Abstract In this paper, enlightened by the effectiveness of Anderson acceleration (AA), a well‐established technique for accelerating fixed‐point solvers, we first present the Anderson accelerating modulus‐based matrix splitting (AAMS) algorithms for a class of horizontal nonlinear complementarity problems. Then, by introducing the strong semi‐smoothness of the absolute value function, we establish the local convergence theory of the AAMS algorithms. More importantly, we provide the optimal parameter for the AAMS algorithms, which is independent of iteration and applicable to other reduced MS algorithms. Finally, the numerical experiments are conducted to clarify the efficiency and practicability of the AAMS algorithms and its optimal parameter, and the effects of two parameters (both derived from AA) of the AAMS algorithms are analyzed.

Abstract In this article, we focus on a two‐dimensional conservative steady‐state Riesz fractional diffusion problem. As is typical for problems in conservative form, we adopt a finite volume (FV)‐based discretization approach. Precisely, we use both classical FVs and the so‐called finite volume elements (FVEs). While FVEs have already been applied in the context of fractional diffusion equations, classical FVs have only been applied in first‐order discretizations. By exploiting the Toeplitz‐like structure of the resulting coefficient matrices, we perform a qualitative study of their spectrum and conditioning through their symbol, leading to the design of a second‐order FV discretization. This same information is leveraged to discuss parameter‐free symbol‐based multigrid methods for both discretizations. Tests on the approximation error and the performances of the considered solvers are given as well.

Abstract Two‐dimensional eigenvalue problems (2DEVP) equations in complex Hermitian case contain complex quadratic forms which is non‐holomorphic. In this case, standard Newton method fails to apply. An existing strategy to solve this problem is to transform them into real problems (TRN). However, this method doubles the size of equations and thus is time consuming. On the other hand, the non‐isolation of the solution set in 2DEVP also complicates the analysis. In this article, we propose a Newton type method which solves the problem caused by non‐holomophism and non‐isolation. It has locally quadratic convergence rate and is about at least twice as much efficient as TRN. We hope our ideas can provide insights for solving other problems including non‐holomorphism and non‐isolation. We apply our algorithm to calculate the distance to instability. Numerical experiments show its advantages in efficiency while keeping good convergence compared with the current state of art algorithms.

Abstract The contour integral method is a class of efficient methods for computing partial eigenvalues of polynomial eigenvalue problem (PEP). Among them, the recently developed Sakurai–Sugiura method with Rayleigh–Ritz projection (SS‐RR) method has received much attention for its effectiveness. However, the SS‐RR method may suffer from numerical instability when the coefficient matrices of the projected PEP vary widely in norm. To improve the numerical stability, we incorporate the tropical scaling technique into the SS‐RR method and establish upper bounds for the backward error of an approximate eigenpair of the original eigenvalue problem. These bounds shed light on mechanism that the tropical scaling improves the numerical stability of the original SS‐RR method. Numerical experiments show that the actual backward errors can be successfully reduced by scaling and the bounds can predict well the errors occurring before and after scaling.

Abstract Dimensionality reduction is an important technique in surrogate modeling and machine learning. In this article, we propose a supervised dimensionality reduction method, “least squares regression principal component analysis” (LSR‐PCA), applicable to both classification and regression problems. To show the efficacy of this method, we present different examples in visualization, classification, and regression problems, comparing it with several state‐of‐the‐art dimensionality reduction methods. Finally, we present a kernel version of LSR‐PCA for problems where the inputs are correlated nonlinearly. The examples demonstrate that LSR‐PCA can be a competitive dimensionality reduction method.

Abstract We introduce the tensor numerical method for solving optimal control problems that are constrained by fractional two‐ (2D) and three‐dimensional (3D) elliptic operators with variable coefficients. We solve the governing equation for the control function which includes a sum of the fractional operator and its inverse, both discretized over large 3D spacial grids. Using the diagonalization of the arising matrix‐valued functions in the eigenbasis of the one‐dimensional Sturm–Liouville operators, we construct the rank‐structured tensor approximation with controllable precision for the discretized fractional elliptic operators and the respective preconditioner. The right‐hand side in the constraining equation (the optimal design function) is supposed to be represented in a form of a low‐rank canonical tensor. Then the equation for the control function is solved in a tensor structured format by using preconditioned CG iteration with the adaptive rank truncation procedure that also ensures the accuracy of calculations, given an ‐threshold. This method reduces the numerical cost for solving the control problem to (plus the quadratic term with a small weight), which outperforms traditional approaches with complexity in the 3D case. The storage for the representation of all 3D nonlocal operators and functions involved is also estimated by . This essentially outperforms the traditional methods operating with fully populated matrices and vectors in . Numerical tests for 2D/3D control problems indicate the almost linear complexity scaling of the rank truncated preconditioned conjugate gradient iteration in the univariate grid sizen.

Abstract In this article, the rank‐1 approximation of a nonnegative tensor is considered. Mathematically, the approximation problem can be formulated as an optimization problem. The Karush–Kuhn–Tucker (KKT) point of the optimization problem can be obtained by computing the nonnegative Z‐eigenvectoryof enlarged tensor . Therefore, we propose an iterative method with prediction and correction steps for computing nonnegative Z‐eigenvectoryof enlarged tensor , called the continuation method. In the theoretical part, we show that the computation requires only flops for each iteration and the computed Z‐eigenvectoryhas nonzero component block, and hence, the KKT point can be obtained. In addition, we show that the KKT point is a local optimizer of the optimization problem. Numerical experiments are provided to support the theoretical results.

Abstract An algorithm for two‐variable rational interpolation is developed. The algorithm is suitable for interpolation cases where neither the number of interpolation points nor the final degrees of the rational interpolant are known a priori. Instead, a maximum degree for the interpolant's numerator and denominator is assumed. By testing the condition number of the interpolation system's matrix at each step, the necessary reductions are made in order to cope with nonnormality and unattainability occasions. The algorithm can be used for applications of the Evaluation–Interpolation technique in matrix manipulations, such as finding the inverse of a matrix with elements rational functions of two variables. Symbolic calculations are completely avoided, thus keeping the execution time very low even if the system size is large. Most importantly, the algorithm achieves accurate function recoveries for greater polynomial degrees than other bivariate rational interpolation methods.

Abstract In this paper, we present explicit expressions for the mixed and componentwise condition numbers of the truncated total least squares (TTLS) solution of under the genericity condition, where A is a m × n real data matrix and is a real m‐vector. Moreover, we reveal that normwise, componentwise, and mixed condition numbers for the TTLS problem can recover the previous corresponding counterparts for the total least squares (TLS) problem when the truncated level of the TTLS problem is n. When A is a structured matrix, the structured perturbations for the structured truncated TLS (STTLS) problem are investigated and the corresponding explicit expressions for the structured normwise, componentwise, and mixed condition numbers for the STTLS problem are obtained. Furthermore, the relationships between the structured and unstructured normwise, componentwise, and mixed condition numbers for the STTLS problem are studied. We devise reliable condition estimation algorithms for the TTLS problem by utilizing small‐sample statistical condition estimation techniques. The proposed condition estimation algorithms employ the singular value decomposition (SVD) of the augmented matrix to reduce the computational complexity, where both unstructured and structured normwise, mixed, and componentwise condition estimations are considered. The proposed condition estimation algorithms can be integrated into the SVD‐based direct solver for the small and medium size TTLS problem to give the error estimation for the numerical TTLS solution. Numerical experiments are reported to illustrate the reliability of the proposed condition estimation algorithms.