| Venue |
Invited
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Titles and
Abstracts |
Logistics |
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Organizing
Committee |
Titles and Abstracts
Modeling Mitral Valve Dynamics by Analysis of 3D-Echocardiographic Movies
Finite Element Methods for a Fourth Order Obstacle Problem
In this talk we will discuss finite element methods for the displacement
obstacle problem of clamped plates. We will present a unified convergence
analysis for $C^1$ finite element methods, classical nonconforming finite
element methods and $C^0$ interior penalty methods. For a $C^2$ obstacle
on a convex domain, the convergence rate in the energy norm is $O(h)$ for
these methods. Corroborating numerical results will also be presented.
Recent Advances in Structure-Preserving Model Order Reduction
In recent years, model order reduction techniques based on Krylov subspaces have
become the methods of choice for generating small-scale macromodels of the
large-scale multi-port RCL networks that arise in VLSI interconnect analysis.
A difficult and not yet completely resolved issue is how to ensure that the
resulting macromodels preserve all the relevant structures of the original
large-scale RCL networks. In this talk, we present a brief review of how
Krylov subspace techniques emerged as the algorithms of choice in VLSI circuit
simulation, describe the current state-of-art of structure preservation,
discuss some recent advances, and mention open problems.
On a conjecture by C. Sundberg: A numerical investigation
Abstract: Carl Sundberg (University of Tennessee-Knoxville) conjectured some time ago that ....
Joint work with: Annalisa Quaini.
Second-Order Optimal Control Methods for Diffeomorphic Shape Dynamics
We present second-order methods for optimal control problems arising in the analysis
of deformable shapes and in associated applications to computational anatomy. In
particular, we implement computationally feasible stagewise Newton steps in
combination with efficient optimization strategies based on Pareto boundaries in
the context of diffeomorphic shape matching. We provide optimization algorithms
for the reconstruction of dynamic deformations matching a given finite time series
of three dimensional biomedical shape snapshots, with some preliminary numerical
experiments.
Reduced Order Modeling for Parametric Nonlinear PDE Constrained Problems Using POD-DEIM
When proper orthogonal decomposition (POD) or another projection
based technique is used to generate reduced order models, the
number of equations and unknowns is typically reduced dramatically.
However, for nonlinear or parametrically varying problems, the cost of
evaluating the reduced order models still depends on the size of the
full order model and is still expensive. To overcome this bottleneck,
Chaturantabut and Sorensen developed the Discrete Empirical Interpolation Method
(DEIM), which generates reduced order models that typically can be evaluated
at a cost that only depends on the size of the reduced order model and therefore
generates a truly useful reduced order model.
We demonstrate why model reduction by POD alone is insufficient and
outline the DEIM. We then extend the POD-DEIM method to finite element
solutions of nonlinear partial differential equations (PDEs) and to the solution of
shape optimization problems governed by PDEs.
Joint work with Harbir Antil and Danny C. Sorensen.
Discontinuous Galerkin Methods for Diffusion-Dominated Radiative
Transfer Problems
It has been noted, that standard upwind discontinuous Galerkin (DG)
discretizations may fail, if the scattering mean free path length is
smaller than the mesh size. Mathematical analysis reveals, that in
this case, convergence is only achieved in a continuous subspace of
the finite element space. Furthermore, if boundary conditions
are not chosen isotropically, convergence can only be expected in
$H^s(\Omega)$ with $s<1/2$. While the latter result seems to be
a property of the transport problem, asymptotic analysis reveals,
that the forcing into a continuous subspace can be avoided. By
choosing a weighted upwinding, the conditions on the diffusion limit
can be weakened. While it has been known for long time, that the
diffusion limit of radiative transfer is a diffusion equation, it
turns out, that by choosing the stabilization carefully, the DG method
can yield either the LDG method or the method by Ern and Guermond
in its diffusion limit.
Mixed Finite Element Methods with Piecewise Constant Fluxes
Numerical Upscaling of Flows in Heterogeneous Media
of High Porosity
The Brinkman equations (or generalized Stokes equations) are used for
modeling flows in highly porous media. Examples of such media are
industrial open foams, filters, and various insulation materials.
Motivated by applications of such materials we have developed a
numerical method for computing flows in heterogeneous highly porous
media with complicated internal structure of the permeability.
We shall present a two-scale finite element approximation of Brinkman
equations. The method uses two main ingredients: (I) discontinuous
Galerkin finite element method for Stokes equations, proposed and
studied by J. Wang and X. Ye (2007) and (II) subgrid approximation
developed by T. Arbogast for Darcy equations (2004). A number of
numerical examples will be presented to demonstrate the performance of
the method.
Joint work with: J. Willems
Numerical simulation of cell/cell and cell/particle interaction in microchannel
Joint work with: Lingling Shi and Roland Glowinski.
Coupling locally mass conservative methods for flow in porous media
A multinumeric approach is proposed to solve the convection-diffusion problem.
Two locally mass conservative methods are coupled, namely the
cell-centered finite volume methods and the primal discontinuous
Galerkin methods.
The coupled scheme takes advantage of both the accuracy of the discontinuous Gal
erkin methods in regions of interest, such as regions containing shales or pinch
-outs, and the efficiency of the finite volume method in the rest of domain.
Cell-centered finite volume methods are currently widely used in most
of the production reservoir simulators. Coupling of finite volume and discontin
uous Galerkin methods produces a more flexible discretization with improved appr
oximations properties. Theoretical and numerical results are presented.
Semismooth Newton methods with multi-dimensional filter globalization
for l_1 optimization
For many applications, it is important to include the requirement of
sparse solutions efficiently into the formulation of optimization problems.
The observation that l_1-regularization promotes sparsity (i.e. few
nonzeros) has resulted in significant recent research with important
applications ranging from image inpainting to actuator placement in optimal
control.
We start with a discussion of important properties of l_1-regularized
(and thus nonsmooth) optimization problems. Then, we focus on methods for
l_1-regularized optimization problems that combine semismooth Newton
algorithms with globally convergent descent methods in a flexible way.
The acceptance of semismooth Newton steps is controlled by a
multi-dimensional filter globalization. The filter approach is beneficial
here since sufficient decrease conditions are difficult to implement for
semismooth Newton steps.
A suitable descent methods is obtained, e.g., via shrinkage steps. We
discuss global convergence of this algorithm and show that under suitable
assumptions the method eventually turns into a semismooth Newton
method that converges locally q-superlinearly. The talk concludes with
numerical illustrations.
Joint work with: Andre Milzarek
GPU Accelerated Discontinuous Galerkin Methods
We will discuss the trend towards many-core architectures
in modern computers and in particular how current non-uniform
memory hierarchy should be factored in to comparisons
of competing finite element formulations for numerically solving partial
differential equations.
We will focus primarily on the discontinuous Galerkin methods,
in particular a customized version that is designed to deliver an
accurate treatment of curvilinear domains with relatively low storage
overhead.
Examples simulations from electromagnetics and gas dynamics
will be shown with benchmarks indicating the performance
obtained on current graphics processing units.