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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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This is devoted to numerical simulation of time domain inverse acoustic scattering problems with a point-like scatterer, multiple point-like scatterers or normal size scatterers. Using Green’s functions and time convolution, we propose direct sampling methods to reconstruct the location of the scatterer. The methods involve only integral calculus without solving any equations and are easy to implement. Numerical experiments show the effectiveness and robustness of the methods.

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A difference mixed finite element method based on the finite element pair $((P_0×P_1),(P_0×P_1),(P_1×P_0))$$×(P_1×P_1)$ for the three dimensional Poisson equation is studied. The method combines a mixed finite element discretization based on $(P_0, P_0, P_1)×P_1$-element in $(x, y)$-plane and a finite difference discretization based on $(P_1, P_1, P_0) × P_1$-element in $z$-direction. This allows to transmit the finite element solution of the three dimensional Poisson equation in the direction $(x, y, z)$ into a series of finite element solution of the two dimensional Poisson equation in the direction $(x, y).$ Moreover, error estimates for the discrete approximation are derived. Numerical tests show the accuracy and efficiency for the method proposed.

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This paper considers a chemotaxis model coupled a stochastic Navier-Stokes equation in two-dimensional case. In the non-convex bounded domain, it is proved that the stochastic chemotaxis-Navier-Stokes system possesses at least one global martingale weak solution when the chemotactic sensitivity function $\chi$ is nonsingular. In the convex bounded domain, under the conditions of $\chi$ and per capita oxygen consumption rate $h$ are appropriately relaxed (where $\chi$ allows singularity), it is proved that the system admits a unique global mild solution. Our results generalize previously known ones.

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An accurate, efficient and robust troubled-cell indicator (TCI) is always required by the discontinuous Galerkin methods for solving the nonlinear system of hyperbolic conservation laws. In this paper, we propose a new framework of TCI on general meshes using simple statistical analysis techniques which is easy to implement and efficient to run. In the framework troubled-cell indication is performed in each local stencil. We first pick the local stencils that contain the troubled cells via the range of the normalized troubled-cell indication values, and then detect the troubled cells in each picked stencil by a simple separation approach. The new TCI framework is implemented on several types of meshes with three indication variables. Numerical experiments conducted through simulating several classical shocked flows demonstrate that the new framework can capture the locations of shock waves accurately and lead to essentially nonoscillatory solutions. The results also verify the robustness of the framework.

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A postprocessing-based a posteriori error estimates for the spectral Galerkin approximation of one-dimensional fourth-order boundary value problems are developed. Our approach begins by introducing a novel postprocessing technique aimed at enhancing the accuracy of the spectral Galerkin approximation. We prove that this postprocessing step improves the convergence rate in both $L^2$- and $H^2$-norm. Using postprocessed superconvergence results, we construct several a posteriori error estimators and prove that they are asymptotically exact as the polynomial degree increases. We further extend the postprocessing technique and error estimators to more general one-dimensional even-order equations and to multidimensional fourth-order equations. The results of numerical experiments illustrate the efficiency of the error estimators.

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In this study, we construct a relaxed second-order time-accurate scheme for the binary phase-field crystal model based on the recent paper [J. Wu, J. Yang and Z. Tan, Eng. Comput. (2023)]. This scheme can significantly improve the consistency between original and modified energies. Using the exponential SAV method, we construct the first- and second-order time-accurate schemes by using the backward Euler formula and the second-order backward difference formula, respectively. Then a new second-order numerical scheme with relaxation is developed. This new numerical scheme satisfies the energy dissipation law. Meanwhile, it is fully decoupled and efficient. Extensive numerical experiments are carried out in 2D and 3D to verify the accuracy and stability of the proposed scheme, as well as its ability to improve the consistency between the original energy and modified energy.

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The momentum-conserving staggered-grid (MCS) scheme for two-layer shallow flow is revisited here. In the previous contribution, the MCS scheme has been successfully used to simulate various steady interfaces of two-layer shallow flow in channels of varying depth and width. In this contribution, the MCS scheme is extended to accommodate a time-dependent barotropic forcing; the barotropic tidal wave is assumed to be a harmonic function. In the MCS scheme, this barotropic forcing is assessed by applying suitable boundary conditions; and its presence would affect the exchange flow. The average exchange transport was then calculated for the barotropic forces with a range of amplitudes and periods, and the results obtained show good agreement with the analytical rigid-lid theory. Next, the MCS scheme is used to study the exchange flow in the Lombok Strait. Taking into account the geographical conditions of the Lombok Strait, a steady interface was simulated, which turned out to be the case of a maximal exchange flow with two controls at Karangasem Narrows (KN) and Nusa Penida Sill (NPS). Next, due to tidal waves in the period of April-May, the average exchange flow in Lombok Strait is estimated. Our findings agree well with the existing results.

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We propose a new non-convex first-order variational model for the image super-resolution problem. The model employs a recently developed regularizer that has proven to be effective in image restoration. Due to this regularizer, the salient feature of our model lies in the fact it can construct sharp edges in those generated super-resolution images from lower-resolution ones. Moreover, it also helps suppress the staircase effect. The maximum-minimum principle is proved, which indicates that there is no need to impose hard constraints on the objective function. Alternating direction method of multipliers with spectral penalty selection (spsADMM) is utilized to minimize the associated functional. Cartoon and real gray and color images are tested to demonstrate the features of our model to show the comparison with state-of-the-art image super-resolution techniques.

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The peridynamic (PD) theory is a reformulation of the classical theory of continuum solid mechanics, which yields an integro-differential equation that does not involve spatial derivatives of the displacement field. Therefore, it is particularly suitable for the description of cracks and their evolution in materials. Due to the nonlocal property of PD models, classical numerical methods usually generate dense stiffness matrices which require $\mathscr{O}(N^3)$ computational work and $\mathscr{O}(N^2)$ memory, where $N$ is the number of spatial unknowns. In this paper, we develop a fast collocation method for a bond-based linear viscoelastic peridynamic model in two space dimensions by exploring the structure of the stiffness matrix. The method has a computational work count of $\mathscr{O}(N{\rm log}N)$ per Krylov subspace iteration and $\mathscr{O}(N)$ memory requirements. In our model, the peridynamic neighborhood is a two-dimensional circular domain, so that one has to deal with the intersection of such circles and rectangular meshes used in the collocation method. We employ an inscribed polygon to approximate the peridynamic neighborhood. This improves the numerical accuracy of solution and reduces programming complexity. Numerical results show the utility of the proposed method.

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