Nonsymmetric Matrices Research Articles

Computationally-efficient locking-free isogeometric discretizations of geometrically nonlinear Kirchhoff–Love shells

Discretizations based on the Bubnov-Galerkin method and the isoparametric concept suffer from membrane locking when applied to Kirchhoff–Love shell formulations. Membrane locking causes not only smaller displacements than expected, but also large-amplitude spurious oscillations of the membrane forces. Continuous-assumed-strain (CAS) elements were originally introduced to remove membrane locking in quadratic NURBS-based discretizations of linear plane curved Kirchhoff rods (Casquero et al., CMAME, 2022). In this work, we propose CASs and CASns elements to overcome membrane locking in quadratic NURBS-based discretizations of geometrically nonlinear Kirchhoff–Love shells. CASs and CASns elements are interpolation-based assumed-strain locking treatments. The assumed strains have C0 continuity across element boundaries and different components of the membrane strains are interpolated at different interpolation points. CASs elements use the assumed strains to obtain both the physical strains and the virtual strains, which results in a global tangent matrix which is a symmetric matrix. CASns elements use the assumed strains to obtain only the physical strains, which results in a global tangent matrix which is a non-symmetric matrix. To the best of the authors’ knowledge, CASs and CASns elements are the first assumed-strain treatments to effectively overcome membrane locking in quadratic NURBS-based discretizations of geometrically nonlinear Kirchhoff–Love shells while satisfying the following important characteristics for computational efficiency: (1) No additional unknowns are added, (2) No additional systems of algebraic equations need to be solved, (3) The same elements are used to approximate the displacements and the assumed strains, (4) No additional matrix operations such as matrix inversions or matrix multiplications are needed to obtain the stiffness matrix, and (5) The nonzero pattern of the stiffness matrix is preserved. Analogously to the interpolation-based assumed-strain locking treatments for Lagrange polynomials that are widely used in commercial FEA software, the implementation of CASs and CASns elements only requires to modify the subroutine that computes the element residual vector and the element tangent matrix. The benchmark problems show that CASs and CASns elements, using either 2 × 2 or 3 × 3 Gauss–Legendre quadrature points per element, are effective locking treatments since these element types result in more accurate displacements for coarse meshes and excise the spurious oscillations of the membrane forces.

A Matrix Approach to Vertex-Degree-Based Topological Indices

A VDB (vertex-degree-based) topological index over a set of digraphs H is a function φ:H→R, defined for each H∈H as φH=12∑uv∈Eφdu+dv−, where E is the arc set of H, du+ and dv− denote the out-degree and in-degree of vertices u and v respectively, and φij=f(i,j) for an appropriate real symmetric bivariate function f. It is our goal in this article to introduce a new approach where we base the concept of VDB topological index on the space of real matrices instead of the space of symmetric real functions of two variables. We represent a digraph H by the p×p matrix αH, where αHij is the number of arcs uv such that du+=i and dv−=j, and p is the maximum value of the in-degrees and out-degrees of H. By fixing a p×p matrix φ, a VDB topological index of H is defined as the trace of the matrix φTα(H). We show that this definition coincides with the previous one when φ is a symmetric matrix. This approach allows considering nonsymmetric matrices, which extends the concept of a VDB topological index to nonsymmetric bivariate functions.

Optimal quadratic full state feedback control of nonlinear affine systems

The solution of the optimal quadratic full-state feedback closed-loop control of a nonlinear affine system problem is presented and solved. This solution uses the State-Dependent Coefficient (SDC) form of a nonlinear system and the Calculus of Variations. Further, it is shown that this solution enables the computation of the value function associated with the respective Hamilton-Jacobi-Bellman equation thus satisfying the necessary and sufficient conditions for optimality. The use of the SDC form representation in the optimal solution for a nonlinear affine system results in an explicit and exact full-state feedback closed-loop control implementation as opposed to the existing open-loop solutions of the optimal control. This solution, like for the linear case, is non-causal and thus cannot be solved in real-time. The closed-loop application involves precomputing a time-varying full-state feedback matrix. The optimal feedback is a linear combination of the current state. The time-varying gain matrix is the backward solution of a non-symmetric matrix Riccati equation. Conditions for the stability of the full-state feedback are presented. This new representation of the optimal solution gives a well-understandable algorithm that enables implementation in the closed loop of the optimal feedback control of a non-linear system which is not possible with the existing optimal open loop representations.

Splitting iteration methods to solve non-symmetric algebraic Riccati matrix equation $$YAY-YB-CY+D=0$$

Splitting iteration methods to solve non-symmetric algebraic Riccati matrix equation $$YAY-YB-CY+D=0$$

Error estimates for quadrature rules based on the Arnoldi process

The Arnoldi process can be applied to inexpensively approximate matrix functionals of the form IA(f)=uTf(A)v, where A∈RN×N is a large nonsymmetric matrix, u,v∈RN, and the superscript T denotes transposition. Here f is a function such that f(A) is a well defined matrix function. The computed approximations may be regarded as quadrature rules. This paper presents a new inexpensive approach, based on an extension of quadrature rules introduced by Spalević (2007), to compute estimates of the error in quadrature rules for the approximation of IA(f). Knowledge of the quadrature error is helpful for determining the number of nodes of the quadrature rule to be used. Numerical experiments illustrate the accuracy of the computed error estimates.

A perturbed averaging operator on finite graphs

On a finite random walk {X,p(x,y)}, an averaging operator A is defined by Au(x)=∑p(x,y)u(y) and its perturbation is Aφu(x)=Au(x)−φ(x)u(x), where φ(x) is a real-valued function on the states of X. The properties of the solutions and the supersolutions of Aφu(x)=0 are studied which fall into three categories depending on the greatest eigenvalue (a term made precise) of the non-symmetric matrix representing Aφ. Relative to the operator Aφ, the Dirichlet-Poisson solution, the Green function, the Equilibrium Principle and the Condenser problem are investigated.

On the Forsythe conjecture

Forsythe formulated a conjecture about the asymptotic behavior of the restarted conjugate gradient method in 1968. We translate several of his results into modern terms, and pose an analogous version of the conjecture (originally formulated only for symmetric positive definite matrices) for symmetric and nonsymmetric matrices. Our version of the conjecture uses a two-sided or cross iteration with the given matrix and its transpose, which is based on the projection process used in the Arnoldi (or for symmetric matrices the Lanczos) algorithm. We prove several new results about the limiting behavior of this iteration, but the conjecture still remains largely open. We hope that our paper motivates further research that eventually leads to a proof of the conjecture.

Decompositions of optimal averaged Gauss quadrature rules

Optimal averaged Gauss quadrature rules provide estimates for the quadrature error in Gauss rules, as well as estimates for the error incurred when approximating matrix functionals of the form uTf(A)v with a large matrix A∈RN×N by low-rank approximations that are obtained by applying a few steps of the symmetric or nonsymmetric Lanczos processes to A; here u,v∈RN are vectors. The latter process is used when the measure associated with the Gauss quadrature rule has support in the complex plane. The symmetric Lanczos process yields a real tridiagonal matrix, whose entries determine the recursion coefficients of the monic orthogonal polynomials associated with the measure, while the nonsymmetric Lanczos process determines a nonsymmetric tridiagonal matrix, whose entries are recursion coefficients for a pair of sets of bi-orthogonal polynomials. Recently, it has been shown, by applying the results of Peherstorfer, that optimal averaged Gauss quadrature rules, which are associated with a nonnegative measure with support on the real axis, can be expressed as a weighted sum of two quadrature rules. This decomposition allows faster evaluation of optimal averaged Gauss quadrature rules than the previously available representation. The present paper provides a new self-contained proof of this decomposition that is based on linear algebra techniques. Moreover, these techniques are generalized to determine a decomposition of the optimal averaged quadrature rules that are associated with the tridiagonal matrices determined by the nonsymmetric Lanczos process. Also, the splitting of complex symmetric tridiagonal matrices is discussed. The new splittings allow faster evaluation of optimal averaged Gauss quadrature rules than the previously available representations. Computational aspects are discussed.

Enhanced domain decomposition Schwarz solution schemes for isogeometric collocation methods

Isogeometric collocation methods have been introduced as an alternative to isogeometric Galerkin formulations, aiming at improving the computational cost of simulation by reducing the cost of assembly of the corresponding matrices. However, in contrast to their Galerkin counterparts, collocation formulations result in non-symmetric matrices of much higher dimensions, for a specified level of accuracy, which require special attention when addressing large-scale simulations. Furthermore, the presence of mixed boundary conditions may lead to indefinite collocation matrices, which hinder the convergence properties of domain decomposition-based iterative solution methods of the corresponding algebraic equations. To address these shortcomings, two-level hybrid, domain decomposition-based preconditioners are proposed, related to both the basic and the enhanced collocation formulations, in conjunction with the generalized minimal residual method and Schwarz-based preconditioners. The proposed solution schemes improve the computational efficiency of the isogeometric collocation simulations and exhibit numerical scalability for an increased number of subdomains. This is attributed to the fact that the iterations are performed on the Schur complement level of the corresponding matrices, leading to stable, computationally efficient, and robust solutions of the corresponding non-symmetric algebraic equations.

A fast preconditioning iterative method for solving the discretized second-order space-fractional advection–diffusion equations

Space-fractional advection–diffusion equations (SFADEs) arise in many application areas. In this paper, we propose fast second-order numerical methods for solving the one- and two-dimensional SFADEs defined on a finite domain. The Crank–Nicolson difference scheme is utilized to discretize the temporal derivatives, the weighted and shifted Grünwald difference operators are employed to discretize the spatial fractional derivatives in SFADEs. We analyze the stability and convergence of the difference schemes by using the matrix analysis method. The coefficient matrix of the discretized system of linear equations has the structure of the sum of an identity matrix and non-symmetric Toeplitz matrices. New τ-matrix approximate preconditioners are proposed for the discretized system of linear equations for both one- and two-dimensional SFADEs, respectively. The generalized minimal residual (GMRES) methods combined with the proposed preconditioners are applied to solve the linear systems. We analyze the convergence rate of the GMRES method for solving the preconditioned linear systems. Numerical results demonstrate the effectiveness of the proposed methods.

The generalized residual cutting method and its convergence characteristics

AbstractIterative methods and especially Krylov subspace methods (KSM) are a very useful numerical tool in solving for large and sparse linear systems problems arising in science and engineering modeling. More recently, the nested loop KSM have been proposed that improve the convergence of the traditional KSM. In this article, we review the residual cutting (RC) and the generalized residual cutting (GRC) that are nested loop methods for large and sparse linear systems problems. We also show that GRC is a KSM that is equivalent to Orthomin with a variable preconditioning. We use the modified Gram–Schmidt method to derive a stable GRC algorithm. We show that GRC presents a general framework for constructing a class of “hybrid” (nested) KSM based on inner loop method selection. We conduct numerical experiments using nonsymmetric indefinite matrices from a widely used library of sparse matrices that validate the efficiency and the robustness of the proposed methods.

Energy approach to the selection of deformation pattern and active slip systems in single crystals

The recently introduced quasi-extremal energy principle for incremental non-potential problems in rate-independent plasticity is applied to select the deformation pattern and active slip systems in single crystals. The standard crystal plasticity framework with a non-symmetric slip-system interaction matrix at finite deformation is used. The incremental work criterion for the formation of deformation bands is combined with the quasi-extremal energy principle for determining the active slip systems and slip increments in the bands. In this way, the incremental energy minimization approach has been extended to the non-potential problem of deformation banding in metal single crystals. It is shown that fulfilment of the mathematical criterion for incipient deformation banding in a homogeneous crystal in the multiple-slip case under certain conditions requires non-positive determinant of the hardening moduli matrix. Numerical examples of energetically preferable patterns of deformation bands are presented for Cu and Ni single crystals.

Matrix prior for data transfer between single cell data types in latent Dirichlet allocation.

Single cell ATAC-seq (scATAC-seq) enables the mapping of regulatory elements in fine-grained cell types. Despite this advance, analysis of the resulting data is challenging, and large scale scATAC-seq data are difficult to obtain and expensive to generate. This motivates a method to leverage information from previously generated large scale scATAC-seq or scRNA-seq data to guide our analysis of new scATAC-seq datasets. We analyze scATAC-seq data using latent Dirichlet allocation (LDA), a Bayesian algorithm that was developed to model text corpora, summarizing documents as mixtures of topics defined based on the words that distinguish the documents. When applied to scATAC-seq, LDA treats cells as documents and their accessible sites as words, identifying "topics" based on the cell type-specific accessible sites in those cells. Previous work used uniform symmetric priors in LDA, but we hypothesized that nonuniform matrix priors generated from LDA models trained on existing data sets may enable improved detection of cell types in new data sets, especially if they have relatively few cells. In this work, we test this hypothesis in scATAC-seq data from whole C. elegans nematodes and SHARE-seq data from mouse skin cells. We show that nonsymmetric matrix priors for LDA improve our ability to capture cell type information from small scATAC-seq datasets.

Linearizing Toda and SVD flows on large phase spaces of matrices with real spectrum

We consider different phase spaces for the Toda flows (Flaschka, 1974; Moser, 1975) and the less familiar SVD flows (Chu, 1986; Li, 1997). For the Toda flow, we handle symmetric and non-symmetric matrices with real simple eigenvalues, possibly with a given profile. Profiles encode, for example, band matrices and Hessenberg matrices. For the SVD flow, we assume simplicity of the singular values. In all cases, an open cover is constructed, as are corresponding charts to Euclidean space. The charts linearize the flows, converting it into a linear differential system with constant coefficients and diagonal matrix. A variant construction transforms the flows into uniform straight line motion. Since limit points belong to the phase space, asymptotic behavior becomes a local issue. The constructions rely only on basic facts of linear algebra, making no use of symplectic geometry.

ADI Method for Pseudoparabolic Equation with Nonlocal Boundary Conditions

This paper deals with the numerical solution of nonlocal boundary-value problem for two-dimensional pseudoparabolic equation which arise in many physical phenomena. A three-layer alternating direction implicit method is investigated for the solution of this problem. This method generalizes Peaceman–Rachford’s ADI method for the 2D parabolic equation. The stability of the proposed method is proved in the special norm. We investigate algebraic eigenvalue problem with nonsymmetric matrices to prove this stability. Numerical results are presented.

The Shigesada–Kawasaki–Teramoto cross-diffusion system beyond detailed balance

The existence of global weak solutions to the cross-diffusion model of Shigesada, Kawasaki, and Teramoto for an arbitrary number of species is proved. The model consists of strongly coupled parabolic equations for the population densities in a bounded domain with no-flux boundary conditions, and it describes the dynamics of the segregation of the population species. The diffusion matrix is neither symmetric nor positive semidefinite. A new logarithmic entropy allows for an improved condition on the coefficients of heavily nonsymmetric diffusion matrices, without imposing the detailed-balance condition that is often assumed in the literature. Furthermore, the large-time convergence of the solutions to the constant steady state is proved by using the relative entropy associated to the logarithmic entropy.

Quantum Markov chains on the line: matrix orthogonal polynomials, spectral measures and their statistics

Inspired by the classical spectral analysis of birth-death chains using orthogonal polynomials, we study an analogous set of constructions in the context of open quantum dynamics and related walks. In such setting, block tridiagonal matrices and matrix-valued orthogonal polynomials are the natural objects to be considered. We recall the problems of the existence of a matrix of measures or weight matrix together with concrete calculations of basic statistics of the walk, such as site recurrence and first passage time probabilities, with these notions being defined in terms of a quantum trajectories formalism. The discussion concentrates on the models of quantum Markov chains, due to S. Gudder, and on the particular class of open quantum walks, due to S. Attal et al. The folding trick for birth-death chains on the integers is revisited in this setting together with applications of the matrix-valued Stieltjes transform associated with the measures, thus extending recent results on the subject. We also consider the case of non-symmetric weight matrices and explore some examples.

New matrix splitting iteration method for generalized absolute value equations

<abstract><p>In this paper, a relaxed Newton-type matrix splitting (RNMS) iteration method is proposed for solving the generalized absolute value equations, which includes the Picard method, the modified Newton-type (MN) iteration method, the shift splitting modified Newton-type (SSMN) iteration method and the Newton-based matrix splitting (NMS) iteration method. We analyze the sufficient convergence conditions of the RNMS method. Lastly, the efficiency of the RNMS method is analyzed by numerical examples involving symmetric and non-symmetric matrices.</p></abstract>

Uniqueness of the Solution of Boundary Value Problems for the Static Equations of Elasticity Theory with a Nonsymmetric Matrix of Elastic Moduli

Uniqueness of the Solution of Boundary Value Problems for the Static Equations of Elasticity Theory with a Nonsymmetric Matrix of Elastic Moduli

Distributed Cooperative Learning for Discrete-Time Strict-Feedback Multi Agent Systems Over Directed Graphs

This paper focuses on the distributed cooperative learning (DCL) problem for a class of discrete-time strict-feedback multi-agent systems under directed graphs. Compared with the previous DCL works based on undirected graphs, two main challenges lie in that the Laplacian matrix of directed graphs is nonsymmetric, and the derived weight error systems exist n-step delays. Two novel lemmas are developed in this paper to show the exponential convergence for two kinds of linear time-varying (LTV) systems with different phenomena including the nonsymmetric Laplacian matrix and time delays. Subsequently, an adaptive neural network (NN) control scheme is proposed by establishing a directed communication graph along with n-step delays weight updating law. Then, by using two novel lemmas on the extended exponential convergence of LTV systems, estimated NN weights of all agents are verified to exponentially converge to small neighbourhoods of their common optimal values if directed communication graphs are strongly connected and balanced. The stored NN weights are reused to structure learning controllers for the improved control performance of similar control tasks by the “mod” function and proper time series. A simulation comparison is shown to demonstrate the validity of the proposed DCL method.