(February 5) Professor Christopher Beattie:
Projection Methods for Structure-preserving Model Reduction -
Dynamical systems are the basic framework for modeling and control of an
enormous variety of complex systems. Direct numerical simulation of the
associated models has been one of the few available means when goals
include accurate prediction or control of complex physical phenomena.
However, the ever increasing need for improved accuracy requires the
inclusion of ever more detail in the modeling stage, leading inevitably to
ever larger-scale, ever more complex dynamical systems. Simulations in such large-scale settings can be overwhelming and make
unmanageably large demands on computational resources; this is the main
motivation for model reduction, which has as its goal production of a much
lower dimensional system having the same input/output characteristics as
the original system. Rational Krylov subspaces are often capable of providing nearly optimal
approximating subspaces for model reduction. A framework for model
reduction is presented that includes rational Krylov-based methods as a
special case. This broader framework allows retention of special structure
in the reduced order models such as symmetry, second order structure,
internal delays, and infinite dimensional subsystems. The main cost to generate the required reduced order bases arises from
solving large scale linear systems of equations having the same order as
the dynamical system itself. Since the need for more detail and accuracy
in the modeling stage causes the system dimension to reach levels on the
order of millions, direct solvers for these linear systems are no longer
feasible. In practice, iterative solvers become necessary and choice of
termination criteria and effect of preconditioning influences the quality
of final reduced order models. The effect on the underlying model
reduction problem from the use of approximate subspaces generated from
iterative solvers will be discussed as well.
(February 12) Professor Claudio Canuto:
High-resolution methods for stochastic boundary-value problems with
geometric
uncertainties -
In modern science and technology, there is an increasing need for numerical
simulations supplemented by quantitative indications of their reliability. A
posteriori error analysis and adaptivity provide an answer. We will rather
focus on problems which are not fully deterministic, but depends on one or
more random variables: in such cases, one is often interested in computing
certain statistical quantities of the solution, which quantify the amount of
uncertainty in the output of the model.
For pde-based models, the use of Monte-Carlo techniques is often ruled out
by the excessive cost of a single "call" to the numerical solver. The viable
alternative is to transform the stochastic problem into a deterministic one,
via suitable expansions of the random variables (Karhunen-Loeve or
Polynomial Chaos expansions). High-resolution numerical methods, which
"minimize" the number of unknowns per given target accuracy, may help in
keeping small the complexity of the problem.
We will illustrate these ideas in the case of boundary-value problems posed
in random domains. We will present various approaches to handle the
geometric uncertainty, such as mapping and fictitious domains. We will
discuss the numerical analysis of the discretization methods, their
implementation issues, as well as certain realistic applications to Wind
Engineering (flow around a bridge deck).
(February 19) Professor Marco Verani:
A Safeguarded Dual Weighted Residual Method
-
The dual weighted residual (DWR) method yields reliable a posteriori
error bounds for linear output functionals provided that the error
incurred by the numerical approximation of the dual solution is
negligible. In that case its performance is generally superior than
that of global energy norm error estimators which are unconditionally
reliable. We present a simple numerical example for which neglecting
the approximation error leads to severe underestimation of the
functional error, thus showing that the DWR method may be unreliable.
We propose a remedy that preserves the original performance, namely a
DWR method safeguarded by additional asymptotically higher order a
posteriori terms. In particular, the enhanced estimator is
unconditionally reliable and asymptotically coincides with the
original DWR method. These properties are illustrated via the
aforementioned example. (joint work with R.H. Nochetto and A. Veeser)
(February 26) Professor Pedro Morin:
Convergence of Adaptive Finite Elements for Eigenvalue Problems -
We will discuss the convergence of adaptive finite element
methods for second order elliptic eigenvalue problems using Lagrange
elements of any degree. Convergence will be proved under a minimal
refinement of marked elements, for all "reasonable" marking
strategies, and starting from any initial mesh.
This work is in collaboration with Eduardo Garau (Santa Fe, Argentina)
and Carlos Zuppa (San Luis, Argentina) (March 4) Professor Xiaobai Sun:
Efficient Encoding of Irregular Geometric Structures
in Matrix Computations
-
In this talk I describe the basic concepts and schemes of encoding
geometric structures in matrices that arise from discretizing certain
differential or integral equations over irregular domains and grids.
The most noticeable examples include the Fast Multipole method and the
Fourier transform, or the inverse Fourier transform, with
non-Cartesian data. These methods are known for their low-order
complexity in memory usage and arithmetic operations with respect to
an arbitrarily specified accuracy tolerance and are typically employed
in iterative solution methods. An efficient encoding scheme involves
not only a compressive mathematical representation of the matrix
defined on an irregular grid, but also a companion ordering for
efficient traversal of the matrix and data in memory with respect to
any particular memory hierarchy. During the talk I introduce a
particular case of the inverse Fourier transform with
three-dimensional radial data from a medical imaging application and
describe the success and impact of the encoding approach. (March 11) Professor Irene Livshits:
Multiscale/Multigrid Method for Finding a Full Eigenbasis of the
Schrodinger Operator -
Many applications involving differential operators require a knowledge of
their many eigenfunctions. In this talk, we will introduce a novel
multigrid technique which, by employing a multiscale structure for
eigenfunctions' approximation, allows calculating all eigenfunctions of
the Schrodinger operator as well as performing many applications typical
for eigenvalue problems in just O(N log N) operations, where N is the
size of the discretized problem. Numerical results and a discussion of
the further extension of the approach will conclude the talk.
(March 25) Professor Shari Moskow:
Asymptotic and Numerical Approximation of the
Resonances of Thin Photonic Structures
-
We study the computation of the resonances of thin dielectric
structures by using both asymptotics and PML. Of
great interest in the study of photonic band gap materials
is when these structures are periodic with a defect. We derive
a limiting resonance problem as a the thickness goes to
zero, and for the case of a simple resonance find a first order
correction. The limiting problem and correction have one
dimension less, which can make the approach very efficient.
Convergence estimates are proved for the asymptotics. We
then proceed to compute these resonances by two very different
methods: the asymptotic method by solving a dense,
but small, nonlinear eigenvalue problem; and PML, which
yields a large, but linear and sparse generalized eigenvalue
problem. Both methods reproduce a photonic band gap
type mode found previously by finite difference time domain
methods. This is joint work with Jay Gopalakrishnan and Fadil Santosa
(April 1) Professor Mili Shah:
Calculating a Symmetric Approximation via the Symmetry Preserving Singular Value Decomposition -
Determining symmetry within a collection of spatially oriented points is a
problem that occurs in many fields. In these applications, large amounts
of data are generally collected, and it is desirable to approximate this
data with a compressed representation. In some situations, the data is
known to obey certain symmetry conditions, and it is profitable to
preserve such symmetry in the compressed approximation. I accomplish this
task by providing a symmetry preserving singular value decomposition. I
will present the formulation of this factorization along with an
application of this method to molecular dynamics.
(April 8) Professor Ricardo Nochetto:
Adaptive FEM for the Laplace-Beltrami Operator and Applications
to Geometric Flows
-
The Laplace-Beltrami operator, or surface Laplacian, is ubiquitous in
geometric problems and has a natural variational structure. This
allows for finite element discretizations of surface and PDE of
arbitrary polynomial degree, and the use of refinement/coarsening
techniques that lead to adaptivity. We first discuss key properties
and present applications such as
surface diffusion, optimal shape control and image segmentation,
along with several simulations exhibiting large deformations as well
as pinching and topological changes in finite time. This part is
joint with G. Dogan, P. Morin and M. Verani. We next discuss a
posteriori error analysis for the Laplace-Beltrami operator and prove
a conditional contraction property of the ensuing AFEM; this is joint
with K. Mekchay and P. Morin. We finally discuss the geometrically
consistent refinement of polyhedral surfaces, which is critical for
large domain deformations and new paradigm in adaptivity. This is
joint with A. Bonito and M.S. Pauletti. (April 15) Professor Jennifer Erway:
Recent Developments in Iterative Methods for Large-Scale
Unconstrained Optimization
-
Numerical optimization involves the minimization or maximization of
an "objective function," possibly subject to functional constraints
on the variables. Over the past decade, advances in methods for
optimization problems with partial differential equation (PDE)
constraints have highlighted the need for optimization algorithms
that can handle problems with huge numbers of variables and
constraints. This has lead to a resurgence of interest in methods
that solve a constrained problem as a sequence of unconstrained
problems. The algorithm for each unconstrained problem must be
iterative in nature, i.e., it must not rely on matrix factorizations.
Two of the biggest challenges for this class of algorithms are
reliability and efficiency. Trust-region methods are one of the most popular approaches to
solving unconstrained optimization problems. This talk will present
recent developments in trust-region methods for large-scale
unconstrained optimization.
(April 21) Professor Vidar Thomee:
On Maximum-Principles in Elliptic and Parabolic
Finite Element Problems
-
We consider finite element discretizations of Dirichlet's
problem for Laplace's equation and the corresponding
initial-boundary value problem for the homogeneous heat equation.
We survey work on discrete versions of the associated maximum-principles,
using piecewise linear finite elements on triangulations
of the underlying spatial domain. In the elliptic case
the maximum-principle holds if
the triangulation is of Delaunay type, but that this is is not
a necessary condition. For the spatially semidiscrete
parabolic problem, the maxiumum-principle does not hold in general,
but, as was shown by Fujii in 1973, it is valid for the lumped
mass variant when the triangulation is of Delaunay type. We show that these
conditions are essentially sharp.
We also study conditions for the solution operator acting on
the discrete initial data, with homogeneous lateral boundary conditions,
to be a contraction in the maximum-norm, or a positive operator.
(April 22) Professor Rick Falk:
A New Approach to Finite Element Methods for the Equations of Linear
Elasticity
-
We exploit the connection between the de Rham sequence and a
corresponding
exact sequence for elasticity to derive new finite element methods for the
approximation
of the equations of linear elasticity. These methods use a Lagrange
multiplier to weakly impose
the symmetry of the stress, thus allowing the use of standard mixed finite
elements
for scalar second order elliptic problems.
(April 29) Professor Houman Owhadi:
From Stochastic Variational Integrators to Ballistic Diffusion at
Uniform Temperature -
We present a continuous and discrete Lagrangian theory for
stochastic Hamiltonian systems on manifolds. Analogously to discrete
mechanics and variational integrators, this theory leads to structure
preserving numerical integrators for noisy mechanical systems by extremizing
a discrete stochastic action. In this talk, we will focus on a particular
one, designed to approximate the solutions of Langevin equations with
uniform friction and noise. This new integrator is characterized by the
following two properties: * It exactly preserves the exponential rate of decay of the symplectic form. * It inherits a near Boltzmann-Gibbs measure preservation property from
the near energy preservation property of its associated noise/friction-free
symplectic Euler integrator.
By applying our integrator to a fluctuation-driven magnetic motor we exhibit
a paradigm for isothermal, mechanical rectification of stochastic
fluctuations. The central idea is to transform energy injected by (random/thermal)
fluctuations into rigid-body rotational kinetic energy. We show that
although directed motion is not possible at uniform temperature, thermal
noise can be used to obtain ballistic diffusion. This is a joint work with
Nawaf Bou-Rabee.
(May 13) Professor Daniele Boffi:
Mimetic Finite Differences and Quadrilateral Finite Elements -
It is well known that quadrilateral finite elements may
produce unsatisfactory results when used on distorted meshes. Many
commonly used finite elements do not achieve optimal convergence
properties: the list of suboptimal elements include serendipity
elements and basically all elements for the approximation of H(div)
(like Raviart-Thomas or Brezzi-Douglas-Marini spaces).
On the other hand, mimetic finite differences have become popular for
the approximation of problems involving H(div) on very general
geometries. We show how to use the ideas of mimetic finite differences
for the stabilization of Raviart-Thomas element on general
quadrilateral meshes.
|