The strongly anisotropic Cahn-Hilliard model is considered. In particular, a highly nonlinear anisotropic surface energy makes the PDE system very challenging at both the analytic and numerical levels. In this talk, a convexity analysis is performed to the surface energy potential, and a careful estimate reveals that all its second order functional derivatives stay uniformly bounded by a global constant. In turn, a linear approximation becomes available for the surface energy part, and a detailed estimate demonstrates the corresponding energy stability. Its combination with the implicit treatment of the nonlinear double well potential terms yields a weakly nonlinear, energy stable scheme for the whole system. Moreover, with a careful application of the global bound for the second order functional derivatives, an optimal rate convergence analysis becomes available, which is the first such result in this area. Some numerical simulation results are also presented in the talk.

In this talk, we will introduce a class of total variation bounded (TVB) spectral methods for solving one dimensional scalar hyperbolic conservation laws. The main idea is to apply a TVB limiter to the existing spectral methods such that the total variation bounded. Numerical results are provided to demonstrate the capability of our proposed schemes.

This talk presents a finite element solver for linear poroelasticity on quadrilateral meshes based on the two-field model (solid displacement and fluid pressure) and its implementation on the deal.II platform. This solver combines the classical Lagrangian elements with reduced integration technique for the displacement in elasticity, the weak Galerkin elements for the pressure in Darcy flow through the implicit Euler discretization. The solver is penalty-free and has fewer unknowns compared to other existing methods. Numerical experiments will be presented to show the solver is free of nonphysical pressure oscillations. This is joint work with Dr. James Liu and Dr. Simon Tavener.

Deep learning is machine learning using neural networks with many hidden layers, and it has become a primary tool in a wide variety of practical learning tasks, such as image classification, speech recognition, driverless cars, or game intelligence. As such, there is a pressing need to provide a solid mathematical framework to analyze various aspects of deep neural networks. This work introduces the mathematical formulation of deep residual neural networks as a PDE optimal control problem. We study the wellposedness, the large time solution behavior, and the characterization of the steady states for the forward. We state and prove optimality conditions for the inverse deep learning problem, using the Hamilton-Jacobi-Bellmann equation and the Pontryagin maximum principle. This serves to establish a mathematical foundation for investigating the algorithmic and theoretical connections between optimal control and deep learning. This is joint work with Peter Markowich (KAUST).

We propose asymptotic methods that combine integral representations of the wavefield and approximations formulated by geometrical optics of the Green’s functions for solving the variable velocity wave equation. The wave is propagated as an integral with Green’s functions whose phase and amplitude terms are described as solutions of an Eikonal equation and a recurrent system of transport equations,respectively. The integral can be efficiently evaluated by the fast Fourier Transform after an appropriate low-rank approximation with Chebyshev polynomial interpolation. To truncate the infinite domain, we apply perfectly matched layers to modify the wave equation. Results and numerical experiments and presented and discussed.

A kernel-independent treecode (KITC) is presented for fast summation of pairwise particle interactions. In general, treecodes replace particle-particle interactions by particle-cluster interactions, and here we employ barycentric Lagrange interpolation at Chebyshev points to compute well-separated particle-cluster interactions. The scheme requires only kernel evaluations, is suitable for non-oscillatory kernels, and utilizes a scale-invariance property of barycentric Lagrange interpolation. For a given level of accuracy, the treecode reduces the operation count for pairwise interactions from \(O(N^2)\) to \(O(N \log N)\), where \(N\) is the number of particles in the system. The algorithm is demonstrated in serial and parallel simulations for systems of regularized Stokeslets and rotlets in 3D, and numerical results show the treecode performance in terms of error, CPU time, and memory consumption. The KITC is a relatively simple algorithm with low memory consumption, and this enables a straightforward parallelization.

The Swift-Hohenberg equation as a central nonlinear model in modern physics has a gradient flow structure. Here we introduce fully discrete discontinuous Galerkin (DG) schemes for a class of fourth order gradient flow problems, including the nonlinear Swift-Hohenberg equation, to produce free-energy-decaying discrete solutions, irrespective of the time step and the mesh size. We exploit and extend the mixed DG method introduced in [H. Liu and P. Yin, J. Sci. Comput., 77: 467--501, 2018] for the spatial discretization, and the ``Invariant Energy Quadratization" method for the time discretization. The resulting IEQ-DG algorithms are linear, thus they can be efficiently solved without resorting to any iteration method. We actually prove that these schemes are unconditionally energy stable. We present several numerical examples that support our theoretical results and illustrate the efficiency, accuracy and energy stability of our new algorithm. The numerical results on two dimensional pattern formation problems indicate that the method is able to deliver comparable patterns of high accuracy.

Population spatial movement is often modeled by random diffusion (to capture the intra-species movement) and directed diffusion (to capture the inter-species movement). In this talk, we will first introduce an S-I-R model with directed spatial movement, particularly total population moving away from crowd and susceptible population moving away from infected group. The model is a degenerated system of second-order partial differential equations, which poses challenges in designing efficient numerical methods to solve it. In the second part of this talk, we propose an efficient numerical scheme, the implicit integration factor (IIF) WENO scheme. We will discuss the order of convergence and its performance in solving this system.

We study viscous fingering formation in an immiscible three-layer Hele-Shaw problem, where two coupled interfaces spread radially. More specifically, we examine how the initial distance between the inner and outer interfaces (i.e., annulus's thickness) influence the shape of the emerging fingering patterns, mainly on its impact on finger tip-broadening and finger tip-splitting events for both the interfaces. The problem is tackled by employing a perturbative mode-coupling approach and we focus our attention on weakly nonlinear stages of the dynamics. By considering different values of viscosity ratios, we perform our analysis in three different scenarios: (a) both interfaces are unstable, (b) inner interface is unstable while the outer is stable, and (c) inner interface is stable and the outer one is unstable. Our weakly nonlinear findings indicate dramatically differences in the dynamics of the coupled interfaces as we explore these three cases of interest. Full nonlinear simulations confirm these findings.

For computing the solution of the high frequency Helmholtz equation, geometrical optics (GO) approximation provides an efficient way to achieve the goal. However, GO approximation is only valid locally near the primary source and fails in the region where caustics occur. To capture the caustics faithfully, we will compute the solution as the steady-state solution of an appropriate time-dependent Schrodinger equation. Numerical experiments will be presented to demonstrate the methods.

This talk establishes the optimal \(H^1\)-norm error estimate for a nonstandard finite element method for approximating \(H^2\) strong solutions of second order linear elliptic PDEs in non-divergence form with continuous coefficients. To circumvent the difficulty of lacking an effective duality argument for this class of PDEs, a new analysis technique is introduced; the crux of it is to establish an \(H^1\)-norm stability estimate for the finite element approximation operator which mimics a similar estimate for the underlying PDE operator recently established by the authors and its proof is based on a freezing coefficient technique and a topological argument. Moreover, both the \(H^1\)-norm stability and error estimate also hold for the linear finite element method.

In this talk, we will discuss a new class of imaging denoising models by using the \(L^p\)-norm of mean curvature of image graphs as regularizers with \( 1 < p \le 2 \). The motivation of introducing such models is to add stronger regularizations than that of the original mean curvature based image denoising model in order to remove noise more efficiently. To minimize these variational models, we develop a novel augmented Lagrangian method, and one thus just needs to solve two linear elliptic equations to find saddle points of the associated augmented Lagrangian functionals. Specifically, we linearize the nonlinear term in one of the two subproblems and minimize a proximal-like functional that can be easily treated. We prove that the minimizer of the substitute functional does reduce the value of the original subproblem under certain conditions. Numerical results are presented to illustrate the features of the proposed models and also the efficiency of the designed algorithm.

The Poisson-Nernst-Planck (PNP) system is a widely accepted model for simulation of ionic channels. We design, analyze, and numerically validate a second order unconditional positivity-preserving scheme for solving a reduced PNP system, which can well approximate the three dimensional ion channel problem. Positivity of numerical solutions is proven to hold true independent of the size of time steps and the choice of the Poisson solver. The scheme is easy to implement without resorting to any iteration method. Several numerical examples further confirm the positivity-preserving property, and demonstrate the accuracy, efficiency, and robustness of the proposed scheme.

Solving time-dependent Schrodinger equation numerically is always challenging in that the wave function is oscillatory and the domain is infinite. To deal with the problem, we present Strang operator splitting based fast huygens sweeping method. Also, the perfectly matched layer approach is used to limit the computation onto a bounded subdomain. The method is asymptotic and combines Strang operator splitting technique, the Huygen's principle and the WKBJ approximation. Implementation details will be explained with formulations, and numerical examples will be used to demonstrate the method.

Helmholtz decomposition is a method of expressing a smooth vector field as the sum of a divergence-free vector field and a curl-free vector field. This decomposition has important applications in physics, engineering, and data science. We will present two methods of numerically implementing the Helmholtz decomposition. The crux of both approaches is to numerically solve the Poisson equation with pure Neumann boundary conditions. Our first approach accomplishes this with a straightforward finite difference method. Our second approach uses the novel weak Galerkin finite element method. The two methods will be illustrated by numerical examples and compared briefly. We will also discuss Matlab and Python implementations of these methods.

We design and analyze some finite difference methods for solving the Poisson--Nernst--Planck (PNP) equations. Central-differencing discretization based finite difference methods are proposed for the Nernst--Planck (NP) with geometric-mean/harmonic-mean types of approximations. The numerical schemes, with proper time discretization, respect three desired properties that are possessed by analytical solutions: I) conservation, II) positivity of solution, and III) free-energy dissipation. The semi-implicit scheme based on the harmonic-mean approximation is further shown to preserve positivity unconditionally and have bounded condition numbers of the associated matrix. Numerical experiments validate the numerical analysis. An application to an electrochemical charging system is also studied to demonstrate the effectiveness of our schemes in solving realistic problems. This is joint work with H. Liu, D. Jie and S. Zhou.