The Extrinsic Geometry of Dynamical Systems Tracking Nonlinear Matrix Projections

Feppon, Florian; Lermusiaux, Pierre F. J.

doi:10.1137/18m1192780

Cited by 15 publications

(9 citation statements)

References 51 publications

(155 reference statements)

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“…In other words, by sending t to zero in ( 45) we obtain a solution of (27). For similar discussions connecting these two approximation methods in closely related contexts see [23,24,33]. To prove consistency between step-truncation and dynamic approximation methods we need to compute T r (u(x, t)) for t infinitesimally close to t 0 .…”

Section: Consistency Of Dynamic Approximation and Step-truncation Methodsmentioning

confidence: 99%

Rank-Adaptive Tensor Methods for High-Dimensional Nonlinear PDEs

2021

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We present a new rank-adaptive tensor method to compute the numerical solution of high-dimensional nonlinear PDEs. The method combines functional tensor train (FTT) series expansions, operator splitting time integration, and a new rank-adaptive algorithm based on a thresholding criterion that limits the component of the PDE velocity vector normal to the FTT tensor manifold. This yields a scheme that can add or remove tensor modes adaptively from the PDE solution as time integration proceeds. The new method is designed to improve computational efficiency, accuracy and robustness in numerical integration of high-dimensional problems. In particular, it overcomes well-known computational challenges associated with dynamic tensor integration, including low-rank modeling errors and the need to invert covariance matrices of tensor cores at each time step. Numerical applications are presented and discussed for linear and nonlinear advection problems in two dimensions, and for a four-dimensional Fokker–Planck equation.

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Section: Consistency Of Dynamic Approximation and Step-truncation Methodsmentioning

confidence: 99%

Rank-Adaptive Tensor Methods for High-Dimensional Nonlinear PDEs

2021

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“…Absil, Mahoney, and Trumpf [3] provide expressions for the sphere, Stiefel manifold, orthogonal group, Grassmann manifold, and low-rank matrix manifold. Feppon and Lermusiaux [34] further extended these results by computing the Weingarten maps and explicit expressions for the principal curvatures of the isospectral manifold and "bi-Grassmann" manifold, among others. Heidel and Schultz [39] computed the Weingarten map of the manifold of higher-order tensors of fixed multilinear rank [65].…”

Section: Measuring Curvature With the Weingarten Map A Central Role In Thismentioning

confidence: 95%

The Condition Number of Riemannian Approximation Problems

Breiding¹,

Vannieuwenhoven²

2021

SIAM J. Optim.

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We consider the local sensitivity of least-squares formulations of inverse problems. The sets of inputs and outputs of these problems are assumed to have the structures of Riemannian manifolds. The problems we consider include the approximation problem of finding the nearest point on a Riemannian embedded submanifold from a given point in the ambient space. We characterize the first-order sensitivity, i.e., condition number, of local minimizers and critical points to arbitrary perturbations of the input of the least-squares problem. This condition number involves the Weingarten map of the input manifold, which measures the amount by which the input manifold curves in its ambient space. We validate our main results through experiments with the n-camera triangulation problem in computer vision.

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“…In other words, by sending ∆t to zero in (53) we obtain a solution of (30). For similar discussions connecting these two approximation methods in closely related contexts see [22,23,31]. To prove consistency between step-truncation and dynamic approximation methods we need to compute T r (u(x, t)) for t infinitesimally close to t 0 .…”

Section: Consistency Between Dynamic Approximation and Step-truncatiomentioning

confidence: 99%

Dynamically orthogonal tensor methods for high-dimensional nonlinear PDEs

Dektor

Venturi

2020

Journal of Computational Physics

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We develop new dynamically orthogonal tensor methods to approximate multivariate functions and the solution of high-dimensional time-dependent nonlinear partial differential equations (PDEs). The key idea relies on a hierarchical decomposition of the approximation space obtained by splitting the independent variables of the problem into disjoint subsets. This process, which can be conveniently be visualized in terms of binary trees, yields series expansions analogous to the classical Tensor-Train and Hierarchical Tucker tensor formats. By enforcing dynamic orthogonality conditions at each level of binary tree, we obtain coupled evolution equations for the modes spanning each subspace within the hierarchical decomposition. This allows us to effectively compute the solution to high-dimensional time-dependent nonlinear PDEs on tensor manifolds of constant rank, with no need for rank reduction methods. We also propose new algorithms for dynamic addition and removal of modes within each subspace. Numerical examples are presented and discussed for high-dimensional hyperbolic and parabolic PDEs in bounded domains.

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The Extrinsic Geometry of Dynamical Systems Tracking Nonlinear Matrix Projections

Cited by 15 publications

References 51 publications

Rank-Adaptive Tensor Methods for High-Dimensional Nonlinear PDEs

Rank-Adaptive Tensor Methods for High-Dimensional Nonlinear PDEs

The Condition Number of Riemannian Approximation Problems

Dynamically orthogonal tensor methods for high-dimensional nonlinear PDEs

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