separated representations built with the help of proper generalized decomposition (PGD) can drastically reduce computational costs associated with solution of a wide variety of problems. However, it is still an open question to know if separated representations can be efficiently used to approximate solutions of hyperbolic evolution problems in space-time domain. In this paper, we numerically address this issue and concentrate on transient elastodynamic models. For such models, the operator associated with the space-time problem is non-symmetric and low-rank approximations are classically computed by minimizing the space-time residual in a natural L 2 sense, yet leading to non optimal approximations in usual solution norms. Therefore, a new algorithm has been recently introduced by one of the authors and allows to find a quasi-optimal low-rank approximation a priori with respect to a target norm. We presently extend this new algorithm to multi-field models. The proposed algorithm is applied to elastodynamics formulated over space-time domain with the Time Discontinuous Galerkin method in displacement and velocity. Numerical examples demonstrate convergence of the proposed algorithm and comparisons are made with classical a posteriori and a priori approaches.

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Greedy algorithms for high-dimensional non-symmetric linear problems

Cited by 16 publications

References 16 publications

Least-Squares Proper Generalized Decompositions for Weakly Coercive Elliptic Problems

Least-Squares Proper Generalized Decompositions for Weakly Coercive Elliptic Problems

Statistical inference for partial differential equations

Ideal minimal residual-based proper generalized decomposition for non-symmetric multi-field models – Application to transient elastodynamics in space-time domain

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