Computational and Applied Mathematics Seminar
Mondays at 4:10 p.m. in 401 Carver
Past Talks Spring 2020
01/15 Carve 401
Energy Stable and Positivity Preserving Scheme for the Quantum Diffusion Equation
ABSTRACT: The quantum diffusion equation is a fourth order parabolic equation. The lack of maximum principle for this equation brings difficulties in solving it numerically while preserve the positivity of solutions. In this talk, we develop a new numerical scheme for the quantum diffusion equation in general dimensions and prove it to be energy stable and positivity-preserving. The difficulty in proving the positivity-preserving property is dealt by reformulating the scheme into an equivalent optimization problem and prove the solutions to the optimization problem cannot vanish, which is because the energy functional develops singularities at zero. We will also give some numerical examples in one and two dimensions to verify the energy stable and positivity-preserving properties.
Efficient, positive, and energy stable schemes for multi-dimensional Poisson-Nernst-Planck systems
ABSTRACT: In this talk, we present positive and energy-dissipating schemes for solving the time-dependent multi-dimensional system of Poisson-Nernst-Planck (PNP) equations. Such equations arise in the modeling of biological membrane channels and semiconductor devices. The PNP system is a strongly coupled system of nonlinear equations, also, as a gradient flow can take long time evolution to reach steady states. Hence, designing efficient and stable methods with comprehensive numerical analysis for the PNP system is highly desirable. We first reformulate the system by using Slotboom variables, such reformulation converts the drift-diffusion operator into a self-adjoint elliptic operator. The new form can be more efficiently solved and suitable for keeping the solution positivity. Our numerical schemes are based on the new formulation. The semi-implicit time discretization results in a well-posed elliptic system, which is shown to be energy dissipating and preserves solution positivity for arbitrary time steps. Our first order (in time) fully-discrete scheme preserves solution positivity and mass conservation unconditionally, and energy dissipation with only a mild O(1) time step restriction. The scheme also preserves the steady-state. We further introduce a second-order (in both time and space) scheme, which has the same computational complexity as the first-order scheme. For such a second-order scheme, we use an accuracy preserving local scaling limiter to restore solution positivity when necessary. A sequence of three-dimensional numerical tests is carried out to verify our theoretical findings.
ABSTRACT: We develop variational methods for nonlinear equations with a gradient
Communication-Efficient Network-Distributed Optimization with Differential-Coded Compressors
Jia (Kevin) Liu
ABSTRACT: Network-distributed optimization has attracted significant attention in recent years due to its ever-increasing applications. However, the classic decentralized gradient descent (DGD) algorithm is communication-inefficient for large-scale and high-dimensional network-distributed optimization problems. To address this challenge, many compressed DGD-based algorithms have been proposed. However, most of the existing works have high complexity and assume compressors with bounded noise power. To overcome these limitations, in this paper, we propose a new differential-coded compressed DGD (DC-DGD) algorithm. The key features of DC-DGD include: i) DC-DGD works with general SNR-constrained compressors, relaxing the bounded noise power assumption; ii) The differential-coded design entails the same convergence rate as the original DGD algorithm; and iii) DC-DGD has the same low-complexity structure as the original DGD due to a self-noise-reduction effect. Moreover, the above features inspire us to develop a hybrid compression scheme that offers a systematic mechanism to minimize the communication cost. Finally, we conduct extensive experiments to verify the efficacy of the proposed DC-DGD and hybrid compressor.