Many practical problems in interdisciplinary sciences can be translated to the multivariable minimization problems of an energy function/functional in mathematics. There are two long-standing, critical problems in computational mathematics: finding the global minimum and finding the relationship between different minima. This paper mainly introduces the recently developed "solution landscape" concept and method. We will review the concept of solution landscape, saddle dynamics method for construction of solution landscape, and its applications on liquid crystals and quasicrystals.

Convolution-type nonlocal potentials are quite common and important in many science and engineering fields. Efficient and accurate evaluations of such potentials often bottleneck the real time simulations. The convolution kernel is usually singular or discontinuous at the origin and/or at the far field. The density is a smooth fast-decaying function, and is naturally well approximated by Fourier spectral method on a bounded rectangular domain, which is uniformly discretized in each spatial direction, with a nearly optimal complexity of O(N log N) that is inherited from the discrete Fast Fourier Transform. In some cases, there exists a strong spatial anisotropy in the density. Nonlocality, singularity and anisotropy are three challenges in convolution evaluation. The numerical problem is to compute the convolutions accurately and efficiently on such uniform mesh grid. In this article, we mainly review the state-of-art fast integral algorithms, including the NonUniform-FFT based method (NUFFT), Gaussian-Sum based method and Kernel Truncation method. All these methods achieve spectral accuracy with a FFT-like complexity O(N log N), and can be rewritten as a discrete convolution structure. The discrete convolution structure helps deal with strong anisotropy perfectly using a pair of FFT and inverse FFT(iFFT) on a twofold zero-padded density. Rigorous error estimates and extensive numerical results are shown to confirm the accuracy, efficiency and anisotropy performance.

In this paper, we design a numerical prediction-correction algorithm based on the finite difference method for the pricing of American options. The algorithm first predicts the free boundary condition using an explicit discretization formula, then discretizes the partial differential equation for the option price after variable substitution in an implicit formula. And we analyze the stability of this discretization formula using the Fourier method. Next, a posteriori error estimator based on the Richardson extrapolation method is introduced. This error estimator allows us to find a suitable grid where the computed solution, both the option price variable and the free boundary, verify a prefixed error tolerance. Finally, the validity, stability and convergence of the proposed algorithm are verified by designing several sets of numerical experiments and comparing them with the numerical results obtained by Fazio^[1] using an explicit discrete formula.

Alternating direction method of multipliers(ADMM) is one of the classical algorithms for solving separable convex optimization problems, but it cannot guarantee the convergence of primal iterates and its subproblems can be computationally demanding. In order to ensure convergence and improve computational efficiency, the golden ratio proximal ADMM using convex combination technique is proposed, where the convex combination factor $\psi$ is the key parameter. Based on the golden ratio proximal ADMM, we enlarge parameter $\psi$ and propose an extended golden ratio proximal ADMM(EgrpADMM). Under very mild assumptions, we establish the global convergence and $\mathcal{O}(1/N)$ ergodic sublinear convergence rate in terms of function value residual and constraint violation of EgrpADMM. Furthermore, the algorithm can achieve $\mathcal{O}(1/N^2)$ ergodic convergence when any of the separable subfunctions of the objective function is strongly convex. Finally, we demonstrate the performance of the proposed algorithms via numerical experiments.

In this paper, the numerical methods for the Riesz space fractional convection-dispersion equation is considered. By applying the weighted shifted Grünwald difference scheme to discretize the spatial fractional derivatives and the Crank-Nicolson difference scheme to discretize the temporal derivative in the Riesz space fractional convection-dispersion equation, respectively, we get the discrete results as a system of linear equations whose coefficient matrix is the sum of an identity matrix and two symmetric positive definite Toeplitz matrices. A $\tau$ preconditioner is constructed and the preconditioned conjugate gradient method is applied to solve the discrete system of linear equations. The spectral distribution of the preconditioned matrix is analyzed and the condition number of the preconditioned matrix is estimated. Numerical experiments show that the constructed preconditioner is very effective when it is combined in the preconditioned conjugate gradient method to solve the discrete system of linear equations.

A multiscale finite element method combined with an adaptively graded mesh is proposed for a convection-diffusion problem with boundary layers in the singular perturbation. The multiscale computation just operates on the coarse-scale level. We present the detailed mapping behaviors among scales through the multiscale basis functions, and provide the enriched data from microscopic scales to macroscopic ones. Further more, the graded meshes are ready for a coarse-scale discretization, and they are capable of approximating the boundary layers adaptively. Mathematical theories for the energy norm error estimate of multiscale solutions are proved to be stable and super-convergent. In numerical experiments, the accurate and efficient uniform super-convergence results are validated by this novel multiscale scheme.

The spectral conjugate gradient method is an effective algorithm for solving the unconstrained optimization problems. In this paper, firstly, the projection corrections of JJSL conjugate parameter [Jiang et al. Computational and Applied Mathematics, 2021, 40(174)] are carried out. Then, two effective spectral conjugate gradient methods are obtained by selecting the appropriate spectral parameters to ensure that the search directions are descending. Under the general assumptions, the global convergence results of two new algorithms are given by using the common inexact line search to calculate the step-size respectively. The numerical test results and the corresponding performance charts further demonstrate the numerical validity of the proposed methods.

In this paper, a new numerical method for solving a class of stochastic partial differential equations driven by Wiener process and Poisson process is derived and analyzed. By means of a splitting-up technique we decompose the stochastic partial differential equation into three simple sub-equations, and construct a splitting-up approximate solution based on three solution operators. We also investigate the convergence and convergence rate of the approximate solution. Then we discretize the spatial and temporal variables with the finite element method and the finite difference scheme, respectively. Combining with the splittingup method, we set up a fully discretized splitting-up approximate solution for solving the stochastic partial differential equations and present its convergence property. Finally, we provide some numerical experiments to verify the theoretical convergence order.

Group Lasso-type problems arise in many data mining and signal applications including signal denoising, compressive sensing and sparse linear regression. In this paper, we propose a proximal sub-gradient method that is based on proximity operator to solve group lasso and sparse group lasso. we prove that the proposed algorithm is linearly convergent without the strong convexity of the objective function. Finally, numerical examples are constructed to describe the advantage of the algorithm.

This work studies a class of nonconvex and nonsmooth nonseparable optimization. Based on the Peaceman-Rachford (PR) splitting algorithm, two linear proximal PR splitting algorithms are proposed by combining the Armijo line search technique and the linear regularization technique. Using the idea of the PR splitting algorithm decomposes the subproblem associated with the augmented Lagrangian method into two small-scale subproblems. In order to facilitate the solution of the just-mentioned subproblems and make them have good theoretical properties, the smooth terms in the objective functions are linearized, and then the regularization terms are added to each subproblem respectively. Under usual conditions, the global convergence and the iteration complexity of the two proposed methods are proved. Finally, numerical experiments show that the two proposed algorithms are effective.

Horizontal linear complementarity problem (HLCP) is one of the important generalization of the famous linear complementarity problem (LCP). The projected iterative method and the modulus-based matrix splitting iterative method are two recent proposed very effective methods for solving the HLCP. The research in this paper shows that although the deriving principles of these two methods are different, they are equivalent under certain conditions. In particular, when the parameter matrix Ω in the modulus-based matrix splitting iteration methods is taken as a specific positive diagonal matrix, the projected Jacobi method, the projected Gauss-Seidel method and the projected SOR method are equivalent to the modulus-based Jacobi iteration method, the accelerated modulus-based Gauss-Seidel iteration method and the accelerated modulus-based SOR iteration method, respectively. In addition, for the general positive diagonal matrix Ω, the equivalence of these two methods is also studied. Finally, a numerical example is presented to verify the obtained theoretical results.

In this paper, an L-BFGS algorithm for solving large-scale $\ell_1$ problems is proposed. The search direction of the algorithm on the active set is the same as that of the ISTA ^[7,9], and the search direction of L-BFGS is used on the free space set. Under some appropriate conditions, we prove that the proposed algorithm with nonmonotonic techniques converges globally. Numerical experiments show that the proposed algorithm is efficient.

When solving the optimization problem of large-scale data, the classical algorithms may be proven inefficient due to the large scale and the high dimension of the data. In this paper, an inexact trust region algorithm and an inexact adaptive cubic regularization algorithm are presented by using inexact gradient and inexact Hessian to reduce the computational cost. Under certain conditions, it is proved that the algorithms both terminate finitely, together with the analysis their computational complexities. In particular, we consider the stochastic trust region algorithm and the stochastic inexact adaptive cubic regularization algorithm, and analysis their computational complexities. At last, some numerical experiments are displayed to show that, in some cases, the proposed algorithms are more effective than the corresponding ones by using exact gradient and inexact Hessian.

This paper constructs a Euler-Maruyama (EM) method for numerically solving a class of variable-order fractional nonlinear stochastic integro-differential equations with initial value. Then, the strong stability and strong convergence of this presented EM method are strictly proved, respectively. The order of strong convergence is max{1 - α^*, 0.5}, where α^* = max{α(t)}, here α(t) is the order of variable-order Riemann-Liouville derivative. Finally, numerical tests are provided to verify the strong convergence of this EM method.

In this article, the backward Euler (BE) fully discrete finite element method of the economical finite difference streamlined diffusion (EFDSD) method for nonlinear convection-dominated diffusion equation is mainly investigated and the superconvergence of order $O(h^2+\tau)$ in $H^1$ norm is derived without the restriction between the time step $\tau$ and the mesh size $h$. Firstly, a time discrete system is established to split the error into two parts, which are the temporal error and spatial error, and with the help of mathematical induction, the regularity of the time discrete system is reduced by the temporal error. Then the finite element solution in $W^{0, \infty}$ norm is bounded by the spatial error and the unconditional superclose and global superconvergence results are gained in $H^1$ norm through interpolation post-processing technique. Lastly, a numerical example is provided to verify the correctness of the theoretical analysis and the effectiveness of the method.

We propse a class of second order explicit energy-perserving schemes for the peridynamical equations based on the E-SAV method and the composite trapezoidal formula. Rigorous error analysis is performed for semi-discrete case. The numerical results verify the energy conservation and convergence order of our method.

The Energy Stable Flux Reconstruction (ESFR) method has the property of energy stability when solving the linear convection equation. However, when solving nonlinear equations, the energy stability property requires L² projection, otherwise alising errors may lead to instability. In this paper, ESFR and over-integration are combined to construct a higher order FR scheme with good dealising effect. The energy stability of the scheme is analyzed theoretically by using the method that the integral point is larger than the solution point (Q > P). The results of using g_DG and g_SD correction functions and three different over-integration methods are compared numerically, and compared with ESFR (Q = P) which is not over-integration. Through the simulation of heterogeneous linear advection equation, isentropic euler vortex and under-resolved vortical flows, the results show that under the g_SD correction function, the ESFR (Q > P) scheme is better than the ESFR (Q = P) scheme, and the numerical error is smaller. Compared with the two correction functions, the g_DG correction function has smaller numerical error and is more stable. When the g_DG correction function is selected, the flux points with Legendre-Gauss-Lobatto(LGL) points or the flux points with Gaussian weight partition have better nonlinear stability, and the flux points with LGL points are optimal.

A two-layer nonlinear finite volume scheme for 2D unstationary diffusion equations is constructed on deformed meshes, which based on a two-point nonlinear discrete scheme of continuous diffusion flux. The scheme uses the idea of Crank-Nicolson (C-N) method to achieve second-order accuracy for time evolution. Since the transpose of the resulting algebraic system of coefficient matrix is an M-matrix, it is guaranteed that the scheme preserves positivity. The existence of discrete solution for the present scheme is proved by using Brouwer fixed-point theorem. Numerical results illustrate that the scheme has secondorder accuracy with a larger time step.

For nonlinear composite stiff impulsive differential equations, explicit Euler method is used to solve the non-stiff part, and implicit Euler method is used to solve the stiff part. Then Euler splitting method is obtained, and the stability and convergence of the method are studied. The correctness of the obtained theory is verified by numerical experiments. It also shows that this method can significantly improve the computational speed.

In this paper, we present a half-space projection algorithm for solving nonmonotone variational inequalities. Under the assumption that the underlying mapping is continuous and the solution set of its dual variational inequality is nonempty, we prove that the infinite sequence generated by the algorithm is globally convergent, and establish the convergence rate analysis under local error and Lipschitz conditions. The effectiveness and feasibility of the proposed algorithm are proved by numerical experiments.

In this paper, the scattering problem of multiple obstacles under the sea surface in twodimensional space is studied theoretically and numerically. By analyzing the characteristics of the scattering problem, using the Helmholtz equation, and combining different boundary conditions and radiation conditions, the mathematical model is established, and the uniqueness of the scattering problem is proved. Based on the potential theory and the indirect integral equation method, the integral representation of the fields in different regions and the integral boundary equation of the density function on the boundary is obtained. By introducing potential operator, the integral domain is truncated, and the operator equation on the bounded domain is obtained. For the established boundary integral equation system, the numerical scheme is constructed using the Nyström method, and the convergence of the numerical solution is proved. Finally, numerical experiments are used to verify the correctness and effectiveness of the theory. Furthermore, numerical experiments are designed to analyze the effects of different parameters on the scattering problem.

A Lagrange quadratic finite volume method scheme for solving the Stokes equation is constructed on triangular meshes in this paper. The piecewise continuous quadratic finite element space and the discontinuous linear finite element space is used as the trial space for velocity and pressure of the Stokes equation respectively, so that the discrete velocity solution of the finite volume method satisfies the local mass conservation on the macro-element triangular element, and the finite element space pair is naturally satisfied with the so-called inf-sup condition. By adopting the special dual partition and the special mapping, the finite volume method scheme for solving the Stokes equation is transformed into the corresponding finite element method. The unconditional stability (or inf-sup condition) of the finite volume method scheme (without the geometric constraints of the triangular meshes) and the optimal-order error estimates in the $\mathbf{H}^1$-norm for velocity are obtained. Finally, numerical experiments show the validity of the theoretical results and the effectiveness of the finite volume method in the numerical simulation of computational fluid dynamics.

The fast multipole method (FMM) can accelerate the iterative solver of the large dense linear equations arising from many physical problems. This article is concerned with the convergence of the FMM for three dimensional potential problems. Firstly, derive the expression of the global error, and then give a novel estimate of the error bound. Secondly, the result is applied to the adaptive octree structure, and the specific convergence order is obtained. Finally, an illustrative example is provided to test the proposed results. The method of this paper can also be used to estimate the error of the FMM for elastostatic problems and Stokes flow problems.