F ¨ Ur Mathematik in Den Naturwissenschaften Leipzig a Projection Method to Solve Linear Systems in Tensor Format a Projection Method to Solve Linear Systems in Tensor Format

Subfields: Numerical Methods & Scientific Computing

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Citations in the corpus

A literature survey of low‐rank tensor approximation techniques (2013, neutral): “• the Richardson method combined with QTT [161] and low-rank matrix decompositions [189]; • a projection method combined with HT [16]; • the conjugate gradient method, BiCGStab and other Krylov subspace methods combined with HT [169, 171,246] and low-rank matrix decompositions [32]; • GMRES…”
ALTERNATING MINIMAL ENERGY METHODS FOR LINEAR SYSTEMS IN HIGHER DIMENSIONS∗ (2014, methodology): “In this section we introduce the notation and deﬁnitions for the high-dimensional arrays (tensors) and the TT format.”
Tensor Networks for Dimensionality Reduction and Large-scale Optimization: Part 2 Applications and Future Perspectives (2017, methodology): “To date, the following cost functions and corresponding gradients have been investigated in connection with tensor networks [Ballani and Grasedyck, 2013, Dolgov, 2013, Lebedeva, 2011] • ∇J(xk)9ATAxk ´ATb for the squared residual J(x) = }Ax´ b}2; • ∇J(xk)9Axk ´ b for J(x) = xTAx ´ 2bTx.”
Alternating Minimal Energy Methods for Linear Systems in Higher Dimensions (2013, unknown): “Following this paradigm, the solution of linear problems in tensor product formats was addressed in [36, 1, 6].”
Numerical solution of the Hartree-Fock equation by multilevel tensor-structured methods (2011, unknown): “6.1 Brief summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137 6.2 Presentations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140”
Algorithm 941 (2014, methodology): “y = ttm(x, {A, B, C}, [2, 3, 4]) successively applies A,B,C to X in modes 2, 3, 4.”
On Local Convergence of Alternating Schemes for Optimization of Convex Problems in the Tensor Train Format (2013, background): “[1, 2, 22, 26, 29]; while the former approach seems to be superior on some problems in the sense that it sometimes provides lower ranks [16], the latter has advantage of being conceptually quite simple, thus often allowing for a simpler analysis from the theoretical point of view, and enabling, from”
An Adaptive Algebraic Multigrid Algorithm for Low-Rank Canonical Tensor Decomposition (2011, unknown): “Note that in this paper we give a complete but concise description of the proposed algorithm.”
A preconditioned low‐rank CG method for parameter‐dependent Lyapunov matrix equations (2014, methodology): “While this paper has focused on Lyapunov equations with symmetric coefficients, the algorithmic extension to the nonsymmetric case is relatively straightforward, by replacing the CG method with BiCGstab or restarted GMRES, as proposed in [17, 19, 27, 49].”
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].”