arXiv subjects

Hans De Sterck

Publications and source records attributed to Hans De Sterck.

40 records · Page 3Linked to original sources

arXiv2011

A Nonlinear GMRES Optimization Algorithm for Canonical Tensor Decomposition

A new algorithm is presented for computing a canonical rank-R tensor approximation that has minimal distance to a given tensor in the Frobenius norm, where the canonical rank-R tensor consists of the sum of R rank-one components. Each iteration of the method consists of three steps. In the first step, a tentative new iterate is generated by a stand-alone one-step process, for which we use alternating least squares (ALS). In the second step, an accelerated iterate is generated by a nonlinear generalized minimal residual (GMRES) approach, recombining previous iterates in an optimal way, and essentially using the stand-alone one-step process as a preconditioner. In particular, the nonlinear extension of GMRES is used that was proposed by Washio and Oosterlee in [ETNA Vol. 15 (2003), pp. 165-185] for nonlinear partial differential equation problems. In the third step, a line search is performed for globalization. The resulting nonlinear GMRES (N-GMRES) optimization algorithm is applied to dense and sparse tensor decomposition test problems. The numerical tests show that ALS accelerated by N-GMRES may significantly outperform both stand-alone ALS and a standard nonlinear conjugate gradient optimization method, especially when highly accurate stationary points are desired for difficult problems. The proposed N-GMRES optimization algorithm is based on general concepts and may be applied to other nonlinear optimization problems.

math.NA↗

arXiv2011

A Self-learning Algebraic Multigrid Method for Extremal Singular Triplets and Eigenpairs

A self-learning algebraic multigrid method for dominant and minimal singular triplets and eigenpairs is described. The method consists of two multilevel phases. In the first, multiplicative phase (setup phase), tentative singular triplets are calculated along with a multigrid hierarchy of interpolation operators that approximately fit the tentative singular vectors in a collective and self-learning manner, using multiplicative update formulas. In the second, additive phase (solve phase), the tentative singular triplets are improved up to the desired accuracy by using an additive correction scheme with fixed interpolation operators, combined with a Ritz update. A suitable generalization of the singular value decomposition is formulated that applies to the coarse levels of the multilevel cycles. The proposed algorithm combines and extends two existing multigrid approaches for symmetric positive definite eigenvalue problems to the case of dominant and minimal singular triplets. Numerical tests on model problems from different areas show that the algorithm converges to high accuracy in a modest number of iterations, and is flexible enough to deal with a variety of problems due to its self-learning properties.

math.NA↗

arXiv2010

A generalized Monte Carlo loop algorithm for frustrated Ising models

We introduce a Generalized Loop Move (GLM) update for Monte Carlo simulations of frustrated Ising models on two-dimensional lattices with bond-sharing plaquettes. The GLM updates are designed to enhance Monte Carlo sampling efficiency when the system's low-energy states consist of an extensive number of degenerate or near-degenerate spin configurations, separated by large energy barriers to single spin flips. Through implementation on several frustrated Ising models, we demonstrate the effectiveness of the GLM updates in cases where both degenerate and near-degenerate sets of configurations are favored at low temperatures. The GLM update's potential to be straightforwardly extended to different lattices and spin interactions allow it to be readily adopted on many other frustrated Ising models of physical relevance.

cond-mat.stat-mech↗

arXiv2009

Monte Carlo study of degenerate groundstates and residual entropy in a frustrated honeycomb lattice Ising model

We study a classical fully-frustrated honeycomb lattice Ising model using Markov chain Monte Carlo methods and exact calculations . The Hamiltonian realizes a degenerate ground state manifold of equal-energy states, where each hexagonal plaquette of the lattice has one and only one frustrated bond, with an extensive residual entropy that grows as the number of spins N. Traditional single-spin flip Monte Carlo methods fail to sample all possible spin configurations in this ground state efficiently, due to their separation by large energy barriers. We develop a non-local "chain-flip" algorithm that solves this problem, and demonstrate its effectiveness on the Ising Hamiltonian with and without perturbative interactions. The two perturbations considered are a slightly weakened bond, and an external magnetic field h. For some cases, the chain-flip move is necessary for the simulation to find an ordered ground state. In the case of the magnetic field, two magnetized ground states with non-extensive entropy are found, and two special values of h exist where the residual entropy again becomes extensive, scaling proportional to N ln phi, where phi is the golden ratio.

cond-mat.stat-mech↗