

Computational and
Applied Mathematics Seminar Fall 2020 Mondays at 4:10 p.m. in 401
Carver
Archive of
previous CAM seminars 3/2 Past Talks Spring 2020 01/15 Carve 401
CEMSE Division 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 positivitypreserving. The difficulty in proving the positivitypreserving 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 positivitypreserving properties.
02/03 Efficient, positive, and energy stable schemes for multidimensional PoissonNernstPlanck systems ABSTRACT: In this talk, we present positive and energydissipating schemes for solving the timedependent multidimensional system of PoissonNernstPlanck (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 driftdiffusion operator into a selfadjoint 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 semiimplicit time discretization results in a wellposed elliptic system, which is shown to be energy dissipating and preserves solution positivity for arbitrary time steps. Our first order (in time) fullydiscrete 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 steadystate. We further introduce a secondorder (in both time and space) scheme, which has the same computational complexity as the firstorder scheme. For such a secondorder scheme, we use an accuracy preserving local scaling limiter to restore solution positivity when necessary. A sequence of threedimensional numerical tests is carried out to verify our theoretical findings.
Li Wang ABSTRACT: We develop variational methods for nonlinear equations with a gradient 2/17
CommunicationEfficient NetworkDistributed Optimization with DifferentialCoded Compressors Jia (Kevin) Liu ABSTRACT: Networkdistributed optimization has attracted significant attention in recent years due to its everincreasing applications. However, the classic decentralized gradient descent (DGD) algorithm is communicationinefficient for largescale and highdimensional networkdistributed optimization problems. To address this challenge, many compressed DGDbased 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 differentialcoded compressed DGD (DCDGD) algorithm. The key features of DCDGD include: i) DCDGD works with general SNRconstrained compressors, relaxing the bounded noise power assumption; ii) The differentialcoded design entails the same convergence rate as the original DGD algorithm; and iii) DCDGD has the same lowcomplexity structure as the original DGD due to a selfnoisereduction 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 DCDGD and hybrid compressor. 02/24 
