In this paper, we present a unified finite volume method preserving discrete maximum principle (DMP) for the conjugate heat transfer problems with general interface conditions. We prove the existence of the numerical solution and the DMP-preserving property. Numerical experiments show that the nonlinear iteration numbers of the scheme in [24] increase rapidly when the interfacial coefficients decrease to zero. In contrast, the nonlinear iteration numbers of the unified scheme do not increase when the interfacial coefficients decrease to zero, which reveals that the unified scheme is more robust than the scheme in [24]. The accuracy and DMP-preserving property of the scheme are also veried in the numerical experiments.

In this paper, ETD3-Padé and ETD4-Padé Galerkin finite element methods are proposed and analyzed for nonlinear delayed convection-diffusion-reaction equations with Dirichlet boundary conditions. An ETD-based RK is used for time integration of the corresponding equation. To overcome a well-known difficulty of numerical instability associated with the computation of the exponential operator, the Padé approach is used for such an exponential operator approximation, which in turn leads to the corresponding ETD-Padé schemes. An unconditional $L^2$ numerical stability is proved for the proposed numerical schemes, under a global Lipshitz continuity assumption. In addition, optimal rate error estimates are provided, which gives the convergence order of $O(k^{3}+h^{r})$ (ETD3-Padé) or $O(k^{4}+h^{r})$ (ETD4-Padé) in the $L^2$ norm, respectively. Numerical experiments are presented to demonstrate the robustness of the proposed numerical schemes.

In this paper, we systematically construct two classes of structure-preserving schemes with arbitrary order of accuracy for canonical Hamiltonian systems. The one class is the symplectic scheme, which contains two new families of parameterized symplectic schemes that are derived by basing on the generating function method and the symmetric composition method, respectively. Each member in these schemes is symplectic for any fixed parameter. A more general form of generating functions is introduced, which generalizes the three classical generating functions that are widely used to construct symplectic algorithms. The other class is a novel family of energy and quadratic invariants preserving schemes, which is devised by adjusting the parameter in parameterized symplectic schemes to guarantee energy conservation at each time step. The existence of the solutions of these schemes is verified. Numerical experiments demonstrate the theoretical analysis and conservation of the proposed schemes.

In this paper, we consider the generalized Nash equilibrium with shared constraints in the stochastic environment, and we call it the stochastic generalized Nash equilibrium. The stochastic variational inequalities are employed to solve this kind of problems, and the expected residual minimization model and the conditional value-at-risk formulations defined by the residual function for the stochastic variational inequalities are discussed. We show the risk for different kinds of solutions for the stochastic generalized Nash equilibrium by the conditional value-at-risk formulations. The properties of the stochastic quadratic generalized Nash equilibrium are shown. The smoothing approximations for the expected residual minimization formulation and the conditional value-at-risk formulation are employed. Moreover, we establish the gradient consistency for the measurable smoothing functions and the integrable functions under some suitable conditions, and we also analyze the properties of the formulations. Numerical results for the applications arising from the electricity market model illustrate that the solutions for the stochastic generalized Nash equilibrium given by the ERM model have good properties, such as robustness, low risk and so on.

In this paper, we apply the Wasserstein-Fisher-Rao (WFR) metric from the unbalanced optimal transport theory to the earthquake location problem. Compared with the quadratic Wasserstein ($W_2$) metric from the classical optimal transport theory, the advantage of this method is that it retains the important amplitude information as a new constraint, which avoids the problem of the degeneration of the optimization objective function near the real earthquake hypocenter and origin time. As a result, the deviation of the global minimum of the optimization objective function based on the WFR metric from the true solution can be much smaller than the results based on the $W_2$ metric when there exists strong data noise. Thus, we develop an accurate earthquake location method under strong data noise. Many numerical experiments verify our conclusions.

By employing $EQ_1^{rot}$ nonconforming finite element, the numerical approximation is presented for multi-term time-fractional mixed sub-diffusion and diffusion-wave equation on anisotropic meshes. Comparing with the multi-term time-fractional sub-diffusion equation or diffusion-wave equation, the mixed case contains a special time-space coupled derivative, which leads to many difficulties in numerical analysis. Firstly, a fully discrete scheme is established by using nonconforming finite element method (FEM) in spatial direction and L1 approximation coupled with Crank-Nicolson (L1-CN) scheme in temporal direction. Furthermore, the fully discrete scheme is proved to be unconditional stable. Besides, convergence and superclose results are derived by using the properties of $EQ_1^{rot}$ nonconforming finite element. What's more, the global superconvergence is obtained via the interpolation postprocessing technique. Finally, several numerical results are provided to demonstrate the theoretical analysis on anisotropic meshes.

This paper concerns the reconstruction of a scalar coefficient of a second-order elliptic equation in divergence form posed on a bounded domain from internal data. This problem finds applications in multi-wave imaging, greedy methods to approximate parameter-dependent elliptic problems, and image treatment with partial differential equations. We first show that  the inverse problem for smooth coefficients can be rewritten as a linear transport equation. Assuming that the coefficient is known near the boundary, we study the well-posedness of associated transport equation as well as its numerical resolution using discontinuous Galerkin method. We propose a regularized transport equation that allow us to derive rigorous convergence rates of the numerical method in terms of the order of the polynomial approximation as well as the regularization parameter. We finally provide numerical examples for the inversion assuming a lower regularity of the coefficient, and using synthetic data.

Numerical integration poses greater challenges in Galerkin meshless methods than finite element methods owing to the non-polynomial feature of meshless shape functions. The reproducing kernel gradient smoothing integration (RKGSI) is one of the optimal numerical integration techniques in Galerkin meshless methods with minimum integration points. In this paper, properties, quadrature rules and the effect of the RKGSI on meshless methods are analyzed. The existence, uniqueness and error estimates of the solution of Galerkin meshless methods under numerical integration with the RKGSI are established. A procedure on how to choose quadrature rules to recover the optimal convergence rate is presented.

We introduce a new algorithm, extended regularized dual averaging (XRDA), for solving regularized stochastic optimization problems, which generalizes the regularized dual averaging (RDA) method. The main novelty of the method is that it allows a flexible control of the backward step size. For instance, the backward step size used in RDA grows without bound, while for XRDA the backward step size can be kept bounded. We demonstrate experimentally that additional control over the backward step size can speed up the convergence of the algorithm while preserving desired properties of the iterates, such as sparsity. Theoretically, we show that the XRDA method achieves the same convergence rate as RDA for general convex objectives.

This note introduces a method for sampling Ising models with mixed boundary conditions. As an application of annealed importance sampling and the Swendsen-Wang algorithm, the method adopts a sequence of intermediate distributions that keeps the temperature fixed but turns on the boundary condition gradually. The numerical results show that the variance of the sample weights is relatively small.