Dynamically orthogonal tensor methods for high-dimensional nonlinear PDEs

AbstractWe 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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Preface to the special issue, CMAM 2011, no. 3.

Computational Methods in Applied Mathematics ◽

10.2478/cmam-2011-0014 ◽

2011 ◽

Vol 11 (3) ◽

pp. 272

Author(s):

Ivan Gavrilyuk ◽

Boris Khoromskij ◽

Eugene Tyrtyshnikov

Keyword(s):

Separation Of Variables ◽

Numerical Algorithms ◽

Multilevel Methods ◽

High Dimensional ◽

Special Issue ◽

High Dimensions ◽

Guiding Principle ◽

Tensor Methods ◽

Closed World ◽

The One

Abstract In the recent years, multidimensional numerical simulations with tensor-structured data formats have been recognized as the basic concept for breaking the "curse of dimensionality". Modern applications of tensor methods include the challenging high-dimensional problems of material sciences, bio-science, stochastic modeling, signal processing, machine learning, and data mining, financial mathematics, etc. The guiding principle of the tensor methods is an approximation of multivariate functions and operators with some separation of variables to keep the computational process in a low parametric tensor-structured manifold. Tensors structures had been wildly used as models of data and discussed in the contexts of differential geometry, mechanics, algebraic geometry, data analysis etc. before tensor methods recently have penetrated into numerical computations. On the one hand, the existing tensor representation formats remained to be of a limited use in many high-dimensional problems because of lack of sufficiently reliable and fast software. On the other hand, for moderate dimensional problems (e.g. in "ab-initio" quantum chemistry) as well as for selected model problems of very high dimensions, the application of traditional canonical and Tucker formats in combination with the ideas of multilevel methods has led to the new efficient algorithms. The recent progress in tensor numerical methods is achieved with new representation formats now known as "tensor-train representations" and "hierarchical Tucker representations". Note that the formats themselves could have been picked up earlier in the literature on the modeling of quantum systems. Until 2009 they lived in a closed world of those quantum theory publications and never trespassed the territory of numerical analysis. The tremendous progress during the very recent years shows the new tensor tools in various applications and in the development of these tools and study of their approximation and algebraic properties. This special issue treats tensors as a base for efficient numerical algorithms in various modern applications and with special emphases on the new representation formats.

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Parallel tensor methods for high-dimensional linear PDEs

Journal of Computational Physics ◽

10.1016/j.jcp.2018.08.057 ◽

2018 ◽

Vol 375 ◽

pp. 519-539 ◽

Cited By ~ 4

Author(s):

Arnout M.P. Boelens ◽

Daniele Venturi ◽

Daniel M. Tartakovsky

Keyword(s):

High Dimensional ◽

Tensor Methods ◽

Linear Pdes

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Deep backward schemes for high-dimensional nonlinear PDEs

Mathematics of Computation ◽

10.1090/mcom/3514 ◽

2020 ◽

Vol 89 (324) ◽

pp. 1547-1579 ◽

Cited By ~ 5

Author(s):

Côme Huré ◽

Huyên Pham ◽

Xavier Warin

Keyword(s):

Nonlinear Pdes ◽

High Dimensional

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Overcoming the Curse of Dimensionality in the Numerical Approximation of Parabolic Partial Differential Equations with Gradient-Dependent Nonlinearities

Foundations of Computational Mathematics ◽

10.1007/s10208-021-09514-y ◽

2021 ◽

Author(s):

Martin Hutzenthaler ◽

Arnulf Jentzen ◽

Thomas Kruse

Keyword(s):

Partial Differential Equations ◽

Approximation Algorithm ◽

Real World ◽

Curse Of Dimensionality ◽

Nonlinear Pdes ◽

High Dimensional ◽

Heat Equations ◽

Time Horizons ◽

Nonlinear Heat Equations ◽

General Time

AbstractPartial differential equations (PDEs) are a fundamental tool in the modeling of many real-world phenomena. In a number of such real-world phenomena the PDEs under consideration contain gradient-dependent nonlinearities and are high-dimensional. Such high-dimensional nonlinear PDEs can in nearly all cases not be solved explicitly, and it is one of the most challenging tasks in applied mathematics to solve high-dimensional nonlinear PDEs approximately. It is especially very challenging to design approximation algorithms for nonlinear PDEs for which one can rigorously prove that they do overcome the so-called curse of dimensionality in the sense that the number of computational operations of the approximation algorithm needed to achieve an approximation precision of size $${\varepsilon }> 0$$ ε > 0 grows at most polynomially in both the PDE dimension $$d \in \mathbb {N}$$ d ∈ N and the reciprocal of the prescribed approximation accuracy $${\varepsilon }$$ ε . In particular, to the best of our knowledge there exists no approximation algorithm in the scientific literature which has been proven to overcome the curse of dimensionality in the case of a class of nonlinear PDEs with general time horizons and gradient-dependent nonlinearities. It is the key contribution of this article to overcome this difficulty. More specifically, it is the key contribution of this article (i) to propose a new full-history recursive multilevel Picard approximation algorithm for high-dimensional nonlinear heat equations with general time horizons and gradient-dependent nonlinearities and (ii) to rigorously prove that this full-history recursive multilevel Picard approximation algorithm does indeed overcome the curse of dimensionality in the case of such nonlinear heat equations with gradient-dependent nonlinearities.

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