Algorithms for Numerical Analysis in High Dimensions

By Gregory Beylkin, Martin J. Mohlenkamp. 2005

402 citations recorded by the source corpus.

Subfields: Computational Mathematics, Computational Theory and Mathematics, Statistical and Nonlinear Physics

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Citations in the corpus

A literature survey of low‐rank tensor approximation techniques (2013, neutral): “Other examples of combining iterative methods with low-rank truncation include: • the (shift-and-invert) power method combined with CP [26,27]; • the (shift-and-invert) power and Lanczos methods combined with CP and Tucker [11,114]; • a restarted Lanczos method combined with TT [126]; • conjugate…”
Alleviating mesh constraints : Model reduction, parallel time integration and high resolution homogenization (2008, methodology): “11 This kind of separated representation can be easily performed using singular value decomposition (in 2D) or using an alternating least squares technique in the general multidimensional case [4].”
On the Convergence of a Greedy Rank-One Update Algorithm for a Class of Linear Systems (2010, methodology): “The concept of separated representation was introduced by Beylkin and Mohlenkamp in [3] and it is related with the problem of constructing the approximate solutions of some classes of problems in high-dimensional spaces by means a separable function.”
On approximation of functions by exponential sums (2005, methodology): “This approximation provides an example of an analytic construction of a separate resentation as introduced in [9] and discussed in [10,23].”
Generic and typical ranks of multi-way arrays (2009, neutral): “Key words: Tensor, Multiway Arrays, Generic Rank, Canonical Decomposition, Parafac, Factor Analysis AMS Classification: 15A69 15A03 15A72 15A57 46M05 Preprint submitted to Linear Algebra Appl. vol.430, pp.2997–3007, June 2009 ha l-0 04 10 05 8,”
Constructive Representation of Functions in Low-Rank Tensor Formats (2012, background): “Let us start from the function It was considered in [1, 2].”
Efficient MATLAB Computations with Sparse and Factored Tensors (2007, neutral): “More recently, these methods have been applied to signal processing [10, 11], image processing [52, 54, 57, 53], data mining [43, 46, 1, 45], scientific computing [6], and elsewhere [26, 36].”
Numerical solution of the Hartree-Fock equation by multilevel tensor-structured methods (2011, neutral): “6.1 Brief summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137 6.2 Presentations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140”
Non-intrusive low-rank separated approximation of high-dimensional stochastic models (2012, methodology): “Additionally, as described in [8, 9], such a normalization of ck may be used to control the loss of significant digits in the evaluation of the separated representation.”
Numerical construction of Green's functions in high dimensional elliptic problems with variable coefficients and analysis of renewable energy data via sparse and separable approximations (2012, background): “In particular, one may perturb the system by an amount λI, where λ is just slightly larger than machine precision and I is the rB × rB identity matrix [104].”
Multivariate Regression and Machine Learning with Sums of Separable Functions (2009, background): “We found in [1, 2, 23] that this extra freedom allows one to find good approximations with surprisingly small r and reveals a much richer structure than one would believe beforehand.”
Tensor Networks for Dimensionality Reduction and Large-scale Optimization: Part 2 Applications and Future Perspectives (2017, methodology): “, Richardson, CG GMRES) and combined with CP format (Beylkin and Mohlenkamp, 2005; Khoromskij and Schwab, 2011), CP and Tucker formats (BillaudFriess et al.”
Non incremental strategies based on separated representations: applications in computational rheology (2010, background): “Another technique able to circumvent, or at least alleviate, the curse of dimensionality consists of using a separated representation of the unknown field (see [17, 5] for some mathematical results on this topic).”
Krylov Subspace Methods for Linear Systems with Tensor Product Structure (2010, background): “This is possible, for example, if f is sufficiently smooth and d is moderate, see, e.g., [4] [5] [9].”
Tensor Decompositions for Signal Processing Applications: From two-way to multiway component analysis (2014, methodology): “Generalizing the matrix SVD, the workhorse of numerical linear algebra, the MLSVD spurred the interest in tensors in applied mathematics and scientiﬁc computing in very high dimensions [14–16].”
F ¨ Ur Mathematik in Den Naturwissenschaften Leipzig Multigrid Accelerated Tensor Approximation of Function Related Multi-dimensional Arrays Multigrid Accelerated Tensor Approximation of Function Related Multidimensional Arrays * (neutral): “…approximation methods have opened challenging perspectives for a feasible numerical solution of multidimensional problems arising in large scale electronic and molecular structure calculations [1] [2] [7] [8] [10] [12] [15] [4] [18] [28], in stochastic PDEs [24], and in financial mathematics.”