NUMERICAL SOLUTION OF PARABOLIC EQUATIONS IN HIGH DIMENSIONS

By Tobias von Petersdorff, Christoph Schwab. 2004

151 citations recorded by the source corpus.

Subfields: Computational Mechanics, Computational Theory and Mathematics, Computer Vision and Pattern Recognition

View Paper: https://www.semanticscholar.org/paper/14913d9fb8dac38ab380d5ad3503a094b05ee277

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Citations in the corpus

Fast Solution of Parabolic Problems in the Tensor Train/Quantized Tensor Train Format with Initial Application to the Fokker-Planck Equation (2012, methodology): “, [30] gives the greedy approximation method, [13] exploits the QTT-Cayley transform, and [15, 44] use the sparse grids approach applied in a t-variable.”
Overcoming the curse of dimensionality in the approximative pricing of financial derivatives with default risks (2019, methodology): “, [30, 94, 95, 97] for deterministic approximation methods for PDEs, cf.”
Sparse tensor discretizations of high-dimensional parametric and stochastic PDEs* (2011, methodology): “The approximation property of sparse grid spaces V̂ (k) L was established, for example, in Schwab and Todor (2003b, Proposition 4.2), Griebel, Oswald and Schiekofer (1999), von Petersdorff and Schwab (2004) and Todor (2009).”
Convergence of a greedy algorithm for high-dimensional convex nonlinear problems (2010, methodology): “One approach consists in using the so-called sparse tensor product representation[13, 14, 15].”
Multi-level Monte Carlo Finite Element method for elliptic PDEs with stochastic coefficients (2011, unknown): “[34]): Lemma 5.”
Two-level Tucker-TT-QTT format for optimized tensor calculus (2012, methodology): “…based on properties of the discretization: the most important components, which can be represented with a low number of parameters are selected and smartly combined to eliminate the errors as far as possible (Sparse Grids, ANOVA decomposition [4] [5] [6] [7], wavelet and multiresolution analysis).”
Approximation of bi-variate functions: singular value decomposition versus sparse grids (2014, background): “The following result has been proven in [25], see also [13].”
Greedy algorithms for high-dimensional non-symmetric linear problems (2012, background): “Let us mention among others sparse grids [21], tensor formats [11], reduced bases [4] and adaptive polynomial approximations [6].”
Approximation of Two-Variate Functions: Singular Value Decomposition Versus Regular Sparse Grids (2010, background): “The following result has been proven in [25], see also [13].”
Quantized-TT-Cayley Transform for Computing the Dynamics and the Spectrum of High-Dimensional Hamiltonians (2011, methodology): “In particular, in the case of high dimensional parabolic problems, this approach was successfully used in [36, 14, 15].”
Machine Learning Approximation Algorithms for High-Dimensional Fully Nonlinear Partial Differential Equations and Second-order Backward Stochastic Differential Equations (2017, methodology): “branching diusions 3 (cf., e.g., [53, 54, 55, 77, 102, 108]), and probabilistic representations based on extensions of the classical Feynman-Kac formula (cf., e.g., [34, 58, 84]). We refer, e.g., to [23, 30, 65, 69, 95, 103, 104, 106] for deterministic approximation methods for PDEs, to [3, 6, 7, 13”
Quantized-TT-Cayley transform to compute dynamics and spectrum of high-dimensional Hamiltonians (2011, methodology): “In particular, in the case of high dimensional parabolic problems this approach has been successfully applied in [36, 14, 15].”