Gamblets for opening the complexity-bottleneck of implicit schemes for hyperbolic and parabolic ODEs/PDEs with rough coefficients

Owhadi, Houman; Zhang, Lei

doi:10.1016/j.jcp.2017.06.037

Cited by 75 publications

(35 citation statements)

References 107 publications

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“…When c ≡ 0, this problem has been studied in [37]. When c = 0, it has been recently studied in [39] independently of our work. The results presented in this second-order case are not new [37].…”

Section: Exponential Decay Of Basis Functions: the Second-order Casementioning

confidence: 86%

Sparse operator compression of higher-order elliptic operators with rough coefficients

Hou

Zhang

2017

Res Math Sci

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We introduce the sparse operator compression to compress a self-adjoint higher-order elliptic operator with rough coefficients and various boundary conditions. The operator compression is achieved by using localized basis functions, which are energy minimizing functions on local patches. On a regular mesh with mesh size h, the localized basis functions have supports of diameter O(h log(1/h)) and give optimal compression rate of the solution operator. We show that by using localized basis functions with supports of diameter O(h log(1/h)), our method achieves the optimal compression rate of the solution operator. From the perspective of the generalized finite element method to solve elliptic equations, the localized basis functions have the optimal convergence rate O(h k ) for a (2k)th-order elliptic problem in the energy norm. From the perspective of the sparse PCA, our results show that a large set of Matérn covariance functions can be approximated by a rank-n operator with a localized basis and with the optimal accuracy. Background Main objectives and the problem settingThe main purpose of this paper is to develop a general strategy to compress a class of self-adjoint higher-order elliptic operators by localized basis functions that give optimal approximation property of the solution operator. To be more specific, suppose L is a self-adjoint elliptic operator in the divergence form Lu = 0≤|σ |,|γ |≤k

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Section: Exponential Decay Of Basis Functions: the Second-order Casementioning

confidence: 86%

Sparse operator compression of higher-order elliptic operators with rough coefficients

Hou

Zhang

2017

Res Math Sci

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“…It is, to some degree, surprising that this decomposition can be achieved in near linear complexity and not in the complexity of an eigenspace decomposition. Naturally [86], this decomposition can be applied to the fast simulation of the wave and parabolic equations associated to (1.1) or to its fast diagonalization.…”

Section: Scientific Discovery As a Decision Theory Problemmentioning

confidence: 99%

Multigrid with Rough Coefficients and Multiresolution Operator Decomposition from Hierarchical Information Games

Owhadi¹

2017

SIAM Rev.

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203

288

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Abstract. We introduce a near-linear complexity (geometric and meshless/algebraic) multigrid/multiresolution method for PDEs with rough (L ∞ ) coefficients with rigorous a priori accuracy and performance estimates. The method is discovered through a decision/game theory formulation of the problems of (1) identifying restriction and interpolation operators, (2) recovering a signal from incomplete measurements based on norm constraints on its image under a linear operator, and (3) gambling on the value of the solution of the PDE based on a hierarchy of nested measurements of its solution or source term. The resulting elementary gambles form a hierarchy of (deterministic) basis functions of H 1 0 (Ω) (gamblets) that (1) are orthogonal across subscales/subbands with respect to the scalar product induced by the energy norm of the PDE, (2) enable sparse compression of the solution space in H 1 0 (Ω), and (3) induce an orthogonal multiresolution operator decomposition. The operating diagram of the multigrid method is that of an inverted pyramid in which gamblets are computed locally (by virtue of their exponential decay) and hierarchically (from fine to coarse scales) and the PDE is decomposed into a hierarchy of independent linear systems with uniformly bounded condition numbers. The resulting algorithm is parallelizable both in space (via localization) and in bandwidth/subscale (subscales can be computed independently from each other). Although the method is deterministic, it has a natural Bayesian interpretation under the measure of probability emerging (as a mixed strategy) from the information game formulation, and multiresolution approximations form a martingale with respect to the filtration induced by the hierarchy of nested measurements.

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“…Another approach by the same authors is presented in [26], where so-called rough polyharmonic splines based on more demanding biharmonic corrector problems are introduced. A more recent approach [25] is based on a decomposition into orthogonal spaces in the spirit of the LOD method and shows the possible generalization of the present approach to a multilevel setting.…”

Section: Introductionmentioning

confidence: 95%

Explicit computational wave propagation in micro-heterogeneous media

Maier

Peterseim

2018

Bit Numer Math

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Explicit time stepping schemes are popular for linear acoustic and elastic wave propagation due to their simple nature which does not require sophisticated solvers for the inversion of the stiffness matrices. However, explicit schemes are only stable if the time step size is bounded by the mesh size in space subject to the so-called CFL condition. In micro-heterogeneous media, this condition is typically prohibitively restrictive because spatial oscillations of the medium need to be resolved by the discretization in space. This paper presents a way to reduce the spatial complexity in such a setting and, hence, to enable a relaxation of the CFL condition. This is done using the Localized Orthogonal Decomposition method as a tool for numerical homogenization. A complete convergence analysis is presented with appropriate, weak regularity assumptions on the initial data.

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Gamblets for opening the complexity-bottleneck of implicit schemes for hyperbolic and parabolic ODEs/PDEs with rough coefficients

Cited by 75 publications

References 107 publications

Sparse operator compression of higher-order elliptic operators with rough coefficients

Sparse operator compression of higher-order elliptic operators with rough coefficients

Multigrid with Rough Coefficients and Multiresolution Operator Decomposition from Hierarchical Information Games

Explicit computational wave propagation in micro-heterogeneous media

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