|Petr Plechac (U of Delaware):
On the accuracy of excited state dynamics
|I will discuss numerical analysis issues that arise in approximations of observables in quantum systems described by many-body Schrödinger equation. I will present quantitative error estimates for molecular dynamics observables compared with observables determined by the time-independent Schrödinger equation, including the case with crossing or nearly crossing electron potential surfaces that can yield large errors. The derivation combines mathematical stability analysis of eigenvalue problems with quantitative numerical Ehrenfest molecular dynamics computations of perturbations. More precisely, we use Egorov's theorem to derive error estimates for time-independent Schrödinger observables for nuclei-electron systems approximated by Born-Oppenheimer molecular dynamics, based on stability analysis of a perturbed eigenvalue problem to estimate the probability to be in an excited state from perturbations in Landau-Zener like dynamic transition probabilities. This is a joint work with Håkon Hoel (KAUST), Ashraful Kadir (KTH), Mattias Sandberg (KTH) and Anders Szepessy (KTH).
|Claudio Torres (George Mason University):
Numerics behind a vertex code for modeling material coarsening
|Polycrystalline materials are extremely important for modern technological applications. Yet, understanding their behavior during coarsening remains a challenge, as none of the existing models are able to capture all of the statistical properties generated by experiments. One commonly used 2D model of microstructure evolution (vertex model) represents a grain boundary network by a collection of straight lines meeting at triple junctions. In this study we will investigate numerical and analytical features of this class of models and reveal sensitivities of their predictions to various modeling and numerical characteristics. The results will be put in detailed comparison with curvature-driven growth, level set models as well as experimental data.
|Ignacio Tomas (U of Maryland, College Park):
The micropolar Navier-Stokes equations: numerical analysis and prospective applications for ferrofluids
|The Micropolar Navier-Stokes equations (MNSE) is a system of partial differential equations that describes the motion of a fluid where the microconstituents have both translational and rotational degrees of freedom. It encompasses the laws of conservation mass, linear and angular momenta. We first outline the derivation of the MNSE from continuum mechanics and their extension to ferrofluids along with some prospective applications. Then, we propose and analyze a first order semi-implicit fully-discrete scheme for the unsteady MNSE, which decouples the computation of the linear and angular velocities. The scheme is unconditionally stable and delivers optimal convergence rates under assumptions analogous to those used for the classical Navier-Stokes equations. Finally, we point the main challenges behind the devise of consistent and stable numerical schemes for the equations of Ferrofluids.
|Roman Andreev (U of Maryland, College Park):
Stable space-time Petrov-Galerkin discretization of parabolic evolution equations
|In view of applications in e.g. control of plasma in tokamaks and massively parallel computations of time-dependent problems, space-time compressive discretizations of parabolic evolution equations are of increasing interest. In this talk we discuss space-time (sparse) tensor product simultaneous Petrov-Galerkin discretizations of parabolic evolution equations, and propose efficient preconditioners for the iterative solution of the resulting single linear system of equations. Therein, space-time stability of the discretization, i.e., the validity of the discrete inf-sup condition with respect to suitable space-time norms uniformly in the discretization parameters, is essential. Viewing the Crank-Nicolson time-stepping scheme as a space-time Petrov-Galerkin discretization, we show that it is conditionally space-time stable. This motivates a general minimal residual Petrov-Galerkin discretization framework along with space-time stable families of trial and test spaces of (sparse) tensor product type, resulting in space-time compressive discretization algorithms. A basic Matlab implementation and references are available at arxiv.org/abs/1212.6037.
|Wujun Zhang (U of Maryland, College Park):
Local analysis of hybridizable discontinuous Galerkin method for convection dominated diffusion problem
|We analyze hybridizable discontinuous Galerkin methods applied to convection dominated diffusion problems. We establish L2 error estimates and point-wise estimates in regions where the solution is smooth.
|Victor Nistor (Penn State University):
Parametric Galerkin method on polyhedral domains
|We establish optimal higher orders of convergence for Galerkin approximations for parametric, second order elliptic partial differential equations on polyhedral domains. This method combines a good refinement strategy for meshes on the given polyhedral domain with an adaptive choice of Finite Element space for each component in a generalized chaos expansion using tensorized Lagendre polynomials. Similar results hold for parametric transmission problems in two dimensions. A common feature of these results is that they require a uniform "shift theorem" in the uncertainty parameter for equations with coefficients of minimal regularity. As in the non-parametric case, our results rely on weighted Sobolev (or Babuska-Kondratiev) spaces. These results are part of joint works with H. Li, Y. Qiao, and C. Schwab.
|LUNCH (will be provided)
|Jan Hesthaven (Brown University):
Reduced order models you can believe in
Models of reduced computational complexity is used extensively throughout science and engineering to enable the fast/real-time/subscale modeling of complex systems for control, design, multi-scale analysis, uncertainty quantification etc While of undisputed value these reduced models are, however, often heuristic in nature and the accuracy of the output is often unknown, hence limiting the predictive value.
We discuss ongoing efforts to develop reduced methods endowed with a rigorous a posteriori theory to certify the accuracy of the model. The focus will be on reduced models for parameterized linear partial differential equations. We outline the basic ideas behind certified reduced basis methods, discuss an offline-online approach to ensure computational efficiency, and emphasize how the error estimator can be exploited to construct an efficient basis at minimal computational off-line cost.
The discussion will draw on examples based both on differential and integral equations formulations. Time permitting, we shall discuss the challenges and some ideas to enable the development of reduced models for high-dimensional parametric problems.
This is work done in collaboration with Y. Chen (UMass Dartmouth), B. Stamm (Paris VI), S. Zhang (Brown), and Y. Maday (Paris VI).
|Stephen Shank (Temple University):
Krylov subspace methods for large-scale constrained Sylvester equations
|We consider the numerical approximation of the solution of a system of matrix equations that appear in the design of reduced order observers. A reformulation of an existing direct method that converts the problem into an unconstrained Sylvester equation will be presented, and the benefits of this new formulation with respect to the large-scale case will be discussed. The structure of the resulting coefficient matrices requires modifications of existing projection methods. We propose new enriched approximation spaces, and provide experimental evidence of their effectiveness on benchmark problems. Joint work with Valeria Simoncini.
|Jyoti Saraswat (U of Maryland, Baltimore County):
Multigrid solution of a distributed optimal control problems constrained by a semilinear elliptic PDEs
|We study a multigrid solution strategy for a distributed optimal control problem constrained by a semilinear elliptic PDE. Working in the discretize-then-optimize framework, we solve the reduced optimal control problem using Newton's method. Further, adjoint methods are used to compute matrix-vector multiplications for the reduced Hessian. In this work we introduce and analyze a matrix-free multigrid preconditioner for the reduced Hessian which proves to be of optimal order with respect to the discretization.
|Minghao Wu (U of Maryland, College Park):
Computational algorithms for stability analysis of incompressible flows
|Linear stability analysis of large-scale dynamical systems requires the computation of the rightmost eigenvalues of a series of large generalized eigenvalue problems. Using the incompressible Navier-Stokes as an example, we show that this can be done efficiently using a new eigenvalue iteration constructed using an idea of Meerbergen and Spence that reformulates the problem into one with a Lyapunov structure. This method requires solution of Lyapunov equations, which in turn entail solutions of subproblems consisting of linearized Navier-Stokes systems. We describe the methodology and demonstrate efficient iterative solution of the Navier-Stokes systems. (Joint work with Howard Elman)
|Sijiang Lu (U of Delaware):
A fully discretized Calderon calculus for two dimensional waves
|In this talk we will present a complete set of fully discretized integral operators for the Helmholtz equation in two dimensions, working on any fine set of smooth closed boundaries. This discretization scheme is basically a simple (assembly-free) black-box low order (2 and 3) method to substitute all operators that appear in the Calderon projector for the Helmholtz equation by fully discrete counterparts. The discretization use Dirac delta distributions and piecewise constant functions on the boundary. We will show that we can create a fully discrete set of integral equations that are of order three in all quantities, meaning that potentials are recovered with order three, but also boundary quantities are obtained with uniform order three approximations. Finally some numerical examples both in frequency and time domain will be shown.
|Gunay Dogan (NIST):
Shape reconstruction in inverse problems
|The reconstructions associated with inverse problems, such as computed tomography, are usually realized through a pixel/voxel grid or the sum of a selection of basis functions. While this is a reasonable approach for many inverse problems, in some problems, the main goal is to identify objects, anomalies or distinct regions in the probed medium. For such problems, a natural approach is shape reconstruction matching the given measurements of the medium, and the mathematical framework to realize this is shape optimization, in which the main variable is a shape, such as a set of curves in 2d or a volume in 3d. In the shape optimization approach, we incorporate the measured data and the expectations (or the prior information) for the shapes in a shape energy, and we try to compute the shape that minimizes this energy. In this work, we introduce a numerical algorithm that enables us to compute the optimal shapes associated with a given inverse problem. For this, we first describe how we can turn the inverse problem into shape optimization problem. Then we develop the numerical discretization to compute the shape updates. Finally, by addressing several pitfalls that arise in practical situations, we are able to demonstrate the effectiveness of our algorithm in various scenarios.
|Yukun Guo (U of Delaware):
A time domain linear sampling method for an inhomogeneous medium
|This talk is concerned with an inverse scattering problem of time-dependent nonharmonic acoustic waves in an inhomogeneous medium. The support of the inhomogeneity is determined from measurements of causal scattered waves by using the linear sampling method in the time domain. The performance of the algorithm will be illustrated with several numerical examples. This includes joint work with David Colton and Peter Monk.
|Benjamin Seibold (Temple University):
Jet schemes and gradient-augmented level set methods
|Jet schemes are semi-Lagrangian advection approaches that evolve parts of the jet of the solution (i.e., function values and higher derivatives) along characteristic curves. Suitable Hermite interpolations give rise to methods that are high order accurate, yet optimally local, i.e., the update for the data at any grid point uses information from a single grid cell only. Moreover, for interface evolution problems, jet schemes give rise to gradient-augmented level set methods (GALSM). These possess sub-grid resolution and yield accurate curvature approximations. In this talk, we investigate the computational efficiency of jet schemes, and demonstrate how the optimal locality of jet schemes gives rise to a straightforward combination with adaptive mesh refinement (AMR).