On the minimal residual methods for solving diffusion-convection SLAEs

The objective of this research is to develop and to study iterative methods in the Krylov subspaces for solving systems of linear algebraic equations (SLAEs) with non-symmetric sparse matrices of high orders arising in the approximation of multi-dimensional boundary value problems on the unstructured grids. These methods are also relevant in many applications, including diffusion-convection equations. The considered algorithms are based on constructing ATA — orthogonal direction vectors calculated using short recursions and providing global minimization of a residual at each iteration. Methods based on the Lanczos orthogonalization, AT — preconditioned conjugate residuals algorithm, as well as the left Gauss transform for the original SLAEs are implemented. In addition, the efficiency of these iterative processes is investigated when solving algebraic preconditioned systems using an approximate factorization of the original matrix in the Eisenstat modification. The results of a set of computational experiments for various grids and values of convective coefficients are presented, which demonstrate a sufficiently high efficiency of the approaches under consideration.

Factorization in Molecular Modeling and Belief Propagation Algorithms

Factorization reduces computational complexity and is therefore an important tool in statistical machine learning of high dimensional systems. Conventional molecular modeling, including molecular dynamics and Monte Carlo simulations of molecular systems, is a large research field based on approximate factorization of molecular interactions. Recently, the local distribution theory was proposed to factorize global joint distribution of a given molecular system into trainable local distributions. Belief propagation algorithms are a family of exact factorization algorithms for trees and are extended to approximate loopy belief propagation algorithms for graphs with loops. Despite the fact that factorization of probability distribution is their common foundation, computational research in molecular systems and machine learning studies utilizing belief propagation algorithms have been carried out independently with respective track of algorithm development. The connection and differences among these factorization algorithms are briefly presented in this perspective, with the hope to intrigue further development in factorization algorithms for physical modeling of complex molecular systems.

Approximate Factorization Method Using Alternating Cell Direction Implicit Method: Comparison of Convergence Characteristics Using Basic Model Equations

In this paper, a fast implicit iteration scheme called the alternating cell directions implicit (ACDI) method is combined with the approximate factorization scheme. The use of fast implicit iteration methods with unstructured grids is hardly. The proposed method allows fast implicit formulations to be used in unstructured meshes, revealing the advantages of fast implicit schemes in unstructured meshes. Fast implicit schemes used in structured meshes have evolved considerably and are much more accurate and robust, and are faster than explicit schemes. It is a crucial novel development that such developed schemes can be applied to unstructured schemes. In steady incompressible potential flow, the convergence character of the scheme is compared with the Runge-Kutta order 4 (RK4), Laasonen, point Gauss–Seidel iteration, old version ACDI, and line Gauss–Seidel iteration methods. The scheme behaves like an approximation of the fully implicit method (Laasonen) up to an optimum pseudo-time-step size. This is a highly anticipated result because the approximate factorization method is an approach to a fully implicit formulation. The results of the numerical study are compared with other fast implicit methods (e.g., the point and line Gauss–Seidel methods), the RK4 method, which is an explicit scheme, and the Laasonen method, which is a fully implicit scheme. The study increased the accuracy of the ACDI method. Thus, the new ACDI method is faster in unstructured grids than other methods and can be used for any mesh construction.

A note on adaptivity in factorized approximate inverse preconditioning

The problem of solving large-scale systems of linear algebraic equations arises in a wide range of applications. In many cases the preconditioned iterative method is a method of choice. This paper deals with the approximate inverse preconditioning AINV/SAINV based on the incomplete generalized Gram–Schmidt process. This type of the approximate inverse preconditioning has been repeatedly used for matrix diagonalization in computation of electronic structures but approximating inverses is of an interest in parallel computations in general. Our approach uses adaptive dropping of the matrix entries with the control based on the computed intermediate quantities. Strategy has been introduced as a way to solve di cult application problems and it is motivated by recent theoretical results on the loss of orthogonality in the generalized Gram– Schmidt process. Nevertheless, there are more aspects of the approach that need to be better understood. The diagonal pivoting based on a rough estimation of condition numbers of leading principal submatrices can sometimes provide inefficient preconditioners. This short study proposes another type of pivoting, namely the pivoting that exploits incremental condition estimation based on monitoring both direct and inverse factors of the approximate factorization. Such pivoting remains rather cheap and it can provide in many cases more reliable preconditioner. Numerical examples from real-world problems, small enough to enable a full analysis, are used to illustrate the potential gains of the new approach.

Localized inverse factorization

We propose a localized divide and conquer algorithm for inverse factorization $S^{-1} = ZZ^*$ of Hermitian positive definite matrices $S$ with localized structure, e.g. exponential decay with respect to some given distance function on the index set of $S$. The algorithm is a reformulation of recursive inverse factorization (Rubensson et al. (2008) Recursive inverse factorization. J. Chem. Phys., 128, 104105) but makes use of localized operations only. At each level of the recursion, the problem is cut into two subproblems and their solutions are combined using iterative refinement (Niklasson (2004) Iterative refinement method for the approximate factorization of a matrix inverse. Phys. Rev. B, 70, 193102) to give a solution to the original problem. The two subproblems can be solved in parallel without any communication and, using the localized formulation, the cost of combining their results is negligible compared to the overall cost for sufficiently large systems and appropriate partitions of the problem. We also present an alternative derivation of iterative refinement based on a sign matrix formulation, analyze the stability and propose a parameterless stopping criterion. We present bounds for the initial factorization error and the number of iterations in terms of the condition number of $S$ when the starting guess is given by the solution of the two subproblems in the binary recursion. These bounds are used in theoretical results for the decay properties of the involved matrices. We demonstrate the localization properties of our algorithm for matrices corresponding to nearest neighbor overlap on one-, two- and three-dimensional lattices, as well as basis set overlap matrices generated using the Hartree–Fock and Kohn–Sham density functional theory electronic structure program Ergo (Rudberg et al. (2018) Ergo: an open-source program for linear-scaling electronic structure. SoftwareX, 7, 107). We evaluate the parallel performance of our implementation based on the chunks and tasks programming model, showing that the proposed localization of the algorithm results in a dramatic reduction of communication costs.

Approximate factorization of positive matrices by using methods of tropical optimization

The problem of rank-one factorization of positive matrices with missing (unspecified) entries is considered where a matrix is approximated by a product of column and row vectors that are subject to box constraints. The problem is reduced to the constrained approximation of the matrix, using the Chebyshev metric in logarithmic scale, by a matrix of unit rank. Furthermore, the approximation problem is formulated in terms of tropical mathematics that deals with the theory and applications of algebraic systems with idempotent addition. By using methods of tropical optimization, direct analytical solutions to the problem are derived for the case of an arbitrary positive matrix and for the case when the matrix does not contain columns (rows) with all entries missing. The results obtained allow one to find the vectors of the factor decomposition by using expressions in a parametric form which is ready for further analysis and immediate calculation. In conclusion, an example of approximate rank-one factorization of a matrix with missing entries is provided.