Results and Questions on a Nonlinear Approximation Approach for Solving High-dimensional Partial Differential Equations

By C. L. Bris, Tony Lelièvre, Yvon Maday. 2008

100 citations recorded by the source corpus.

Subfields: Computational Mechanics, Modeling and Simulation

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Citations in the corpus

A literature survey of low‐rank tensor approximation techniques (2013, neutral): “Convergence results in this direction can be found in [3, 40, 41, 85–87, 180].”
Tensor-product approach to global time-space-parametric discretization of chemical master equation (2012, methodology): “As such reduced grid-based methods, we would like to mention the sparse grids technique [13], as well as the initial tensor-structured approaches: the greedy algorithm in the canonical tensor format [14, 15, 16, 17, 18], and the solver in the Tucker format, based on the manifold dynamics via the pro”
Recent Developments in Spectral Stochastic Methods for the Numerical Solution of Stochastic Partial Differential Equations (2009, methodology): “In fact, for the case of elliptic symmetric problems (see Appendix B), this algorithm appears as a particular case of a Greedy algorithm [ 69 ].”
Low-rank methods for high-dimensional approximation and model order reduction (2015, unknown): “u,w) = ku−wk, conditions for the convergence of greedy algorithms in a general setting can be found in [81]. In the case of the minimization of convex functionals, convergence results can be found in [11, 13, 39, 82]. Note that this greedy construction is not speciﬁc to the particular setting of ten”
Fast Solution of Parabolic Problems in the Tensor Train/Quantized Tensor Train Format with Initial Application to the Fokker-Planck Equation (2012, methodology): “Various solution methods, based on the Monte Carlo approach [43], direct treatment via the spectral methods for lowdimensional examples [32, 7]; greedy algorithms to compute sums of separable tensor products [11, 1, 31, 5] (the so-called canonical tensor format or PARAFAC [18, 17, 6]) were developed”
Proper generalized decomposition for nonlinear convex problems in tensor Banach spaces (2011, unknown): “The PGD method has been successfully applied to this class of problems in di erent contexts: separation of spatial coordinates for the solution of Poisson equation in high dimension [2,6], separation of physical variables and random parameters for the solution of parameterized stochastic partial di ”
An error estimator for separated representations of highly multidimensional models (2010, methodology): “Its simpler version was analyzed from a numerical analysis point of view in [13].”
The Proper Generalized Decomposition for Advanced Numerical Simulations: A Primer (2013, methodology): “One then applies the PGD strategy in together with an appropriate penalty formulation in \ [3].”
Greedy algorithms for high-dimensional non-symmetric linear problems (2012, background): “This numerical strategy has been extensively studied for the resolution of (linear or nonlinear) elliptic problems [20, 13, 10, 5, 18].”
Proper Generalized Decompositions and Separated Representations for the Numerical Solution of High Dimensional Stochastic Problems (2010, background): “For some variants of algorithms and some very particular frameworks, some mathematical results are available [7,16].”
A priori model reduction through Proper Generalized Decomposition for solving time-dependent partial differential equations (2010, methodology): “PGD methods have also been introduced in other contexts: separation of physical variables and parameters (or random variables) in the context of parametrized (or stochastic ) PDEs [27] [28] [32] [29] 3 , with a possible additional separation of parameters [11] [30] 4 , separation of coordinate varia”
A Proper Generalized Decomposition for the solution of elliptic problems in abstract form by using a functional Eckart–Young approach (2011, methodology): “A proof of convergence for this particular PGD has been introduced in [18], for the case of a second order elliptic symmetric partial differential equation defined in a 2-dimensional domain, and in [3], for the case of linear systems with a full rank square matrix.”
Alternating Minimal Energy Methods for Linear Systems in Higher Dimensions (2013, unknown): “The application of greedy algorithms to optimization in tensor formats was rigorously studied in [12, 2, 26].”
Non-intrusive low-rank separated approximation of high-dimensional stochastic models (2012, background): “Alternatively, following the work in [4, 13], a greedy approach may be applied to perform updates only on the new separation term while freezing all the previous ones.”