In earlier work, two of the authors developed a high order accurate numerical boundary condition procedure for hyperbolic conservation laws, which allows the computation using high order finite difference schemes on Cartesian meshes to solve problems in arbitrary physical domains whose boundaries do not coincide with grid lines. This procedure is based on the so-called inverse Lax-Wendroff (ILW) procedure for inflow boundary conditions and high order extrapolation for outflow boundary conditions. However, the algebra of the ILW procedure is quite heavy for two dimensional (2D) hyperbolic systems, which makes it difficult to implement the procedure for order of accuracy higher than three. In this paper, we first discuss a simplified and improved implementation for this procedure, which uses the relatively complicated ILW procedure only for the evaluation of the first order normal derivatives. Fifth order WENO type extrapolation is used for all other derivatives, regardless of the direction of the local characteristics and the smoothness of the solution. This makes the implementation of a fifth order boundary treatment practical for 2D systems with source terms. For no-penetration boundary condition of compressible inviscid flows, a further simplification is discussed, in which the evaluation of the tangential derivatives involved in the ILW procedure is avoided. We test our simplified and improved boundary treatment for Euler equations with or without source terms representing chemical reactions in detonations. The results demonstrate the designed fifth order accuracy, stability, and good performance for problems involving complicated interactions between detonation/shock waves and solid boundaries.

We study a robust and efficient eigensolver for computing a few smallest positive eigenvalues of the three-dimensional Maxwell’s transmission eigenvalue problem. The discretized governing equations by the Nedelec edge element result in a large-scale quadratic eigenvalue problem (QEP) for which the spectrum contains many zero eigenvalues and the coefficient matrices consist of patterns in the matrix form XY^{−1}Z, both of which prevent existing eigenvalue solvers from being efficient. To remedy these difficulties, we rewrite the QEP as a particular nonlinear eigenvalue problem and develop a secant-type iteration, together with an indefinite locally optimal block preconditioned conjugate gradient method (LOBPCG), to sequentially compute the desired positive eigenvalues. Furthermore, we propose a novel method to solve the linear systems in each iteration of LOBPCG. Intensive numerical experiments show that our proposed method is robust, although the desired real eigenvalues are surrounded by complex eigenvalues.

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This paper focuses on coordinate update methods, which are useful for solving problems involving large or high-dimensional datasets. They decompose a problem into simple subproblems, where each updates one, or a small block of, variables while fixing others. These methods can deal with linear and nonlinear mappings, smooth and nonsmooth functions, as well as convex and nonconvex problems. In addition, they are easy to parallelize. The great performance of coordinate update methods depends on solving simple sub-problems. To derive simple subproblems for several new classes of applications, this paper systematically studies coordinate-friendly operators that perform low-cost coordinate updates. Based on the discovered coordinate friendly operators, as well as operator splitting techniques, we obtain new coordinate update algorithms for a variety of problems in machine learning, image processing, as well as sub-areas of optimization. Several problems are treated with coordinate update for the first time in history. The obtained algorithms are scalable to large instances through parallel and even asynchronous computing. We present numerical examples to illustrate how effective these algorithms are.