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In this paper, we use an implicit two-derivative deferred correction time discretization approach and combine it with a spatial discretization of the discontinuous Galerkin spectral element method to solve (non-)linear PDEs. The resulting numerical method is high order accurate in space and time. As the novel scheme handles two time derivatives, the spatial operator for both derivatives has to be defined. This results in an extended system matrix of the scheme. We analyze this matrix regarding possible simplifications and an efficient way to solve the arising (non-)linear system of equations. It is shown how a carefully designed preconditioner and a matrix-free approach allow for an efficient implementation and application of the novel scheme. For both, linear advection and the compressible Euler equations, up to eighth order of accuracy in time is shown. Finally, it is illustrated how the method can be used to approximate solutions to the compressible Navier-Stokes equations.

In the paper [Hainaut, D. and Colwell, D.B., A structural model for credit risk with switchingprocesses and synchronous jumps, The European Journal of Finance44(33) (4238):3262-3284],the authors exploit a synchronous-jump regime-switching model to compute the default probabilityof a publicly traded company. Here, we first generalize the proposed L\'evy model to more generalsetting of tempered stable processes recently introduced into the finance literature. Based on thesingularity of the resulting partial integro-differential operator, we propose a general frameworkbased on strictly positive-definite functions to de-singularize the operator. We then analyze anefficient meshfree collocation method based on radial basis functions to approximate the solution ofthe corresponding system of partial integro-differential equations arising from the structural creditrisk model. We show that under some regularity assumptions, our proposed method naturallyde-sinularizes the problem in the tempered stable case. Numerical results of applying the methodon some standard examples from the literature confirms the accuracy of our theoretical results andnumerical algorithm.

In this paper, we develop an oscillation free local discontinuous Galerkin (OFLDG) method for solving nonlinear degenerate parabolic equations. Following the idea of our recent work [J. Lu, Y. Liu, and C.-W. Shu, SIAM J. Numer. Anal. 59(2021), pp. 1299-1324.], we add the damping terms to the LDG scheme to control the spurious oscillations when solutions have a large gradient. The $L^2$-stability and optimal priori error estimates for the semi-discrete scheme are established. The numerical experiments demonstrate that the proposed method maintains the high-order accuracy and controls the spurious oscillations well.

In this paper, we develop a general framework to design differentially private expectation-maximization (EM) algorithms in high-dimensional latent variable models, based on the noisy iterative hard-thresholding. We derive the statistical guarantees of the proposed framework and apply it to three specific models: Gaussian mixture, mixture of regression, and regression with missing covariates. In each model, we establish the near-optimal rate of convergence with differential privacy constraints, and show the proposed algorithm is minimax rate optimal up to logarithm factors. The technical tools developed for the high-dimensional setting are then extended to the classic low-dimensional latent variable models, and we propose a near rate-optimal EM algorithm with differential privacy guarantees in this setting. Simulation studies and real data analysis are conducted to support our results.

A linearized numerical scheme is proposed to solve the nonlinear time fractional parabolic problems with time delay. The scheme is based on the standard Galerkin finite element method in the spatial direction, the fractional Crank-Nicolson method and extrapolation methods in the temporal direction. A novel discrete fractional Gr\"{o}nwall inequality is established. Thanks to the inequality, the error estimate of fully discrete scheme is obtained. Several numerical examples are provided to verify the effectiveness of the fully discrete numerical method.

Partial differential equations on manifolds have been widely studied and plays a crucial role in many subjects. In our previous work, a class of nonlocal models was introduced to approximate the Poisson equation on manifolds that embedded in high dimensional Euclid spaces with Dirichlet and Neumann boundaries. In this paper, we improve the accuracy of such model under Dirichlet boundary by adding a higher order term along a layer adjacent to the boundary. Such term is explicitly expressed by the normal derivative of solution and the mean curvature of the boundary, while the normal derivative is regarded as a variable. All the truncation errors that involve or do not involve such term have been re-analyzed and been significantly reduced. Our concentration is on the well-posedness analysis of the weak formulation corresponding to the nonlocal model and the convergence analysis to its PDE counterpart. The main result of our work is that, such manifold nonlocal model converges to the local Poisson problem in a rate of \mathcal{O}(\delta^2) in H^1 norm, where {\delta} is the parameter that denotes the range of support for the kernel of the nonlocal operators. Such convergence rate is currently optimal among all the nonlocal models according to the literature. Two numerical experiments are included to illustrate our convergence results on the other side.

Sampling algorithms based on discretizations of Stochastic Differential Equations (SDEs) compose a rich and popular subset of MCMC methods. This work provides a general framework for the non-asymptotic analysis of sampling error in 2-Wasserstein distance, which also leads to a bound of mixing time. The method applies to any consistent discretization of contractive SDEs. When applied to Langevin Monte Carlo algorithm, it establishes $\tilde{\mathcal{O}}\left( \frac{\sqrt{d}}{\epsilon} \right)$ mixing time, without warm start, under the common log-smooth and log-strongly-convex conditions, plus a growth condition on the 3rd-order derivative of the potential of target measures at infinity. This bound improves the best previously known $\tilde{\mathcal{O}}\left( \frac{d}{\epsilon} \right)$ result and is optimal (in terms of order) in both dimension $d$ and accuracy tolerance $\epsilon$ for target measures satisfying the aforementioned assumptions. Our theoretical analysis is further validated by numerical experiments.

Regula Falsi, or the method of false position, is a numerical method for finding an approximate solution to f(x) = 0 on a finite interval [a, b], where f is a real-valued continuous function on [a, b] and satisfies f(a)f(b) < 0. Previous studies proved the convergence of this method under certain assumptions about the function f, such as both the first and second derivatives of f do not change the sign on the interval [a, b]. In this paper, we remove those assumptions and prove the convergence of the method for all continuous functions.

We study the numerical error in solitary wave solutions of nonlinear dispersive wave equations. A number of existing results for discretizations of solitary wave solutions of particular equations indicate that the error grows quadratically in time for numerical methods that do not conserve energy, but grows only linearly for conservative methods. We provide numerical experiments suggesting that this result extends to a very broad class of equations and numerical methods.

In this note we prove that the version of Newton algorithm with line search we used in [2] converges quadratically.