Journal of Scientific Computing最新文献

We investigate how to solve smooth matrix optimization problems with general linear inequality constraints on the eigenvalues of a symmetric matrix. We present solution methods to obtain exact global minima for linear objective functions, i.e., (F(varvec{X}) = langle varvec{C}, varvec{X}rangle ), and perform exact projections onto the eigenvalue constraint set. Two first-order algorithms are developed to obtain first-order stationary points for general non-convex objective functions. Both methods are proven to converge sublinearly when the constraint set is convex. Numerical experiments demonstrate the applicability of both the model and the methods.

In this paper, we propose and analyze an efficient preconditioning method for the elliptic problem based on the reconstructed discontinuous approximation method. This method is originally proposed in Li et al. (J Sci Comput 80(1):268–288, 2019) that an arbitrarily high-order approximation space with one unknown per element is reconstructed by solving a local least squares fitting problem. This space can be directly used with the symmetric/nonsymmetric interior penalty discontinuous Galerkin methods. The least squares problem is modified in this paper, which allows us to establish a norm equivalence result between the reconstructed high-order space and the piecewise constant space. This property further inspires us to construct a preconditioner from the piecewise constant space. The preconditioner is shown to be optimal that the upper bound of the condition number to the preconditioned symmetric/nonsymmetric system is independent of the mesh size. In addition, we can enjoy the advantage on the efficiency of the approximation in number of degrees of freedom compared with the standard DG method. Numerical experiments are provided to demonstrate the validity of the theory and the efficiency of the proposed method.

In this paper, we propose a derivative-free Levenberg-Marquardt algorithm for nonlinear least squares problems, where the Jacobian matrices are approximated via orthogonal spherical smoothing. It is shown that the gradient models which use the approximate Jacobian matrices are probabilistically first-order accurate. The high probability complexity bound of the algorithm is also given.

In this work, we present the construction of two distinct finite element approaches to solve the porous medium equation (PME). In the first approach, we transform the PME to a log-density variable formulation and construct a continuous Galerkin method. In the second approach, we introduce additional potential and velocity variables to rewrite the PME into a system of equations, for which we construct a mixed finite element method. Both approaches are first-order accurate, mass conserving, and proved to be unconditionally energy stable for their respective energies. The mixed approach is shown to preserve positivity under a CFL condition, while a much stronger property of unconditional bound preservation is proved for the log-density approach. A novel feature of our schemes is that they can handle compactly supported initial data without the need for any perturbation techniques. Furthermore, the log-density method can handle unstructured grids in any number of dimensions, while the mixed method can handle unstructured grids in two dimensions. We present results from several numerical experiments to demonstrate these properties.

We construct a quasi-conservative alternative WENO finite difference scheme respectively coupled with the global Lax-Friedrichs (AWENO-GLF) and the contact restored Harten-Lax-van Leer approximate Riemann solver (AWENO-HLLC) for solving compressible multicomponent flows. The mass equation, the momentum equation, and the energy equation are discretized by a fully conservative AWENO-GLF or AWENO-HLLC finite difference scheme from which a consistent nonconservative discretization of the topological equation is derived according to the velocity and pressure equilibrium principle proposed by Agrall (J Comput Phys 125:150–160, 1996). We prove that, coupling with the constructed scheme, WENO interpolations with common weights for conservative variables or standard WENO interpolations with independent weights for primitive quantities can maintain velocity and pressure equilibrium. Numerical examples demonstrate that AWENO-HLLC scheme is not only less dissipative but also less oscillatory than classical WENO-GLF scheme for compressible multicomponent flows.

A numerical search approach is used to design high-order diagonally implicit Runge-Kutta (DIRK) time stepping schemes equipped with embedded error estimators, some of which have identical diagonal elements (i.e., SDIRK) and explicit first stage (i.e., ESDIRK). In each of these classes, we present new A-stable schemes of order six (the highest order of previously known A-stable DIRK-type schemes) up to order eight. For each order, we include one scheme that is only A-stable as well as several schemes that are L-stable, stiffly accurate, and/or have stage order two. The latter types require more stages, but yield better convergence rates for differential-algebraic equations (DAEs), and particularly those which have stage order two result in better accuracy for moderately stiff problems. The development of the eighth-order schemes requires, in addition to imposing A-stability, finding highly accurate numerical solutions for a system of 200 equations in over 100 variables, which is accomplished via a combination of global and local optimization strategies. The accuracy, stability, and adaptive stepsize control of the schemes are demonstrated on diverse problems.

We propose an operator-splitting scheme to approximate scalar conservation equations with stiff source terms having multiple (at least two) stable equilibrium points. The scheme combines a (reaction-free) transport substep followed by a (transport-free) reaction substep. The transport substep is approximated using the forward Euler method with continuous finite elements and graph viscosity. The reaction substep is approximated using an exponential integrator. The crucial idea of the paper is to use a mesh-dependent cutoff of the reaction time-scale in the reaction substep. We establish a bound on the entropy residual motivating the design of the scheme. We show that the proposed scheme is invariant-domain preserving under the same CFL restriction on the time step as in the nonreactive case. Numerical experiments in one and two space dimensions using linear, convex, and nonconvex fluxes with smooth and nonsmooth initial data in various regimes show that the proposed scheme is asymptotic preserving.

In this work, we present a high-order discontinuous Galerkin (DG) method for solving the one-fluid two-temperature Euler equations for non-equilibrium hydrodynamics. In order to achieve optimal order of accuracy as well as suppress potential numerical oscillations behind strong shocks, special jump terms are applied in the DG spatial discretization for the nonconservative equation of electronic internal energy. Moreover, inspired by the solution procedure of Riemann problem, we develop a new HLLC (Harten–Lax–van Leer Contact) approximate Riemann solver for the one-fluid two-temperature Euler equations and use it as a building block for the high-order discontinuous Galerkin method. Several key features of the proposed HLLC approximate Riemann solver are analyzed. Finally, we design typical test cases to numerically verify and demonstrate the performance of the proposed method.

We design a finite element method for the quad-curl problem on three dimensional Lipschitz polyhedral domains with general topology that is based on the Hodge decomposition for divergence-free vector fields. Error estimates and corroborating numerical results are presented.

High-dimensional data classification is a fundamental task in machine learning and imaging science. In this paper, we propose an efficient and versatile multi-class semi-supervised classification method for classifying high-dimensional data and unstructured point clouds. To begin with, a warm initialization is generated by using a fuzzy classification method such as the standard support vector machine or random labeling. Then an unconstraint convex variational model is proposed to purify and smooth the initialization, followed by a step which is to project the smoothed partition obtained previously to a binary partition. These steps can be repeated, with the latest result as a new initialization, to keep improving the classification quality. We show that the convex model of the smoothing step has a unique solution and can be solved by a specifically designed primal–dual algorithm whose convergence is guaranteed. We test our method and compare it with the state-of-the-art methods on several benchmark data sets. Thorough experimental results demonstrate that our method is superior in both the classification accuracy and computation speed for high-dimensional data and point clouds.