Low-rank approximation of linear parabolic equations by space-time tensor Galerkin methods

Boiveau, Thomas; Ehrlacher, Virginie; Ern, Alexandre; Nouy, Anthony

doi:10.1051/m2an/2018073

Cited by 12 publications

(12 citation statements)

References 37 publications

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“…, with the inner product (v hn , w hn ) L 2 (Jn;V h ) := Jn (A h (v hn ), w hn ) L dt for all v hn , w hn ∈ P k (J n ; V h ). The least-squares minimization viewpoint is adopted, e.g., in Nouy [231], Andreev [9, 10], Boiveau et al [37]. We refer the reader to Smears [262] for further insight on how to precondition efficiently the symmetric system (69.38).…”

Section: General Casementioning

confidence: 99%

Finite Elements III

Ern

Guermond

2021

Texts in Applied Mathematics

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In Part XII, composed of Chapters 56 to 63, we study the finite element approximation of PDEs where a coercivity property is not available, so that the analysis solely relies on inf-sup conditions. Stability can be obtained by employing various stabilization techniques (residual-based or fluctuation-based). In the present chapter, we introduce the prototypical model problem we are going to work on: it is a system of first-order linear PDEs introduced in 1958 by Friedrichs [131]. This system enjoys symmetry and positivity properties and is often referred to in the literature as Friedrichs' system. Friedrichs wanted to handle within a single functional framework PDEs that are partly elliptic and partly hyperbolic, and for this purpose he developed a formalism that goes beyond the traditional classification of PDEs into elliptic, parabolic, and hyperbolic types. Friedrichs' formalism is very powerful and encompasses several model problems. Important examples are the advection-reaction equation, the div-grad problem related to Darcy's equations, and the curl-curl problem related to Maxwell's equations. This theory will be used systematically in the following chapters. All the theoretical arguments in this chapter are presented assuming that the functions are complex-valued. The real-valued case can be obtained by replacing the field C by R, by replacing the Hermitian transpose Z H by the transpose Z T , and by removing the real part symbol ℜ. Basic ideasLet D be a Lipschitz domain in R d . We consider functions defined on D with values in C m for some integer m ≥ 1. The (Hermitian) inner product inWe denote by I m the identity matrix in C m×m .M , we observe that u − I h0 (u) ∈ V 0 . Using (56.26), we infer that for all v ∈ V 0 = ker(M − N ),with φ := max(ℓ * , µ 0 h). If the boundary of D is not smooth and/or if the coefficients σ, µ are discontinuous, it is in general preferable to use H(curl)-conforming finite element subspaces based on edge elements (see Chapter 15) instead of using H 1 -conforming finite element subspaces; see §45.4.

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Section: General Casementioning

confidence: 99%

Finite Elements III

Ern

Guermond

2021

Texts in Applied Mathematics

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“…Let us also mention that space‐time methods for the solution of the linear parabolic evolution equations have been developed by Reference 37 and extended to its low rank approximation by Reference 38. One can also use other approaches for space discretization of the problem such as classical multiplicative or additive domain decomposition method (see e.g., Reference 39), the S‐method (Reference 40), the tilling method (Reference 41), or the recent S‐variant‐method (Reference 42).…”

Section: Introductionmentioning

confidence: 99%

A finite addition of matter elements method for modeling and solution of an SLM thermal problem by a multiscale method

Ruyssen

Dhia

2022

Numerical Meth Engineering

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In this paper, we are interested in flexible modeling and performing solution of transient thermal Selective Laser Melting Problems. For this, we first introduce a Finite Addition of Matter Elements Method (FAMEM) to generate any wished finite sequence of thermal problems, defined in additively constructed domains. Second, we use the Multiscale Arlequin Frame-work to develop a three-level Arlequin weakstrong formulation of each problem of the finite sequence. Two Arlequin patches are used in the latter to localize the steepest thermal gradients and the nonlinear phase-change phenomena, allowing for fine local approximations and the localized nonlinearity treatment by an algorithm we develop. These patches are identified via the solution of a representative mono-domain transient thermal problem. The latter is also solved with our three-Level Arlequin method for comparison of the solutions and respective performances of both approaches.Moreover, two dimensional tests consisting in the creation of a 316 L stainless steel wall and a two AlSi10Mg layers, are carried out to further enlighten our global approach and to position it with respect to literature.

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“…At present there is an active research toward further progress of tensor numerical methods in scientific computing [59,60,36,40,50]. In particular, there are considerable achievements of tensor-based approaches in computational chemistry [26,27,34,12,25], in bio-molecular modeling [3,4,31,38], in optimal control problems (including the case of fractional control) [22,11,13,53], and in many other fields [2,5,15,42,7,41].…”

Section: Introductionmentioning

confidence: 99%

Reduced Higher Order SVD: ubiquitous rank-reduction method in tensor-based scientific computing

Khoromskaia¹,

Khoromskij²

2022

Preprint

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Tensor numerical methods, based on the rank-structured tensor representation of d-variate functions and operators discretized on large n ⊗d grids, are designed to provide O(dn) complexity of numerical calculations contrary to O(n d ) scaling by conventional grid-based methods. However, multiple tensor operations may lead to enormous increase in the tensor ranks (curse of ranks) of the target data, making calculation intractable. Therefore one of the most important steps in tensor calculations is the robust and efficient rank reduction procedure which should be performed many times in the course of various tensor transforms in multidimensional operator and function calculus. The rank reduction scheme based on the Reduced Higher Order SVD (RHOSVD) introduced in [33] played a significant role in the development of tensor numerical methods. Here, we briefly survey the essentials of RHOSVD method and then focus on some new theoretical and computational aspects of the RHOSVD demonstrating that this rank reduction technique constitutes the basic ingredient in tensor computations for reallife problems. In particular, the stability analysis of RHOSVD is presented. We recall the performance of the RHOSVD in tensor-based calculation of the Hartree potential in computational quantum chemistry. We introduce the multilinear algebra of tensors represented in the range-separated (RS) tensor format. This allows to apply the RHOSVD rank-reduction techniques to non-regular functional data with many singularities, for example, to the rank-structured computation of the collective multi-particle interaction potentials in bio-molecular modeling, as well as to complicated composite radial functions. The new theoretical and numerical results on application of the RHOSVD in scattered data modeling are presented. We finally underline that RHOSVD proved to be the efficient rank reduction technique in numerous applications ranging from numerical treatment of multi-particle systems in material sciences up to a numerical solution of PDE constrained control problems in R d .

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Low-rank approximation of linear parabolic equations by space-time tensor Galerkin methods

Cited by 12 publications

References 37 publications

Finite Elements III

Finite Elements III

A finite addition of matter elements method for modeling and solution of an SLM thermal problem by a multiscale method

Reduced Higher Order SVD: ubiquitous rank-reduction method in tensor-based scientific computing

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