Welcome to Phuong Hoang's homepage

        T. T. Phuong Hoang
       Auburn University

        Location and time: Parker Hall 328, 2:00 pm - 3:00 pm
        Title: High Order Explicit Local Time-Stepping Methods For Hyperbolic Conservation Laws

Abstract: We present and analyze a general framework for constructing high order explicit local time stepping (LTS) methods for hyperbolic conservation laws. In particular, we consider the model problem discretized by Runge-Kutta discontinuous Galerkin (RKDG) methods and design LTS algorithms based on the strong stability preserving Runge-Kutta (SSP-RK) schemes that allow spatially variable time step sizes to be used for time integration in different regions of the computational domain. The proposed algorithms are of predictor-corrector type, in which the interface information along the time direction is first predicted based on the SSP-RK approximations and Taylor expansions, and then the fluxes over the region of the interface are corrected to conserve mass exactly at each time step. Following the proposed framework, we derive the corresponding LTS schemes with accuracy up to the fourth order, and prove their conservation property and nonlinear stability for the scalar conservation laws. Numerical results on various test cases are also presented to demonstrate the performance of the proposed LTS algorithms.

Location and time: Parker Hall 328, 2:00 pm - 3:00 pm
Title: Enriched Galerkin Finite Element Methods for Coupling Flow and Transport in Porous Media

Abstract: We present and analyze enriched Galerkin finite element methods (EG) to solve coupled flow and transport system in porous media such as viscosity and density-driven flows. The EG is formulated by enriching the conforming continuous Galerkin finite element method (CG) with piecewise constant functions. This approach is shown to be locally and globally conservative while keeping fewer degrees of freedom in comparison with discontinuous Galerkin finite element methods (DG). Linear solvers and dynamic mesh adaptivity techniques using entropy residual and hanging nodes will be discussed. Some numerical tests in two and three dimensions are presented to confirm our theoretical results as well as to demonstrate the advantages of the EG.

        Yingda Cheng
       Department of Mathematics, Michigan State University

       Location and time: Parker Hall 328, 2:00 pm - 3:00 pm
       Title: Sparse Grid Discontinuous Galerkin Methods for High-Dimensional Transport Equations

       Abstract: In this talk, we present sparse grid discontinuous Galerkin (DG) schemes for solving high-dimensional PDEs. We will discuss the construction of the scheme based on sparse finite element spaces built from multiwavelets, its properties and applications in kinetic transport equations, such as Vlasov equations in plasma physics.

        Braxton Osting
       Department of Mathematics, University of Utah

        Location and time: Parker Hall 328, 2:00 pm - 3:00 pm
       Title: Diffusion generated methods for target-valued maps

       Abstract: A variety of tasks in inverse problems and data analysis can be formulated as the variational problem of minimizing the Dirichlet energy of a function that takes values in a certain target set and possibly satisfies additional constraints. These additional constraints may be used to enforce fidelity to data or other structural constraints arising in the particular problem considered. I'll present diffusion generated methods for solving this problem for a wide class of target sets and prove some stability and convergence results. I’ll give examples of how these methods can be used for the geometry processing task of generating quadrilateral meshes, finding Dirichlet partitions, constructing smooth orthogonal matrix valued functions, and solving inverse problems for target-valued maps. This is joint work with Dong Wang and Ryan Viertel.

Location and time: Parker Hall 328, 2:00 pm - 3:00 pm
Title: Decentralized consensus optimization on networks with delayed and stochastic gradients

Abstract: Decentralized consensus optimization has extensive applications in many emerging big data, machine learning, and sensor network problems. In decentralized computing, nodes in a network privately hold parts of the objective function and need to collaboratively solve for the consensual optimal solution of the total objective, while they can only communicate with their immediate neighbors during updates. In real-world networks, it is often difficult and sometimes impossible to synchronize these nodes, and as a result they have to use stale and stochastic gradient information which may steer their iterates away from the optimal solution. In this talk, we focus on a decentralized consensus algorithm by taking the delays of gradients into consideration. We show that, as long as the random delays are bounded in expectation and a proper diminishing step size policy is employed, the iterates generated by this algorithm still converge to a consensual optimal solution. Convergence rates of both objective and consensus are derived. Numerical results on some synthetic optimization problems and on real seismic image reconstruction will also be presented.

        Minglei Yang
       Department of Mathematics and Statistics, Auburn University

        Location and time: Parker Hall 328, 2:00 pm - 3:00 pm
       Title: Probabilistic schemes for semi-linear nonlocal diffusion equations with application in predicting runaway electron dynamics

Abstract: The semi-linear nonlocal diffusion equations have been applied in a wide variety of applications, including, e.g., contaminant flow in groundwater, the dynamics of financial markets and plasma physics. In this work, we focus on developing and analyzing novel probabilistic numerical approaches for solving several types of semi-linear nonlocal diffusion equations in both unbounded and bounded high dimensional spaces. And we also introduce one application of initial-boundary value partial integro-differential equation problem, the approximation of the runaway probability of electrons in fusion tokamak simulation. Analysis of the approximation errors of the proposed schemes and several numerical examples will be presented to verify the accuracy and effectiveness of our approaches.

        Somak Das
       Department of Mathematics and Statistics, Auburn University

        Location and time: Parker Hall 328, 2:00 pm - 3:00 pm
       Title: Stochastic Gradient Descent and Adaptive Gradient Descent methods in control of stochastic Partial Differential Equations

Abstract: Most of our contemporary mathematical models are based on partial differential equations. However, the varied levels of randomness pose difficulties for such systems to be accurately modeled using deterministic partial differential equations. In such settings we use stochastic partial differential equations to incorporate the randomness. To determine the optimal control for the stochastic system in this project, we adopt the stochastic gradient descent algorithm. With vast data-sets being customary for training of most machine learning algorithms, the stochastic gradient descent method is one of the efficient ways to obtain the optimal control. Another class of algorithms, adaptive gradient, has also widespread applications in large scale stochastic optimizations. The algorithm adjusts its step-size at every iteration depending on the current gradient value unlike stochastic gradient where we need to re-tune the step-size manually. In this talk we show the results obtained from these algorithms.