In this paper, we propose a positivity-preserving finite element method for solving the three-dimensional quantum drift-diffusion model. The model consists of five nonlinear elliptic equations, and two of them describe quantum corrections for quasi-Fermi levels. We propose an interpolated-exponential finite element (IEFE) method for solving the two quantum-correction equations. The IEFE method always yields positive carrier densities and preserves the positivity of second-order differential operators in the Newton linearization of quantum-correction equations. Moreover, we solve the two continuity equations with the edge-averaged finite element (EAFE) method to reduce numerical oscillations of quasi-Fermi levels. The Poisson equation of electrical potential is solved with standard Lagrangian finite elements. We prove the existence of solution to the nonlinear discrete problem by using a fixed-point iteration and solving the minimum problem of a new discrete functional. A Newton method is proposed to solve the nonlinear discrete problem. Numerical experiments for a three-dimensional nano-scale FinFET device show that the Newton method is robust for source-to-gate bias voltages up to 9V and source-to-drain bias voltages up to 10V.

The main objective of this paper is to present an efficient structure-preserving scheme, which is based on the idea of the scalar auxiliary variable approach, for solving the twodimensional space-fractional nonlinear Schrödinger equation. First, we reformulate the equation as an canonical Hamiltonian system, and obtain a new equivalent system via introducing a scalar variable. Then, we construct a semi-discrete energy-preserving scheme by using the Fourier pseudo-spectral method to discretize the equivalent system in space direction. After that, applying the Crank-Nicolson method on the temporal direction gives a linearly-implicit scheme in the fully-discrete version. As expected, the proposed scheme can preserve the energy exactly and more efficient in the sense that only decoupled equations with constant coefficients need to be solved at each time step. Finally, numerical experiments are provided to demonstrate the efficiency and conservation of the scheme.

In this paper, we first present the optimal error estimates of the semi-discrete ultra-weak discontinuous Galerkin method for solving one-dimensional linear convection-diffusion equations. Then, coupling with a kind of Runge-Kutta type implicit-explicit time discretization which treats the convection term explicitly and the diffusion term implicitly, we analyze the stability and error estimates of the corresponding fully discrete schemes. The fully discrete schemes are proved to be stable if the time-step τ ≤ τ₀, where τ₀ is a constant independent of the mesh-size h. Furthermore, by the aid of a special projection and a careful estimate for the convection term, the optimal error estimate is also obtained for the third order fully discrete scheme. Numerical experiments are displayed to verify the theoretical results.

This paper concerns the convex optimal control problem governed by multiscale elliptic equations with arbitrarily rough L^∞ coefficients, which has not only complex coupling between nonseparable scales and nonlinearity, but also important applications in composite materials and geophysics. We use one of the recently developed numerical homogenization techniques, the so-called Rough Polyharmonic Splines (RPS) and its generalization (GRPS) for the efficient resolution of the elliptic operator on the coarse scale. Those methods have optimal convergence rate which do not rely on the regularity of the coefficients nor the concepts of scale-separation or periodicity. As the iterative solution of the nonlinearly coupled OCP-OPT formulation for the optimal control problem requires solving the corresponding (state and co-state) multiscale elliptic equations many times with different right hand sides, numerical homogenization approach only requires one-time pre-computation on the fine scale and the following iterations can be done with computational cost proportional to coarse degrees of freedom. Numerical experiments are presented to validate the theoretical analysis.

In this article, we propose a new finite element space Λ_h for the expanded mixed finite element method (EMFEM) for second-order elliptic problems to guarantee its computing capability and reduce the computation cost. The new finite element space Λ_h is designed in such a way that the strong requirement V_h ⊂ Λ_h in [9] is weakened to {v_h ∈ V_h; divv_h=0} ⊂ Λ_h so that it needs fewer degrees of freedom than its classical counterpart. Furthermore, the new Λ_h coupled with the Raviart-Thomas space satisfies the inf-sup condition, which is crucial to the computation of mixed methods for its close relation to the behavior of the smallest nonzero eigenvalue of the stiff matrix, and thus the existence, uniqueness and optimal approximate capability of the EMFEM solution are proved for rectangular partitions in $\mathbb{R}^d$, d=2, 3 and for triangular partitions in $\mathbb{R}^2$. Also, the solvability of the EMFEM for triangular partition in $\mathbb{R}^3$ can be directly proved without the inf-sup condition. Numerical experiments are conducted to confirm these theoretical findings.

We use a narrow-band approach to compute harmonic maps and conformal maps for surfaces embedded in the Euclidean 3-space, using point cloud data only. Given a surface, or a point cloud approximation, we simply use the standard cubic lattice to approximate its ∈-neighborhood. Then the harmonic map of the surface can be approximated by discrete harmonic maps on lattices. The conformal map, or the surface uniformization, is achieved by minimizing the Dirichlet energy of the harmonic map while deforming the target surface of constant curvature. We propose algorithms and numerical examples for closed surfaces and topological disks. To the best of the authors' knowledge, our approach provides the first meshless method for computing harmonic maps and uniformizations of higher genus surfaces.

In this paper, we propose using the tailored finite point method (TFPM) to solve the resulting parabolic or elliptic equations when minimizing the Huber regularization based image super-resolution model using the augmented Lagrangian method (ALM). The Huber regularization based image super-resolution model can ameliorate the staircase for restored images. TFPM employs the method of weighted residuals with collocation technique, which helps get more accurate approximate solutions to the equations and reserve more details in restored images. We compare the new schemes with the Marquina-Osher model, the image super-resolution convolutional neural network (SRCNN) and the classical interpolation methods: bilinear interpolation, nearest-neighbor interpolation and bicubic interpolation. Numerical experiments are presented to demonstrate that with the new schemes the quality of the super-resolution images has been improved. Besides these, the existence of the minimizer of the Huber regularization based image super-resolution model and the convergence of the proposed algorithm are also established in this paper.

For fractional Volterra integro-differential equations (FVIDEs) with weakly singular kernels, this paper proposes a generalized Jacobi spectral Galerkin method. The basis functions for the provided method are selected generalized Jacobi functions (GJFs), which can be utilized as natural basis functions of spectral methods for weakly singular FVIDEs when appropriately constructed. The developed method’s spectral rate of convergence is determined using the L^∞-norm and the weighted L²-norm. Numerical results indicate the usefulness of the proposed method.

A simple criterion is studied for the first time for identifying the discrete energy dissipation of the Crank-Nicolson scheme for Maxwell's equations in a Cole-Cole dispersive medium. Several numerical formulas that approximate the time fractional derivatives are investigated based on this criterion, including the L1 formula, the fractional BDF-2, and the shifted fractional trapezoidal rule (SFTR). Detailed error analysis is provided within the framework of time domain mixed finite element methods for smooth solutions. The convergence results and discrete energy dissipation law are confirmed by numerical tests. For nonsmooth solutions, the method SFTR can still maintain the optimal convergence order at a positive time on uniform meshes. Authors believe this is the first appearance that a second-order time-stepping method can restore the optimal convergence rate for Maxwell's equations in a Cole-Cole dispersive medium regardless of the initial singularity of the solution.

Shape gradient flows are widely used in numerical shape optimization algorithms. We investigate the accuracy and effectiveness of approximate shape gradients flows for shape optimization of elliptic problems. We present convergence analysis with a priori error estimates for finite element approximations of shape gradient flows associated with a distributed or boundary expression of Eulerian derivative. Numerical examples are presented to verify theory and show that using the volume expression is effective for shape optimization with Dirichlet and Neumann boundary conditions.

In this paper, we develop an active set identification technique. By means of the active set technique, we present an active set adaptive monotone projected Barzilai-Borwein method (ASAMPBB) for solving nonnegative matrix factorization (NMF) based on the alternating nonnegative least squares framework, in which the Barzilai-Borwein (BB) step sizes can be adaptively picked to get meaningful convergence rate improvements. To get optimal step size, we take into account of the curvature information. In addition, the larger step size technique is exploited to accelerate convergence of the proposed method. The global convergence of the proposed method is analysed under mild assumption. Finally, the results of the numerical experiments on both synthetic and real-world datasets show that the proposed method is effective.

Stability and global error bounds are studied for a class of stepsize-dependent linear multistep methods for nonlinear evolution equations governed by ω-dissipative vector fields in Banach space. To break through the order barrier p ≤ 1 of unconditionally contractive linear multistep methods for dissipative systems, strongly dissipative systems are introduced. By employing the error growth function of the methods, new contractivity and convergence results of stepsize-dependent linear multistep methods on infinite integration intervals are provided for strictly dissipative systems (ω＜0) and strongly dissipative systems. Some applications of the main results to several linear multistep methods, including the trapezoidal rule, are supplied. The theoretical results are also illustrated by a set of numerical experiments.

The Peaceman-Rachford splitting method is efficient for minimizing a convex optimization problem with a separable objective function and linear constraints. However, its convergence was not guaranteed without extra requirements. He et al. (SIAM J. Optim. 24: 1011 - 1040, 2014) proved the convergence of a strictly contractive Peaceman-Rachford splitting method by employing a suitable underdetermined relaxation factor. In this paper, we further extend the so-called strictly contractive Peaceman-Rachford splitting method by using two different relaxation factors. Besides, motivated by the recent advances on the ADMM type method with indefinite proximal terms, we employ the indefinite proximal term in the strictly contractive Peaceman-Rachford splitting method. We show that the proposed indefinite-proximal strictly contractive Peaceman-Rachford splitting method is convergent and also prove the o(1/t) convergence rate in the nonergodic sense. The numerical tests on the l₁ regularized least square problem demonstrate the efficiency of the proposed method.

In this paper, we consider two kinds of extragradient methods to solve the pseudomonotone stochastic variational inequality problem. First, we present the modified stochastic extragradient method with constant step-size (MSEGMC) and prove the convergence of it. With the strong pseudo-monotone operator and the exponentially growing sample sequences, we establish the R-linear convergence rate in terms of the mean natural residual and the oracle complexity O(1/∈). Second, we propose a modified stochastic extragradient method with adaptive step-size (MSEGMA). In addition, the step-size of MSEGMA does not depend on the Lipschitz constant and without any line-search procedure. Finally, we use some numerical experiments to verify the effectiveness of the two algorithms.

In this paper, a multirate time iterative scheme with multiphysics finite element method is proposed and analyzed for the nonlinear poroelasticity model. The original problem is reformulated into a generalized nonlinear Stokes problem coupled with a diffusion problem of a pseudo pressure field by a new multiphysics approach. A multiphysics finite element method is adopted for the spatial discretization, and the generalized nonlinear Stokes problem is solved in a coarse time step and the diffusion problem is solved in a finer time step. The proposed algorithm is a decoupled algorithm, which is easily implemented in computation and reduces greatly computation cost. The stability analysis and the convergence analysis for the multirate iterative scheme with multiphysics finite element method are given. Some numerical tests are shown to demonstrate and validate the analysis results.

This work is concerned with the convergence and stability of the truncated EulerMaruyama (EM) method for super-linear stochastic differential delay equations (SDDEs) with time-variable delay and Poisson jumps. By constructing appropriate truncated functions to control the super-linear growth of the original coefficients, we present two types of the truncated EM method for such jump-diffusion SDDEs with time-variable delay, which is proposed to be approximated by the value taken at the nearest grid points on the left of the delayed argument. The first type is proved to have a strong convergence order which is arbitrarily close to 1/2 in mean-square sense, under the Khasminskii-type, global monotonicity with U function and polynomial growth conditions. The second type is convergent in q-th (q < 2) moment under the local Lipschitz plus generalized Khasminskii-type conditions. In addition, we show that the partially truncated EM method preserves the mean-square and H_∞ stabilities of the true solutions. Lastly, we carry out some numerical experiments to support the theoretical results.

By combination of iteration methods with the partition of unity method (PUM), some finite element parallel algorithms for the stationary incompressible magnetohydrodynamics (MHD) with different physical parameters are presented and analyzed. These algorithms are highly efficient. At first, a global solution is obtained on a coarse grid for all approaches by one of the iteration methods. By parallelized residual schemes, local corrected solutions are calculated on finer meshes with overlapping sub-domains. The subdomains can be achieved flexibly by a class of PUM. The proposed algorithm is proved to be uniformly stable and convergent. Finally, one numerical example is presented to confirm the theoretical findings.

In this paper, we study the strong convergence of a jump-adapted implicit Milstein method for a class of jump-diffusion stochastic differential equations with non-globally Lipschitz drift coefficients. Compared with the regular methods, the jump-adapted methods can significantly reduce the complexity of higher order methods, which makes them easily implementable for scenario simulation. However, due to the fact that jump-adapted time discretization is path dependent and the stepsize is not uniform, this makes the numerical analysis of jump-adapted methods much more involved, especially in the non-globally Lipschitz setting. We provide a rigorous strong convergence analysis of the considered jump-adapted implicit Milstein method by developing some novel analysis techniques and optimal rate with order one is also successfully recovered. Numerical experiments are carried out to verify the theoretical findings.

In this paper, we consider a uniformly accurate compact finite difference method to solve the quantum Zakharov system (QZS) with a dimensionless parameter 0 < ε ≤ 1, which is inversely proportional to the acoustic speed. In the subsonic limit regime, i.e., when 0 < ε ? 1, the solution of QZS propagates rapidly oscillatory initial layers in time, and this brings significant difficulties in devising numerical algorithm and establishing their error estimates, especially as 0 < ε ? 1. The solvability, the mass and energy conservation laws of the scheme are also discussed. Based on the cut-off technique and energy method, we rigorously analyze two independent error estimates for the well-prepared and ill-prepared initial data, respectively, which are uniform in both time and space for ε ∈ (0, 1] and optimal at the fourth order in space. Numerical results are reported to verify the error behavior.

Minimax optimization problems are an important class of optimization problems arising from modern machine learning and traditional research areas. While there have been many numerical algorithms for solving smooth convex-concave minimax problems, numerical algorithms for nonsmooth convex-concave minimax problems are rare. This paper aims to develop an efficient numerical algorithm for a structured nonsmooth convex-concave minimax problem. A semi-proximal point method (SPP) is proposed, in which a quadratic convex-concave function is adopted for approximating the smooth part of the objective function and semi-proximal terms are added in each subproblem. This construction enables the subproblems at each iteration are solvable and even easily solved when the semiproximal terms are cleverly chosen. We prove the global convergence of our algorithm under mild assumptions, without requiring strong convexity-concavity condition. Under the locally metrical subregularity of the solution mapping, we prove that our algorithm has the linear rate of convergence. Preliminary numerical results are reported to verify the efficiency of our algorithm.

The purpose of this paper is to verify that the computational scheme from[Heid et al., Gradient flow finite element discretizations with energy-based adaptivity for the Gross-Pitaevskii equation, J. Comput. Phys. 436 (2021)] for the numerical approximation of the ground state of the Gross-Pitaevskii equation can equally be applied for the effective approximation of excited states of Schrödinger's equation. That procedure employs an adaptive interplay of a Sobolev gradient flow iteration and a novel local mesh refinement strategy, and yields a guaranteed energy decay in each step of the algorithm. The computational tests in the present work highlight that this strategy is indeed able to approximate excited states, with (almost) optimal convergence rate with respect to the number of degrees of freedom.

In this paper, a two-grid mixed finite element method (MFEM) of implicit Backward Euler (BE) formula is presented for the fourth order time-dependent singularly perturbed Bi-wave problem for d-wave superconductors by the nonconforming EQ₁^rot element. In this approach, the original nonlinear system is solved on the coarse mesh through the Newton iteration method, and then the linear system is computed on the fine mesh with Taylor’s expansion. Based on the high accuracy results of the chosen element, the uniform superclose and superconvergent estimates in the broken H¹- norm are derived, which are independent of the negative powers of the perturbation parameter appeared in the considered problem. Numerical results illustrate that the computing cost of the proposed two-grid method is much less than that of the conventional Galerkin MFEM without loss of accuracy.

In this paper, we extend the work of Brenner and Sung [Math. Comp. 59, 321–338 (1992)] and present a regularity estimate for the elastic equations in concave domains. Based on the regularity estimate we prove that the constants in the error estimates of the nonconforming Crouzeix-Raviart element approximations for the elastic equations/eigenvalue problem are independent of Lamé constant, which means the nonconforming Crouzeix-Raviart element approximations are locking-free. We also establish two kinds of two-grid discretization schemes for the elastic eigenvalue problem, and analyze that when the mesh sizes of coarse grid and fine grid satisfy some relationship, the resulting solutions can achieve the optimal accuracy. Numerical examples are provided to show the efficiency of two-grid schemes for the elastic eigenvalue problem.

In this paper, we consider the initial-boundary value problem (IBVP) for the micropolar Naviers-Stokes equations (MNSE) and analyze a first order fully discrete mixed finite element scheme. We first establish some regularity results for the solution of MNSE, which seem to be not available in the literature. Next, we study a semi-implicit time-discrete scheme for the MNSE and prove L²-H¹ error estimates for the time discrete solution. Furthermore, certain regularity results for the time discrete solution are establishes rigorously. Based on these regularity results, we prove the unconditional L²-H¹ error estimates for the finite element solution of MNSE. Finally, some numerical examples are carried out to demonstrate both accuracy and efficiency of the fully discrete finite element scheme.

We propose and analyze a single-interval Legendre-Gauss-Radau (LGR) spectral collocation method for nonlinear second-order initial value problems of ordinary differential equations. We design an efficient iterative algorithm and prove spectral convergence for the single-interval LGR collocation method. For more effective implementation, we propose a multi-interval LGR spectral collocation scheme, which provides us great flexibility with respect to the local time steps and local approximation degrees. Moreover, we combine the multi-interval LGR collocation method in time with the Legendre-Gauss-Lobatto collocation method in space to obtain a space-time spectral collocation approximation for nonlinear second-order evolution equations. Numerical results show that the proposed methods have high accuracy and excellent long-time stability. Numerical comparison between our methods and several commonly used methods are also provided.

Implicit determinant method is an effective method for some linear eigenvalue optimization problems since it solves linear systems of equations rather than eigenpairs. In this paper, we generalize the implicit determinant method to solve an Hermitian eigenvalue optimization problem for smooth case and non-smooth case. We prove that the implicit determinant method converges locally and quadratically. Numerical experiments confirm our theoretical results and illustrate the efficiency of implicit determinant method.

In this paper, we develop and analyze a finite difference method for linear second-order stochastic boundary-value problems (SBVPs) driven by additive white noises. First we regularize the noise by the Wong-Zakai approximation and introduce a sequence of linear second-order SBVPs. We prove that the solution of the SBVP with regularized noise converges to the solution of the original SBVP with convergence order O(h) in the meansquare sense. To obtain a numerical solution, we apply the finite difference method to the stochastic BVP whose noise is piecewise constant approximation of the original noise. The approximate SBVP with regularized noise is shown to have better regularity than the original problem, which facilitates the convergence proof for the proposed scheme. Convergence analysis is presented based on the standard finite difference method for deterministic problems. More specifically, we prove that the finite difference solution converges at O(h) in the mean-square sense, when the second-order accurate three-point formulas to approximate the first and second derivatives are used. Finally, we present several numerical examples to validate the efficiency and accuracy of the proposed scheme.

In [20], a semi-implicit spectral deferred correction (SDC) method was proposed, which is efficient for highly nonlinear partial differential equations (PDEs). The semi-implicit SDC method in [20] is based on first-order time integration methods, which are corrected iteratively, with the order of accuracy increased by one for each additional iteration. In this paper, we will develop a class of semi-implicit SDC methods, which are based on second-order time integration methods and the order of accuracy are increased by two for each additional iteration. For spatial discretization, we employ the local discontinuous Galerkin (LDG) method to arrive at fully-discrete schemes, which are high-order accurate in both space and time. Numerical experiments are presented to demonstrate the accuracy, efficiency and robustness of the proposed semi-implicit SDC methods for solving complex nonlinear PDEs.

Topology optimization (TO) has developed rapidly recently. However, topology optimization with stress constraints still faces many challenges due to its highly non-linear properties which will cause inefficient computation, iterative oscillation, and convergence guarantee problems. At the same time, isogeometric analysis (IGA) is accepted by more and more researchers, and it has become one important tool in the field of topology optimization because of its high fidelity. In this paper, we focus on topology optimization with stress constraints based on isogeometric analysis to improve computation efficiency and stability. A new hybrid solver combining the alternating direction method of multipliers and the method of moving asymptotes (ADMM-MMA) is proposed to solve this problem. We first generate an initial feasible point by alternating direction method of multipliers (ADMM) in virtue of the rapid initial descent property. After that, we adopt the method of moving asymptotes (MMA) to get the final results. Several benchmark examples are used to verify the proposed method, and the results show its feasibility and effectiveness.

This note introduces the double flip move to accelerate the Swendsen-Wang algorithm for Ising models with mixed boundary conditions below the critical temperature. The double flip move consists of a geometric flip of the spin lattice followed by a spin value flip. Both symmetric and approximately symmetric models are considered. We prove the detailed balance of the double flip move and demonstrate its empirical efficiency in mixing.

In this work, we focus on the conforming and nonconforming leap-frog virtual element methods for the generalized nonlinear Schrödinger equation, and establish their unconditional stability and optimal error estimates. By constructing a time-discrete system, the error between the solutions of the continuous model and the numerical scheme is separated into the temporal error and the spatial error, which makes the spatial error τ-independent. The inverse inequalities in the existing conforming and new constructed nonconforming virtual element spaces are utilized to derive the L^∞-norm uniform boundedness of numerical solutions without any restrictions on time-space step ratio, and then unconditionally optimal error estimates of the numerical schemes are obtained naturally. What needs to be emphasized is that if we use the pre-existing nonconforming virtual elements, there is no way to derive the L^∞-norm uniform boundedness of the functions in the nonconforming virtual element spaces so as to be hard to get the corresponding inverse inequalities. Finally, several numerical examples are reported to confirm our theoretical results.

In this paper, we consider the Cahn-Hilliard-Hele-Shaw (CHHS) system with the dynamic boundary conditions, in which both the bulk and surface energy parts play important roles. The scalar auxiliary variable approach is introduced for the physical system; the mass conservation and energy dissipation is proved for the CHHS system. Subsequently, a fully discrete SAV finite element scheme is proposed, with the mass conservation and energy dissipation laws established at a theoretical level. In addition, the convergence analysis and error estimate is provided for the proposed SAV numerical scheme.

This paper proposes an interior-point technique for detecting the nondominated points of multi-objective optimization problems using the direction-based cone method. Cone method decomposes the multi-objective optimization problems into a set of single-objective optimization problems. For this set of problems, parametric perturbed KKT conditions are derived. Subsequently, an interior point technique is developed to solve the parametric perturbed KKT conditions. A differentiable merit function is also proposed whose stationary point satisfies the KKT conditions. Under some mild assumptions, the proposed algorithm is shown to be globally convergent. Numerical results of unconstrained and constrained multi-objective optimization test problems are presented. Also, three performance metrics (modified generational distance, hypervolume, inverted generational distance) are used on some test problems to investigate the efficiency of the proposed algorithm. We also compare the results of the proposed algorithm with the results of some other existing popular methods.

In recent years, the nuclear norm minimization (NNM) as a convex relaxation of the rank minimization has attracted great research interest. By assigning different weights to singular values, the weighted nuclear norm minimization (WNNM) has been utilized in many applications. However, most of the work on WNNM is combined with the l²-data-fidelity term, which is under additive Gaussian noise assumption. In this paper, we introduce the L1-WNNM model, which incorporates the l¹-data-fidelity term and the regularization from WNNM. We apply the alternating direction method of multipliers (ADMM) to solve the non-convex minimization problem in this model. We exploit the low rank prior on the patch matrices extracted based on the image non-local self-similarity and apply the L1-WNNM model on patch matrices to restore the image corrupted by impulse noise. Numerical results show that our method can effectively remove impulse noise.

Neural networks are powerful tools for approximating high dimensional data that have been used in many contexts, including solution of partial differential equations (PDEs). We describe a solver for multiscale fully nonlinear elliptic equations that makes use of domain decomposition, an accelerated Schwarz framework, and two-layer neural networks to approximate the boundary-to-boundary map for the subdomains, which is the key step in the Schwarz procedure. Conventionally, the boundary-to-boundary map requires solution of boundary-value elliptic problems on each subdomain. By leveraging the compressibility of multiscale problems, our approach trains the neural network offline to serve as a surrogate for the usual implementation of the boundary-to-boundary map. Our method is applied to a multiscale semilinear elliptic equation and a multiscale p-Laplace equation. In both cases we demonstrate significant improvement in efficiency as well as good accuracy and generalization performance.

This paper aims to construct and analyze the conforming and nonconforming virtual element methods for a class of fourth order nonlinear Schrödinger equations with trapped term. We mainly consider three types of virtual elements, including H² conforming virtual element, C⁰ nonconforming virtual element and Morley-type nonconforming virtual element. The fully discrete schemes are constructed by virtue of virtual element methods in space and modified Crank-Nicolson method in time. We prove the mass and energy conservation, the boundedness and the unique solvability of the fully discrete schemes. After introducing a new type of the Ritz projection, the optimal and unconditional error estimates for the fully discrete schemes are presented and proved. Finally, two numerical examples are investigated to confirm our theoretical analysis.

We treat infinite horizon optimal control problems by solving the associated stationary Bellman equation numerically to compute the value function and an optimal feedback law. The dynamical systems under consideration are spatial discretizations of non linear parabolic partial differential equations (PDE), which means that the Bellman equation suffers from the curse of dimensionality. Its non linearity is handled by the Policy Iteration algorithm, where the problem is reduced to a sequence of linear equations, which remain the computational bottleneck due to their high dimensions. We reformulate the linearized Bellman equations via the Koopman operator into an operator equation, that is solved using a minimal residual method. Using the Koopman operator we identify a preconditioner for operator equation, which deems essential in our numerical tests. To overcome computational infeasability we use low rank hierarchical tensor product approximation/tree-based tensor formats, in particular tensor trains (TT tensors) and multi-polynomials, together with high-dimensional quadrature, e.g. Monte-Carlo. By controlling a destabilized version of viscous Burgers and a diffusion equation with unstable reaction term numerical evidence is given.

In this paper, we propose a novel Hermite weighted essentially non-oscillatory (HWENO) fast sweeping method to solve the static Hamilton-Jacobi equations efficiently. During the HWENO reconstruction procedure, the proposed method is built upon a new finite difference fifth order HWENO scheme involving one big stencil and two small stencils. However, one major novelty and difference from the traditional HWENO framework lies in the fact that, we do not need to introduce and solve any additional equations to update the derivatives of the unknown function φ. Instead, we use the current φ and the old spatial derivative of φ to update them. The traditional HWENO fast sweeping method is also introduced in this paper for comparison, where additional equations governing the spatial derivatives of φ are introduced. The novel HWENO fast sweeping methods are shown to yield great savings in computational time, which improves the computational efficiency of the traditional HWENO scheme. In addition, a hybrid strategy is also introduced to further reduce computational costs. Extensive numerical experiments are provided to validate the accuracy and efficiency of the proposed approaches.

We present Alikhanov linearized Galerkin methods for solving the nonlinear time fractional Schrödinger equations. Unconditionally optimal estimates of the fully-discrete scheme are obtained by using the fractional time-spatial splitting argument. The convergence results indicate that the error estimates hold without any spatial-temporal stepsize restrictions. Numerical experiments are done to verify the theoretical results.

We propose a deep learning based discontinuous Galerkin method (D2GM) to solve hyperbolic equations with discontinuous solutions and random uncertainties. The main computational challenges for such problems include discontinuities of the solutions and the curse of dimensionality due to uncertainties. Deep learning techniques have been favored for high-dimensional problems but face difficulties when the solution is not smooth, thus have so far been mainly used for viscous hyperbolic system that admits only smooth solutions. We alleviate this difficulty by setting up the loss function using discrete shock capturing schemes–the discontinous Galerkin method as an example–since the solutions are smooth in the discrete space. The convergence of D2GM is established via the Lax equivalence theorem kind of argument. The high-dimensional random space is handled by the Monte-Carlo method. Such a setup makes the D2GM approximate high-dimensional functions over the random space with satisfactory accuracy at reasonable cost. The D2GM is found numerically to be first-order and second-order accurate for (stochastic) linear conservation law with smooth solutions using piecewise constant and piecewise linear basis functions, respectively. Numerous examples are given to verify the efficiency and the robustness of D2GM with the dimensionality of random variables up to 200 for (stochastic) linear conservation law and (stochastic) Burgers’ equation.