Rank-adaptive tensor methods for high-dimensional nonlinear PDEs

Dektor, Alec; Rodgers, Abram; Venturi, Daniele

doi:10.48550/arxiv.2012.05962

Cited by 4 publications

(5 citation statements)

References 39 publications

(81 reference statements)

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“…With this notation at hand, we can define a finite volume scheme for (10) in compact notation. Let us define the sparse diffusion stencil matrices L…”

Section: Spatial Discretizationmentioning

confidence: 99%

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A robust collision source method for rank adaptive dynamical low-rank approximation in radiation therapy

Kusch

Stammer

2023

ESAIM: M2AN

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Deterministic models for radiation transport describe the density of radiation particles moving through a background material. In radiation therapy applications, the phase space of this density is composed of energy, spatial position and direction of flight. The resulting six-dimensional phase space prohibits fine numerical discretizations, which are essential for the construction of accurate and reliable treatment plans. In this work, we tackle the high dimensional phase space through a dynamical low-rank approximation of the particle density. Dynamical low-rank approximation (DLRA) evolves the solution on a low-rank manifold in time. Interpreting the energy variable as a pseudo-time lets us employ the DLRA framework to represent the solution of the radiation transport equation on a low-rank manifold for every energy. Stiff scattering terms are treated through an efficient implicit energy discretization and a rank adaptive integrator is chosen to dynamically adapt the rank in energy. To facilitate the use of boundary conditions and reduce the overall rank, the radiation transport equation is split into collided and uncollided particles through a collision source method. Uncollided particles are described by a directed quadrature set guaranteeing low computational costs, whereas collided particles are represented by a low-rank solution. It can be shown that the presented method is L 2 -stable under a time step restriction which does not depend on stiff scattering terms. Moreover, the implicit treatment of scattering does not require numerical inversions of matrices. Numerical results for radiation therapy configurations as well as the line source benchmark underline the efficiency of the proposed method.

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“…With this notation at hand, we can define a finite volume scheme for (10) in compact notation. Let us define the sparse diffusion stencil matrices L…”

Section: Spatial Discretizationmentioning

confidence: 99%

“…Furthermore, a fixed choice of the rank does not capture the time evolution of the solution complexity. Rank adaptive DLRA integrators which pick the rank in an automated fashion during run time have for example been proposed in [10,39,8]. This work presents a dynamical low-rank approximation for radiation therapy.…”

Section: Introductionmentioning

confidence: 99%

A robust collision source method for rank adaptive dynamical low-rank approximation in radiation therapy

Kusch

Stammer

2023

ESAIM: M2AN

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“…Strategies of rank adaptivity for dynamical low-rank approximation in the context of the projector-splitting integrator of [18,19] have recently been proposed by Dektor, Rodgers & Venturi [4] and Yang & White [30], and we are aware of ongoing work by Schrammer [28]. Those approaches are substantially different from what is proposed here.…”

Section: Introductionmentioning

confidence: 95%

A rank-adaptive robust integrator for dynamical low-rank approximation

Ceruti¹,

Kusch²,

Lubich³

2021

Preprint

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A rank-adaptive integrator for the dynamical low-rank approximation of matrix and tensor differential equations is presented. The fixed-rank integrator recently proposed by two of the authors is extended to allow for an adaptive choice of the rank, using subspaces that are generated by the integrator itself. The integrator first updates the evolving bases and then does a Galerkin step in the subspace generated by both the new and old bases, which is followed by rank truncation to a given tolerance. It is shown that the adaptive low-rank integrator retains the exactness, robustness and symmetry-preserving properties of the previously proposed fixed-rank integrator. Beyond that, up to the truncation tolerance, the rank-adaptive integrator preserves the norm when the differential equation does, it preserves the energy for Schrödinger equations and Hamiltonian systems, and it preserves the monotonic decrease of the functional in gradient flows. Numerical experiments illustrate the behaviour of the rank-adaptive integrator. Keywordsdynamical low-rank approximation • rank adaptivity • structurepreserving integrator • matrix and tensor differential equations Mathematics Subject Classification (2000) 65L05 • 65L20 • 65L70 • 15A69

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“…Most of these methods are formulated for constant rank r . Rank adaptivity can be incorporated without much difficulty for splitting and for projected schemes; see [3,6,9,35]. Finally, given the importance of DLRA in problems from physics (like the Schrödinger and Vlasov equation), the integrators in [7,29] also preserve certain invariants, like energy.…”

Section: Introductionmentioning

confidence: 99%

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

2023

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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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Rank-adaptive tensor methods for high-dimensional nonlinear PDEs

Cited by 4 publications

References 39 publications

A robust collision source method for rank adaptive dynamical low-rank approximation in radiation therapy

A robust collision source method for rank adaptive dynamical low-rank approximation in radiation therapy

A rank-adaptive robust integrator for dynamical low-rank approximation

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

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