arXiv subjects

Benjamin Carrel

Publications and source records attributed to Benjamin Carrel.

5 recordsLinked to original sources

arXiv2026

Sketch low-rank dynamics: orthogonal vs. oblique projections

We study how sketching techniques from randomized numerical linear algebra can be incorporated into the dynamical low-rank approximation (DLRA) of large-scale matrix differential equations. A natural approach is to sketch the Galerkin condition that defines the DLRA, which leads to an oblique tangent space projection. We show that this oblique projection approximately reproduces the standard DLRA only under restrictive conditions on the vector field, and that it fails on problems with a large perpendicular residual. As an alternative, we propose an orthogonal sketch DLRA that evolves sketch-orthogonal bases while using standard orthogonal projections for the dynamics. This approach preserves the geometric structure of the classical DLRA and is numerically stable. The computational advantage of randomized Gram--Schmidt over Householder QR lies in fewer global synchronizations on a row-distributed basis, at a comparable flop count; when the basis is well conditioned, randomized Gram--Schmidt can be replaced by randomized Cholesky QR, which additionally shifts the basis update from BLAS-2 to BLAS-3 kernels, making it well-suited to modern accelerators. We derive sketch versions of the projector-splitting and BUG integrators, and demonstrate the approach on the Allen--Cahn, Fokker--Planck, and Vlasov--Poisson equations.

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arXiv2025

Interpolatory Dynamical Low-Rank Approximation: Theoretical Foundations and Algorithms

Dynamical low-rank approximation (DLRA) is a widely used paradigm for solving large-scale matrix differential equations, as they arise, for example, from the discretization of time-dependent partial differential equations on tensorized domains. Through orthogonally projecting the dynamics onto the tangent space of a low-dimensional manifold, DLRA achieves a significant reduction of the storage required to represent the solution. However, the need for evaluating the velocity field can make it challenging to attain a corresponding reduction of computational cost in the presence of nonlinearities. In this work, we address this challenge by replacing orthogonal tangent space projections with oblique, data-sparse projections selected by a discrete empirical interpolation method (DEIM). At the continuous-time level, this leads to DLRA-DEIM, a well-posed differential inclusion (in the Filippov sense) that captures the discontinuities induced by changes in the indices selected by DEIM. We establish an existence result, exactness property and error bound for DLRA-DEIM that match existing results for DLRA. For the particular case of QDEIM, a popular variant of DEIM, we provide an explicit convex-polytope characterization of the differential inclusion. Building on DLRA-DEIM, we propose a new class of projected integrators, called PRK-DEIM, that combines explicit Runge--Kutta methods with DEIM-based projections. We analyze the convergence order of PRK-DEIM and show that it matches the accuracy of previously proposed projected Runge-Kutta methods, while being significantly cheaper. Extensions to exponential Runge--Kutta methods and low-order tensor differential equations demonstrate the versatility of our framework.

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arXiv2024

Randomized methods for dynamical low-rank approximation

We introduce novel dynamical low-rank methods for solving large-scale matrix differential equations, motivated by algorithms from randomized numerical linear algebra. In terms of performance (cost and accuracy), our methods overperform existing dynamical low-rank techniques. Several applications to stiff differential equations demonstrate the robustness, accuracy and low variance of the new methods, despite their inherent randomness. Allowing augmentation of the range and corange, the new methods have a good potential for preserving critical physical quantities such as the energy, mass and momentum. Numerical experiments on the Vlasov-Poisson equation are particularly encouraging. The new methods comprise two essential steps: a range estimation step followed by a post-processing step. The range estimation is achieved through a novel dynamical rangefinder method. Subsequently, we propose two methods for post-processing, leading to two time-stepping methods: dynamical randomized singular value decomposition (DRSVD) and dynamical generalized Nystr\"om (DGN). The new methods naturally extend to the rank-adaptive framework by estimating the error via Gaussian sampling.

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arXiv2023

Projected exponential methods for stiff dynamical low-rank approximation problems

The numerical integration of stiff equations is a challenging problem that needs to be approached by specialized numerical methods. Exponential integrators form a popular class of such methods since they are provably robust to stiffness and have been successfully applied to a variety of problems. The dynamical low- \rank approximation is a recent technique for solving high-dimensional differential equations by means of low-rank approximations. However, the domain is lacking numerical methods for stiff equations since existing methods are either not robust-to-stiffness or have unreasonably large hidden constants. In this paper, we focus on solving large-scale stiff matrix differential equations with a Sylvester-like structure, that admit good low-rank approximations. We propose two new methods that have good convergence properties, small memory footprint and that are fast to compute. The theoretical analysis shows that the new methods have order one and two, respectively. We also propose a practical implementation based on Krylov techniques. The approximation error is analyzed, leading to a priori error bounds and, therefore, a mean for choosing the size of the Krylov space. Numerical experiments are performed on several examples, confirming the theory and showing good speedup in comparison to existing techniques.

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arXiv2022

Low-rank Parareal: a low-rank parallel-in-time integrator

In this work, the Parareal algorithm is applied to evolution problems that admit good low-rank approximations and for which the dynamical low-rank approximation (DLRA) can be used as time stepper. Many discrete integrators for DLRA have recently been proposed, based on splitting the projected vector field or by applying projected Runge--Kutta methods. The cost and accuracy of these methods are mostly governed by the rank chosen for the approximation. These properties are used in a new method, called low-rank Parareal, in order to obtain a time-parallel DLRA solver for evolution problems. The algorithm is analyzed on affine linear problems and the results are illustrated numerically.

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