10:00 Harbir Antil (George Mason University):
Galerkin v. discrete-optimal projection in nonlinear model reduction
Discrete-optimal model-reduction techniques such as the Gauss-Newton with Approximated Tensors (GNAT) method have shown promise, as they have generated stable, accurate solutions for large-scale turbulent, compressible flow problems where standard Galerkin techniques have failed. However, there has been limited comparative analysis of the two approaches. This is due in part to difficulties arising from the fact that Galerkin techniques perform projection at the time-continuous level, while discrete-optimal techniques do so at the time-discrete level. This talk provides a detailed theoretical and experimental comparison of the two techniques for two common classes of time integrators: linear multistep schemes and Runge-Kutta schemes. We present a number of new findings, including conditions under which the discrete-optimal ROM (reduced order model) has a time-continuous representation, conditions under which the two techniques are equivalent, and time-discrete error bounds for the two approaches. Perhaps most surprisingly, we demonstrate both theoretically and experimentally that decreasing the time step does not necessarily decrease the error for the discrete-optimal ROM; instead, the time step should be `matched' to the spectral content of the reduced basis. In numerical experiments carried out on a turbulent compressible-flow problem with over one million unknowns, we show that increasing the time step to an intermediate value decreases both the error and the simulation time of the discrete-optimal reduced-order model by an order of magnitude.
10:20 Virginia Forstall (U of Maryland, College Park):
Preconditioners for reduced-order models
Reduced-order models (ROMs) efficiently solve related linear systems arising in many-query applications. The full linear system is projected onto a smaller dimensional subspace resulting in a ROM whose solution approximates the solution of the original system. The reduced model size is problem dependent and could be smaller than the full problem, but large enough to make direct solution costly. In this scenario, the iterative solution of the ROM can be more efficient. We construct preconditioners for reduced iterative methods which are derived from preconditioners for the full problem.
10:40 Gunay Dogan (Theiss Research, NIST):
A fast algorithm to compute the shape dissimilarity of 2d curves
In many scientific applications, one needs to compare the shapes of curves from different sources of data in a quantitative manner. These curves might correspond to real objects in imaging applications, e.g. cell populations in microscopy, or they might be a representation of an important physical characteristic, such as the spectrum of material specimens. A researcher working with such applications would often need to state of how similar or dissimilar the shapes are, and might also want to use the shape dissimilarity numbers (or distances) to pursue large-scale statistical analyses of the shape samples. A rigorous and quantitative tool to compare curves and to compute their dissimilarity is shape geodesics. For this, one chooses an appropriate shape representation for the curves and identifies the Riemannian manifolds where the shapes lie and then tries to compute the shape geodesics between the shapes. The computation of shape geodesics usually reduces to optimization on Riemannian manifolds, specifically manifolds of curves. In this work, we propose an efficient algorithm to compute the geodesic distances of shapes on a practical realization of such a shape manifold. The shape representation that we consider is the square root velocity representation of Srivastava et al (IEEE PAMI, 2011). For this choice, the Riemannian manifold is the L^2 unit hypersphere of curves and the computation of the geodesic distances can be reduced to minimizing the L^2 distance between the derived curves. As we require this distance to be invariant under various transformations of the curves, such as rotations and reparameterizations, this setup induces an optimization problem, in which the starting point on the curve, possible rotations and reparameterizations are the variables. We derive the optimality conditions of the corresponding energy formulation, and use the optimality conditions to devise an effective optimization scheme. We test and demonstrate our method on several synthetic and real examples, and achieve significant improvement in both speed and accuracy of the geodesic distance computations.
11:00 BREAK
11:30 Matthew Hassell (U of Delaware):
Coupling BEM and FEM for the wave equation
We study a symmetric coupling scheme between BEM and FEM for wave transmission and scattering by inhomogeneous obstacles. Elementary evolution equation theory is applied to produce energy estimates for the semi-discrete formulation, which are used to demonstrate stability of the fully discrete CQ-BEM-FEM scheme. We provide a number of numerical examples to demonstrate the flexibility of the method.
11:50 Kihyo Moon (Virginia Tech):
An immersed discontinuous Galerkin method for acoustic wave propagation in nonhomogeneous media
We present an immersed discontinuous Galerkin finite element method on Cartesian meshes for two dimensional acoustic wave propagation problems in nonhomogeneous media where elements are allowed to be cut by the material interface. The proposed method uses the standard discontinuous Galerkin finite element formulation with polynomial approximation on elements that contain one material while on interface elements containing more than one material or fluid it uses a specially-built piecewise polynomial shape functions that satisfy appropriate interface jump conditions. The finite element spaces on interface elements satisfy physical interface conditions from the acoustic problem in addition to extended conditions derived from the system of partial differential equations. We present several computational results that suggest that the proposed method has optimal convergence rates. Several computational examples are included with linear and curved interfaces.
12:10 Pablo Venegas (U of Maryland, College Park):
Numerical solution of transient nonlinear axisymmetric eddy current models with hysteresis
This work deals with the mathematical analysis and the computation of transient electromagnetic fields in nonlinear magnetic media with hysteresis. The results obtained complement previous results, where the mathematical and numerical analysis of a 2D nonlinear axisymmetric eddy current model was performed under fairly general assumptions on the H–B curve but without considering hysteresis effects. In our case, the constitutive relation between H and B is given by a hysteresis operator, i.e., the values of the magnetic induction depend not only on the present values of the magnetic field but also on its past history. We assume axisymmetry of the fields and then we consider two kinds of boundary conditions. Firstly the magnetic field is given on the boundary (Dirichlet boundary condition). Secondly, the magnetic flux through a meridional plane is given, leading to a non-standard boundary-value problem. For both problems, an existence result is achieved under suitable assumptions. For the numerical solution, we consider the Preisach model as hysteresis operator, a finite element discretization by piecewise linear functions, and the backward Euler time-discretization.We report a numerical test in order to assess the order of convergence of the proposed numerical method. Finally, we validate the numerical scheme with experimental results. With this aim, we consider a physical application: the numerical computation of eddy current losses in laminated media as those used in transformers or electric machines.
12:30 LUNCH (will be provided)
  1:30 Sören Bartels (University of Freiburg):             KEYNOTE SPEAKER
Numerical methods for total variation regularized problems
Various phenomena involving free boundaries such as damage or plasticity require the description of physical quantities with discontinuous functions. One approach to their mathematical modeling is based on the space of functions of bounded variation which includes functions that are discontinuous and may jump across lower dimensional subsets. Numerical methods for their approximate solution are often based on regularizations which typically lead to restrictive conditions on discretization parameters. We try to avoid such modifications and discuss the convergence of discretizations with different finite element spaces, the iterative solution of the resulting finite-dimensional nonlinear systems of equations, and adaptive mesh-refinement techniques based on rigorous a~posteriori error estimates for a model problem related to image processing. The application of the techniques to total variation flow, very singular diffusion processes, and segmentation problems will be addressed. Part of this talk is based on joint work with Ricardo H. Nochetto (University of Maryland, USA) and Abner J. Salgado (University of Tennessee, USA).
  2:30 BREAK
  3:00 Ignacio Tomas (U of Maryland, College Park):
Convergence analysis for a simplified PDE model for two-phase ferrofluid flows
We present a simplified model describing the behavior of two-phase ferrofluid flows using phase-field techniques. For this PDE model, and the corresponding numerical scheme we prove, in addition to stability, convergence and as by-product existence of solutions.
  3:20 Wujun Zhang (U of Maryland, College Park):
A finite element method for nematic liquid crystals with variable degree of orientation
We consider the simplest one-constant model, put forward by J. Ericksen, for nematic liquid crystals with variable degree of orientation. The equilibrium state is described by a director field and its degree of orientation, which minimize a sum of Frank-like energies and a double well potential. In particular, the Euler-Lagrange equations for the minimizer contain a degenerate elliptic equation for the director field, which allows for line and plane defects to have finite energy. We present a structure preserving discretization of the liquid crystal energy without regularization, and show that it is consistent. We prove convergence of the continuous piecewise linear finite solutions as the mesh size goes to zero. We develop a weighted gradient flow scheme for computing discrete equilibrium solutions and prove that it has a strict energy decrease property. We present simulations in two and three dimensions that exhibit both line and plane defects and illustrate key features of the method.
  3:40 Enrique Otarola (U of Maryland, College Park):
Piecewise polynomial interpolation in weighted Sobolev spaces and applications to optimal control problems
We develop a piecewise polynomial approximation theory in weighted Sobolev spaces with Muckenhoupt weights. We construct a quasi-interpolation operator and derive optimal error estimates on simplicial shape regular meshes and anisotropic rectangular meshes. The interpolation theory extends to cases when the error and function regularity require different weights. Concerning the applications to the optimal control theory, we consider: a linear quadratic optimal control problem involving a nonuniformly elliptic equation; a problem with a pointwise tracking objective and one where the control is the amplitude of secondary forces, modeled as point masses.
  4:00 BREAK
  4:30 Araz Hashemi (University of Delaware):
Sensitivity Analysis of Multiscale Reaction Networks with Stochastic Averaging
Monte Carlo methods, such as Gillespie’s direct SSA, are widely used to simulate chemical reaction networks. These simulations can then be used to estimate properties such as the mean value of an observable at some time horizon, or the sensitivity of mean values with respect to system parameters. However, when there are multiscale dynamics in the reaction network, direct simulation methods become ineffective because they can only advance the system on the smallest scale. This results in a prohibitive computational burden to reach a time horizon for the large scale dynamics. We shall show how stochastic averaging may be employed to speed computations and obtain estimates of mean values and sensitivities with respect to the steady state distribution. Further, we shall establish bounds which show the bias induced by the averaging method decays to zero as the disparity between the scales increases. This talk presents joint work with P. Plechac, M. Nunez, and D. G. Vlachos.
  4:50 Ting Wang (U of Maryland, Baltimore County):
Efficiency of Girsanov transformation method for parametric sensitivities of stochastic reaction network
For stochastic reaction networks consisting of several species, Monte Carlo methods are the most suitable for parametric sensitivity analysis. Most of the Monte Carlo methods for sensitivity analysis can be classified into three categories, the pathwise derivative (PD) method, the finite difference (FD) method and the Girsanov transformation (GT) method. It has been numerically observed in the literature that when applicable, the PD method and FD method tend to be more efficient than the GT method. In this talk, I will provide a theoretical justification for this observation in terms of system size asymptotic analysis.
  5:10 Gideon Simpson (Drexel University):
A Relative Entropy Formulation of Diffusive Molecular Dynamics
Diffusive Molecular Dynamics (DMD) is a novel approach to problems in molecular dynamics that aims to reach the diffusive time scale of milliseconds and beyond. To accomplish this, DMD “averages out” the vibrational time scale of femtoseconds and evolves probability densities at atomistic sites. This requires the approximation of a probability distribution in an extended state space by a synthetic approximate distribution, which can easily be sampled. We will present an analysis that this approximation can be interpreted as a minimization of the relative entropy distance between distributions. This has consequences for the implementation of the algorithm, and it allows for flexibility in future applications. This is joint work with M. Luskin (U. Minnesota) and D. Srolovitz (Penn).
  5:30 Erik von Schwerin (U of Delaware):
Optimal mesh hierarchies in Multilevel Monte Carlo methods
Multilevel Monte Carlo methods (MLMC) can dramatically reduce the computational cost of Monte Carlo simulations where each sample is computed using a discretization based numerical method. For example, when computing the expected value of a quantity of interest depending on the solution to an Ito stochastic differential equation or a partial differential equation with stochastic data. I will discuss how to choose optimal mesh hierarchies in MLMC simulations based on uniform discretization methods with general approximation orders and computational costs. I will compare optimized geometric and non-geometric hierarchies and discuss how enforcing some domain constraints on parameters of MLMC hierarchies affects the optimality of these hierarchies. I will also discuss the optimal tolerance splitting between the bias and the statistical error contributions and its asymptotic behavior. This talk presents joint work with N.Collier, A.-L.Haji-Ali, F.Nobile, and R.Tempone.
  5:50 END