• Abstract

This paper is concerned with the inverse scattering problems of simultaneously determining the unknown potential and unknown source for the biharmonic wave equation. We first derive an increasing stability estimate for the inverse potential scattering problem without a priori knowledge of the source function by multi-frequency active boundary measurements. The stability estimate consists of the Lipschitz type data discrepancy and the high frequency tail of the potential function, where the latter decreases as the upper bound of the frequency increases. The key ingredients in the analysis are employing scattering theory to derive an analytic domain and resolvent estimates and an application of the quantitative analytic continuation principle. Utilizing the derived stability for the inverse potential scattering, we further prove an increasing stability estimate for the inverse source problem. The main novelty of this paper is that both the source and potential functions are unknown.

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In this paper, we study a coupled problem of Darcy’s law and Stokes equation with nonlinear slip boundary conditions. We derive a variational inequality for the coupled problem in detail. Then we introduce and analyze a weak Galerkin method to solve the coupled problem numerically. Under proper regularity assumptions, we obtain the optimal error estimate $\mathcal{O}(h)$ in newly defined $h$-norms. Finally, we give some numerical results to support the theoretical conclusions.

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We propose an ODE approach to solving multiple choice polynomial programming (MCPP) after assuming that the optimum point can be approximated by the expected value of so-called thermal equilibrium as usually did in simulated annealing. The explicit form of the feasible region and the affine property of the objective function are both fully exploited in transforming an MCPP problem into an ODE system. We also show theoretically that a local optimum of the former can be obtained from an equilibrium point of the latter. Numerical experiments on two typical combinatorial problems, MAX-$k$-CUT and the calculation of star discrepancy, demonstrate the validity of the ODE approach, and the resulting approximate solutions are of comparable quality to those obtained by the state-of-the-art heuristic algorithms but with much less cost. When compared with the numerical results obtained by using Gurobi to solve MCPP directly, our ODE approach is able to produce approximate solutions of better quality in most instances. This paper also serves as the first attempt to use a continuous algorithm for approximating the star discrepancy.

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A high-accuracy and unconditional energy stable numerical scheme for a phase field model for diblock copolymers (PF-BCP model) is developed. The PF-BCP model is reformulated into an equivalent model, which based on scaler auxiliary variable (SAV) formulation. After that a stable Runge-Kutta (RK) method and a Fourier-spectral method are applied to the SAV-reformulated PF-BCP model to discretize on the temporal and spatial dimensions respectively. The fully discretized numerical scheme is computed by fixed-point iterations. Meanwhile, the unconditional energy decay property is proved rigorously. Finally, we present the results of numerical experiments to show the accuracy and efficiency of the RK scheme used and discuss the influence of physical parameters and initial conditions on the phase separation in the simulation of the PF-BCP model. In addition, the energy decay property of the numerical solutions is verified.

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The paper focuses on the solvability and computability of the parameterized polynomial inverse eigenvalue problem (PPIEP). Employing multiparameter eigenvalue problems, we establish a sufficient solvability condition for the PPIEP. Three numerical methods are used to solve PPIEPs. The first one is the Newton method based on locally smooth $QR$-decomposition with the column pivoting and the second the Newton method based on the smallest singular value. In order to reduce the computational cost of computing the smallest singular values and the corresponding unit left and right singular vectors in each iteration, we approximate these values by using one-step inverse iterations. Subsequently, we introduce another method — viz. a Newton-like method based on the smallest singular value. Each of three methods exhibits locally quadratic convergence under appropriate conditions. Numerical examples demonstrate the effectiveness of the methods proposed.

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A relaxation two-step Newton-based matrix splitting (RTNMS) iteration method is proposed to solve the generalized absolute value equations associated with circular cones (CCGAVE). The convergence of the RTNMS iteration method is investigated under suitable conditions. Numerical results illustrate that the RTNMS iteration method is feasible and effective for solving the CCGAVE. Moreover, some sufficient conditions for the unique solvability of the CCGAVE are given.

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This article is concerned with a matrix splitting preconditioning technique with two selective relaxations and algebraic multigrid subsolves for $(G+2)×(G+2)$ block-structured sparse linear systems derived from the three-dimensional flux-limited multi-group radiation diffusion equations, where $G$ is the number of photon energy groups. We introduce an easy-to-implement algebraic selection strategy for the sole contributing parameter, report a spectral analysis and investigate the degree of the minimal polynomial of its left and right preconditioned matrices, and discuss its sequential practical implementation together with the two-level parallelization. Experiments are run with the representative real-world unstructured capsule implosion test cases and it is found that the numerical robustness, computational efficiency and parallel scalability of the proposed preconditioner evaluated on the Tianhe-2A supercomputer with up to 2,816 processor cores are superior to some existing popular monolithic and block preconditioning approaches.

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Resorting to the discrete zero-curvature equation and the Lenard recursion equations, a hierarchy of integrable semi-discrete nonlinear evolution equations is derived from a $3×3$ matrix spectral problem with three potentials. Based on the characteristic polynomial of the Lax matrix for the hierarchy, a trigonal curve is introduced, and the properties of the corresponding three-sheeted Riemann surface are studied, including the genus, three kinds of Abelian differentials, Riemann theta functions. The asymptotic properties of the Baker-Akhiezer function and fundamental meromorphic functions defined on the trigonal curve are analyzed with the established theory of trigonal curves. As a result, finite genus solutions of the whole integrable semi-discrete nonlinear evolution hierarchy are obtained.

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