Mathematik in den Naturwissenschaften Leipzig Tensor-Structured Galerkin Approximation of parametric and stochastic Elliptic PDEs

By Boris N. Khoromskij, Christoph Schwab. 2010

Subfields: Numerical Methods & Scientific Computing

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Citations in the corpus

A literature survey of low‐rank tensor approximation techniques (2013, neutral): “Other techniques include the use of approximate inverses for high-dimensional Laplace operators [151, 165, 172, 210], low-rank manipulation of the PDE coefﬁcients [60], low-rank tensor approximation of multilevel preconditioners [8], and low-rank tensor diagonal preconditioners for wavelet…”
Low-rank tensor structure of linear diffusion operators in the TT and QTT formats☆ (2013, background): “…Tensor [19, 20] representations has been demonstrated in many papers, including [21, 22, 23] on elliptic PDEs and eigenvalue problems, [24, 25, 26] on parabolic PDEs, [27] on nonlinear EVPs, [28] on multi-parametric problems, [29] on stochastic PDEs, [30] on problems in quantum molecular dynamics.”
The geometry of algorithms using hierarchical tensors (2013, background): “Besides their longstanding tradition in the computational physics community (see [20] [23] for recent overviews), the (Q)TT and HT formats have more recently been used, amongst others, in [24] [25] [26] [27] [28] [29] [30] [31] for solving a variety of high-dimensional problems.”
ALTERNATING MINIMAL ENERGY METHODS FOR LINEAR SYSTEMS IN HIGHER DIMENSIONS∗ (2014, methodology): “, algorithms for the solution of linear systems [51, 21, 47, 38, 34, 31, 39, 2, 8, 4, 3], partial eigenproblems [41, 40, 24, 20], and Fourier transform [11].”
Non-intrusive low-rank separated approximation of high-dimensional stochastic models (2012, result): “Similar representations, although with different solution strategies, have been later investigated in [60, 41, 28].”
An Error Analysis of Galerkin Projection Methods for Linear Systems with Tensor Product Structure (2013, methodology): “5) can be used as preconditioners in iterative methods for more general linear systems [19, 4] and eigenvalue problems [22].”
Approximation rates for the hierarchical tensor format in periodic Sobolev spaces (2014, unknown): “This problem has been attacked in [12, 24].”
Algorithm 941 (2014, background): “Examples for this strategy can be found in [4, 14, 20, 21, 25].”
Adaptive low rank wavelet methods and applications to two-electron Schrödinger equations (2012, background): “, in [101, 5, 96, 140], and for eigenvalue problems in [92, 6, 74, 102, 95, 140].”
Alternating Minimal Energy Methods for Linear Systems in Higher Dimensions (2013, unknown): “, algorithms for the solution of linear systems [51, 21, 47, 38, 34, 31, 39, 2, 8, 4, 3], partial eigenproblems [41, 40, 24, 20], and Fourier transform [11].”
Low-rank tensor methods for linear systems and eigenvalue problems (2012, methodology): “MSc ETH CSE born July 31, 1984 citizen of Thal SG accepted on the recommendation of Prof. Dr. Daniel Kressner, examiner Prof. Dr. Lars Grasedyck, co-examiner Prof. Dr. Christoph Schwab, co-examiner 2012”
DMRG+QTT approach to computation of the ground state for the molecular Schrödinger operator (2010, background): “However, in the case of moderate spacial dimension the proper combination of both models leads to e cient tensor-structured numerical methods [21, 25, 26, 19, 18].”
Model order reduction based on proper generalized decomposition for the propagation of uncertainties in structural dynamics (2012, background): “Note that other definitions and algorithms have been recently proposed for the construction of such separated representations of multiparametric stochastic functions [7, 12].”
Tensors-structured Numerical Methods in Scientific Computing: Survey on Recent Advances (2012, background): “Section 4 presents various numerical examples on the QTT approximation of N -d tensors applied to the elliptic/parabolic equations and to stochastic/multiparametric PDEs.”
Low-Rank Explicit QTT Representation of the Laplace Operator and Its Inverse (2012, methodology): “Note that the straightforward discretizations of the Laplace operator we consider turn out to be very efficient in tensor-structured solvers based on the TT, QTT, and hierarchical tensor [15, 13] formats and yield convincing experimental results when constructed on fine meshes; see the papers [24, 8”
Quantics-TT Collocation Approximation of Parameter-Dependent and Stochastic Elliptic PDEs (2010, unknown): “AMS Subject Classification: 65F30, 65F50, 65N35, 65F10 Key words: elliptic equations, stochastic PDEs, the Karhunen-Loève expansion, separable approximation, quantics-TT tensors, preconditioners, tensor-truncated iteration.”
Tensor Networks for Dimensionality Reduction and Large-scale Optimization: Part 2 Applications and Future Perspectives (2017, methodology): “…equations and discretization of PDEs by iterative algorithms (e.g., Richardson, CG GMRES) and combined with CP format [Beylkin and Mohlenkamp, 2005, Khoromskij and Schwab, 2011], CP and Tucker formats [Billaud-Friess et al., 2014], TT format [Dolgov, 2013, Khoromskij and Oseledets, 2010], HT…”
A preconditioned low‐rank CG method for parameter‐dependent Lyapunov matrix equations (2014, background): “In the parameter-dependent case, the results are promising but also indicate that the rank growth in the transient phase of the method may pose a bottleneck for more complex problems, which can only be overcome by the development of more effective preconditioners, see [26] for an example.”