Rank-Adaptive Tensor Methods for High-Dimensional Nonlinear PDEs

Dektor, Alec; Rodgers, Abram; Venturi, Daniele

doi:10.1007/s10915-021-01539-3

Cited by 26 publications

(25 citation statements)

References 54 publications

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“…The second method we present allows the solution to leave the tensor manifold H r , and then maps it back onto the manifold at each time step. Applying either (5) or (6) to the right hand side of (10) results in a step-truncation method…”

Section: Step-truncation Methods (B-st Svd-st)mentioning

confidence: 99%

“…) denote an order-p increment function defining a one-step temporal integration scheme as in (10) and let f k ∈ H r . We have that…”

Section: Proposition 1 (Consistency Of B-st and B-tsp) Let φ(N F δTmentioning

confidence: 99%

“…be a convergent one-step scheme 2 approximating the solution to the initial value problem (2). In (10), u k denotes the numerical solution to (2) at time instant t k .…”

Section: Introductionmentioning

confidence: 99%

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Adaptive Integration of Nonlinear Evolution Equations on Tensor Manifolds

2022

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We develop new adaptive algorithms for temporal integration of nonlinear evolution equations on tensor manifolds. These algorithms, which we call step-truncation methods, are based on performing one time step with a conventional time-stepping scheme, followed by a truncation operation onto a tensor manifold. By selecting the rank of the tensor manifold adaptively to satisfy stability and accuracy requirements, we prove convergence of a wide range of step-truncation methods, including explicit one-step and multi-step methods. These methods are very easy to implement as they rely only on arithmetic operations between tensors, which can be performed by efficient and scalable parallel algorithms. Adaptive step-truncation methods can be used to compute numerical solutions of high-dimensional PDEs, which, have become central to many new areas of application such optimal mass transport, random dynamical systems, and mean field optimal control. Numerical applications are presented and discussed for a Fokker-Planck equation with spatially dependent drift on a flat torus of dimension two and four.

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Section: Step-truncation Methods (B-st Svd-st)mentioning

confidence: 99%

“…) denote an order-p increment function defining a one-step temporal integration scheme as in (10) and let f k ∈ H r . We have that…”

Section: Proposition 1 (Consistency Of B-st and B-tsp) Let φ(N F δTmentioning

confidence: 99%

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Adaptive Integration of Nonlinear Evolution Equations on Tensor Manifolds

2022

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“…In the case where the initial condition is imposed softly, the term L I defined as in (7) will be added as a part of the loss L PINN and minimized through training. Meanwhile, the physical constraint of solution can be imposed by parametrization:…”

Section: Pinn Formulationmentioning

confidence: 99%

“…One natural idea is to restrict to a solution ansatz. For example, using the tensor train (TT) format to approximate the solutions of high-dimensional PDEs [6][7][8][9][10][11]. While such methods are quite successful if the solution can be well represented by the tensor train, the representability is not guaranteed.…”

Section: Introductionmentioning

confidence: 99%

Neural Network Based Variational Methods for Solving Quadratic Porous Medium Equations in High Dimensions

Lu¹,

Wang²

2022

Preprint

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In this paper, we propose and study neural network based methods for solutions of high-dimensional quadratic porous medium equation (QPME). Three variational formulations of this nonlinear PDE are presented: a strong formulation and two weak formulations. For the strong formulation, the solution is directly parameterized with a neural network and optimized by minimizing the PDE residual. It can be proved that the convergence of the optimization problem guarantees the convergence of the approximate solution in the L 1 sense. The weak formulations are derived following [1] which characterizes the very weak solutions of QPME. Specifically speaking, the solutions are represented with intermediate functions who are parameterized with neural networks and are trained to optimize the weak formulations. Extensive numerical tests are further carried out to investigate the pros and cons of each formulation in low and high dimensions. This is an initial exploration made along the line of solving high-dimensional nonlinear PDEs with neural network based methods, which we hope can provide some useful experience for future investigations.

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Macro-micro decomposition for consistent and conservative model order reduction of hyperbolic shallow water moment equations: a study using POD-Galerkin and dynamical low-rank approximation

Koellermeier,

Krah,

Kusch

2024

Adv Comput Math

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Geophysical flow simulations using hyperbolic shallow water moment equations require an efficient discretization of a potentially large system of PDEs, the so-called moment system. This calls for tailored model order reduction techniques that allow for efficient and accurate simulations while guaranteeing physical properties like mass conservation. In this paper, we develop the first model reduction for the hyperbolic shallow water moment equations and achieve mass conservation. This is accomplished using a macro-micro decomposition of the model into a macroscopic (conservative) part and a microscopic (non-conservative) part with subsequent model reduction using either POD-Galerkin or dynamical low-rank approximation only on the microscopic (non-conservative) part. Numerical experiments showcase the performance of the new model reduction methods including high accuracy and fast computation times together with guaranteed conservation and consistency properties.

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Rank-Adaptive Tensor Methods for High-Dimensional Nonlinear PDEs

Cited by 26 publications

References 54 publications

Adaptive Integration of Nonlinear Evolution Equations on Tensor Manifolds

Adaptive Integration of Nonlinear Evolution Equations on Tensor Manifolds

Neural Network Based Variational Methods for Solving Quadratic Porous Medium Equations in High Dimensions

Macro-micro decomposition for consistent and conservative model order reduction of hyperbolic shallow water moment equations: a study using POD-Galerkin and dynamical low-rank approximation

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