9:00	BREAKFAST
9:30	Omar Lakkis (University of Sussex): Saint-Venant's shallow water equations on a rainy day: Modeling, Analysis and Computations
	In flood risk assessment modeling Saint-Venant's shallow water equations (SWEs) must be coupled with other systems providing recharge (inflow and internal) from the interaction of meteorological and hydrological phenomena such as rain and groundwater flows. These terms must be ultimately modelled as stochastic processes and can be quite rough in nature; in this talk we address the deterministic problem with bounded and integrable source in space-time. The SWEs must therefore be appropriately modified to accommodate source and sink terms. While modifications of the SWEs including rain have been proposed in the past, e.g., Delestre (2010), we provide, to the best of our knowledge, the first derivation of such models starting from the Navier--Stokes equations with appropriate free and bottom surface boundary conditions taking into account the extra terms. Crucially to obtain physically meaningful momentum balance, our approach requires the introduction of novel ``friction'' terms accounting for the inertial effects of recharge. The resulting equations, which are nonlinear hyperbolic balance laws, provide a natural extension of the SWEs and consistent both in momentum and energy. They therefore respects the formal connections of conservation laws to kinetic theory. For such equations, among the shock caputring schemes, we focus on kinetic schemes by Perthame et al. (1999-2004) which were inspired by kinetic theory and can be suitably adapted to include source terms. In this talk I will discuss the model's derivation, the existence of weak solutions and show numerical simulations. (based on joint work with Philip Townsend, Mehmet Ersoy and Donatella Donatelli)
9:50	Wenbo Li (University of Maryland, College Park): Computation of Nonlocal Minimal Graphs
	In this talk, we propose and analyze a finite element discretization for the computation of nonlocal minimal graphs of order \(\sim s \in (0,1/2)\) on a bounded domain \(\Omega\). Such a Plateau problem of order \(s\) can be reinterpreted as a Dirichlet problem for a nonlocal, nonlinear, degenerate operator of order \(s + 1/2\). The natural energy space is closely related to \(W^{2s}_1(\Omega)\); we prove that our numerical scheme converges in \(W^{2r}_1(\Omega)\) for all \(r\lt s\). Moreover, we introduce a geometric notion of error that, for any pair of \(H^1\) functions, in the limit \(s \to 1/2\) recovers a weighted \(L^2\)-discrepancy between the normal vectors to their graphs. We derive error bounds with respect to this novel geometric quantity as well. In spite of performing approximations with continuous, piecewise linear, Lagrangian finite elements, the so-called stickiness phenomenon becomes apparent in the numerical experiments we present.
10:10	Deepanshu Verma (George Mason University): EXTERNAL OPTIMAL CONTROL OF FRACTIONAL PARABOLIC PDES
	In this talk, we introduce a new notion of optimal control, or source identification in inverse, problems with fractional parabolic PDEs as constraints. This new notion allows a source/control placement outside the domain where the PDE is fulfilled. We tackle the Dirichlet, the Neumann and the Robin cases. The need for these novel optimal control concepts stems from the fact that the classical PDE models only allow placing the source/control either on the boundary or in the interior where the PDE is satisfied. However, the nonlocal behavior of the fractional operator now allows placing the control in the exterior. We introduce the notions of weak and very-weak solutions to the parabolic Dirichlet problem. We present an approach on how to approximate the parabolic Dirichlet solutions by the parabolic Robin solutions (with convergence rates). A complete analysis for the Dirichlet and Robin optimal control problems has been discussed. The numerical examples confirm our theoretical findings and further illustrate the potential benefits of nonlocal models over the local ones.
10:30	Abhijit Biswas (Temple University): Infinite Time Solution of Numerical Schemes for Advection Problems
	Solving advection problems accurately for a large number of time steps plays an important role. We can think of this as a sub-problem to solve moving interface problems, time-dependent Navier-Stokes or radiative transfer. We present a method called jet scheme with nonlinear interpolation, which can produce accurate solutions over a large number of time steps.
11:20	Juan Pablo Borthagaray (University of Maryland, College Park): A structure-preserving finite element method for uniaxial nematic liquid crystals
	In the Landau-de Gennes Q-tensor model for uniaxial nematic liquid crystals with variable degree of orientation, molecule distribution is given by a rank-one tensor and its degree of orientation. The tensor field accounts for the head-to-tail symmetry of the molecules: the states of director n and -n are indistinguishable. The Euler-Lagrange equations for the minimizer contain a degenerate elliptic for the rank-one tensor, which, in particular, allows for half-integer defects to have finite energy. We present a structure-preserving discretization of the liquid crystal energy with piecewise linear finite elements that can handle the degenerate elliptic part without regularization, and show that it is consistent and stable. We prove Gamma-convergence of discrete global minimizers to continuous ones as the mesh size goes to zero. In order to compute discrete equilibrium configurations, we develop a quasi-gradient flow scheme and prove it has a strictly monotone energy decreasing property. We present simulations in two and three dimensions to illustrate the method's ability to handle non-trivial defects as well as colloidal and electric field effects.
11:40	Shuo Yang (University of Maryland, College Park): LDG method to large bending prestrained plates
	We develop a numerical method to study the large bending deformation of plates with incompatible prestrain. We apply a local discontinuous Galerkin (LDG) type approach to discretize a proper energy functional, and design a gradient flow to compute a minimizer of the discrete energy. We prove Gamma-convergence of the discrete energy. Some numerical experiments are conducted as well.
12:00	Wujun Zhang (Rutgers University): A rate of convergence of numerical optimal transport problem with quadratic cost
	The goal in optimal transportation is to transport a measure \(\mu(x)\) into a measure \(\nu(y)\) with minimal total effort with respect to a given cost function \(c(x,y)\). In recent years, optimal transport has many applications in evolutionary dynamics, statistics, and machine learning. On way to approximate the optimal transport solution is to approximate the measure \(mu\) by the convex combination of Dirac measure \(\mu_h\) on equally spaced nodal set and solve the discrete optimal transport between \(\mu_h\) and \(\nu\). If the cost function is quadratic, i.e. \(c(x,y) = |x-y|^2\), the optimal transport mapping is related to an important concept from computational geometry, namely Laguerre cells. In this talk, we study the rate of convergence of the discrete optimal mapping using tools in computational geometry, such as Brunn-Minkowski inequality. We show that the rate of convergence of the discrete mapping measured in \(W^1_1\) norm is of order \(O(h^2)\) under suitable assumptions on the regularity of the optimal mapping. We will also discuss the rate of convergence in the case that the optimal mapping is degenerate.
3:00	Shukai Du (University of Delaware): Projection-based analysis of HDG methods with reduced stabilization
	The standard HDG methods (use degree k polynomial spaces for all variables) can lose optimal convergence in some cases, and one way to recover is to use a variant (which we call HDG+ for simplicity), where the degree of polynomials approximating the primal unknown is increased by 1, and the numerical flux is redefined with a reduced stabilization function. The existing analysis of HDG+ is quite different from the projection-based analysis of the standard HDG methods, where specific projections are defined to make the analysis simple and concise. We show that with some new analytical tools, we can similarly devise projections for HDG+ methods. This enables us to recycle many existing analyses of the standard HDG methods for the analyses of HDG+, and help understand better the connections between these several variants.
3:20	Ahmed Zytoon (University of Pittsburgh): Low-order Raviart-Thomas approximations of axisymmetric Darcy flow
	We study low-order mixed finite element approximations of three-dimensional Darcy flow in axisymmetric domains and with axisymmetric data. Due to the symmetry of the problem, the original three-dimensional problem is reduced to a two-dimensional one, and therefore this reduction offers considerable less computational effort to approximate the solutions. On the other hand, the axisymmetric formulation necessitates the use of weighted function spaces and modified (singular) differential operators leading to theoretical difficulties. This is especially true for mixed finite element methods because their structure-preserving properties often do not hold in the axisymmetric setting. We derive error estimates of the direct mixed finite element method using the lowest-order RT elements. The main difficulty in the analysis is that, in contrast to the Cartesian setting, the axisymmetric divergence operator acting on the RT space is not surjective onto the space of piecewise constants. Therefore, in this sense, the method is non-conforming. The approach we take in the analysis is classical. We simply apply Strang's second lemma to obtain abstract error estimates in terms of the approximation properties of the RT space and the inconsistency (or non-conformity) of the method. We then derive several estimates of the (local) inconsistency of the method, each tailored for different regions of the domain. Using a convex combination of these estimates, and by applying a dyadic decomposition of the domain with respect to the r-variable, we show that the inconsistency of the method is almost first-order provided that the domain is convex.
3:40	Giuseppe Vacca (University of Milano Bicocca): Virtual Element Method in the Presence of Curved Edges
	In this talk we introduce the analysis of Virtual Elements in the presence of curved faces. We consider in particular the case of a fixed curved boundary in two dimensions, as it happens in the approximation of problems posed on a curved domain. We show (both theoretically and numerically) that the proposed curved VEM lead to an optimal rate of convergence, without any approximation of the boundary.
4:30	Nawaf Bou-Rabee (Rutgers University Camden): Numerical Methods for Sticky Diffusions
	Feller's boundary condition for Brownian motion extends the classical Dirichlet, Neumann and Robin conditions - corresponding to killed, reflecting and elastic Brownian motions, respectively - to second-order derivative boundary conditions which give rise to 'sticky' boundary behavior. In this talk, we revisit Brownian motions with Feller's boundary condition, and show how to capture their dynamics using a simple modification of the random walk approximation of Brownian motion. We then generalize this result to high-dimensional sticky diffusion processes. Our approximation turns out to be thousands of times faster than a penalty method. This approximation builds on ideas in "Continuous-time Random Walks for the Numerical Solution of Stochastic Differential Equations", AMS Memoir, ISBN: 978-1-4704-3181-5. The talk itself is based on a forthcoming paper with Miranda Holmes-Cerfon (Courant Institute, NYU).
4:50	Gideon Simpson (Drexel University): Sampling from Rough Energy Landscapes
	Rough energy landscapes appear in a variety of applications including disordered media and soft matter. In this work, we examine challenges to sampling from Boltzmann distributions associated with rough energy landscapes. Here, the roughness will correspond to highly oscillatory, but bounded, perturbations of a fundamentally smooth landscape. Through a combination of numerical experiments and asymptotic analysis, we demonstrate that the performance of Metropolis Adjusted Langevin Algorithm can be severely attenuated as the roughness increases. In contrast, we prove, rigorously, that Random Walk Metropolis is insensitive to such roughness. We also formulate two alternative sampling strategies that incorporate large scale features of the energy landscape, while resisting the impact of roughness; these also outperform Random Walk Metropolis. Numerical experiments on these landscapes are presented that confirm our predictions. Open analysis questions and numerical challenges are also highlighted. This is joint work with P. Plechac (Delaware).
5:10	Xiaofeng Cai (University of Delaware): A high order semi-Lagrangian discontinuous Galerkin method coupled with Runge-Kutta exponential integrators for Vlasov simulations
	In this talk, we will consider the nonlinear Vlasov-Poisson model for plasma applications. Based on a communicator-free exponential integrator proposed by Celledoni, et al. (FCGS 19, 341-352,2003), a nonlinear convection equation can be evolved as a composition of linearized transport equations. We propose to solve these linearized equations by the semi-Lagrangian discontinuous Galerkin method proposed in Cai, et al. (J. Sci. Comput. 514-542, 2017), which enjoy many advantages such as allowing for huge time-stepping size, mass conservative, no splitting error, highly accurate, and positivity-preserving. Extensive numerical experiments for some benchmarks will be presented to verify the high order accuracy of the proposed schemes in both space and time discretization, and to show their ability in resolving complex solution structures.
5:30	Muruhan Rathinam (UMBC): Homogenization Theory for Random Walks on Graphs
	We propose random walks on suitably defined graphs as a framework for finescale modeling of particle motion in an obstructed environment where the particle may have interactions with the obstructions and the mean pathlength of the particle may not be negligible in comparison with the finescale. As we allow for jump rates to be irreversible, our framework allows for the modeling of very general forms of interactions of the particle with the obstructions such as attraction, repulsion and bonding. This motivates our study of a periodic, directed and weighted graph embedded in a Euclidean space and the scaling limit of the associated continuous time random walk on the graph's nodes which jumps along the graph's edges with jump rates given by the edge weights. We show that a suitably scaled version of the process converges to a linear drift, and the case of interest to us is that of null drift. In this case, we show that a suitably rescaled process converges weakly to a Brownian motion. The diffusivity of the limiting Brownian motion can be computed by solving a set of linear algebra problems. These linear algebra problems are analogous to the unit-cell problems in the homogenization theory for PDEs. We also provide a formal asymptotic derivation of the effective diffusivity in analogy with homogenization theory for PDEs.
5:50	END