Title: High-order solutions of three-dimensional rough surface scattering problems at high -frequencies
Abstract:
We present a new numerical scheme for three-dimensional acoustic and
electromagnetic rough-surface scattering simulations that can deliver highly
accurate results from low to high frequencies at a cost that is independent of the
wavelength of the incoming radiation. The method is based on high-order asymptotic
expansions of the oscillatory integrals that enter potential theoretic
formulations of the scattering problems in the high-frequency regime, in a manner
that bypasses the need to resolve the fields on the scale of the wavelength of
radiation. Indeed, we seek the solutions of the integral equations (e.g. the
normal velocity in acoustics and the current in electromagnetics) in the form of
slow modulations of highly oscillatory exponentials of known phases, and we choose
series expansions in inverse powers of the wavenumber to represent the unknown
slowly varying envelopes. As we show, this framework can be made to yield an
efficiently computable recursion for the terms of the series to any arbitrary
order. The resulting algorithms generally provide a very significant improvement
over classical (e.g. Kirchhoff's) approximations in both accuracy and
applicability and they can, in fact, efficiently produce results with full
double-precision accuracy for configurations of practical interest.
Title: Elliptic Variational Problems on Constricted Networks with Applications to Ginzburg-Landau Theory
Abstract:
I will analyze variational problems set on a network of thin constricted
tubes. In the asymptotic regime where the tubes collapse to a graph, one
can identify a one dimensional variational problem with interesting
natural boundary conditions at the nodes which in particular encourage
jumps in the (limit of) minimizers across the various branches of the
graph. An application to tunneling across weak links in a superconductor
will be discussed.
Title: Stress-driven grain boundary diffusion: modeling, analysis and numerical methods
Abstract:
Microchips often fail when the metallic interconnects between
transistors and diodes on the chip degrade due to extremely high
current densities. The physics of this process is quite interesting;
it is a non-local moving interface problem involving elastic
deformation and diffusion. Stress singularities can develop which
make boundary conditions difficult to understand and numerical
simulation difficult to implement reliably.
After describing the model, I will outline our recent proof of
well-posedness, which uses techniques from semigroup theory and
requires an analysis of a type of Dirichlet to Neumann map involving
the equations of elasticity. I will also briefly describe my recent
work on computing stable asymptotics for singularities of
Agmon-Douglis-Nirenberg elliptic systems near corners and interface
junctions, and show how to adjoin these singular functions to the
finite element basis to accurately and efficiently resolve stress
singularities without mesh refinement. If time permits, I will also
show that my least squares finite element formulation for elasticity
transitions gracefully to the Stokes equations in the incompressible
limit, and show how to incorporate the convection term to obtain an
efficient Navier-Stokes solver for low to moderate Reynolds numbers.
Title: Differential equations in optical design
Abstract:
Classically, differential equations play an indirect role in optical
design. Indeed, optics is fundamentally based on Maxwell's equation.
Even geometrical optics, which governs the bulk of optical design, is
based on a differential (eikonal) equation. Yet, optical designers
devised methods that are based to a large extent on ray tracing.
Therefore, although PDEs are somewhere at the background, they are not
used explicitly. In this talk I shall present examples in optical design
where PDEs play a direct role.
Title: The Time Dimension: A New Unified Theory Underlying Linear First Order Systems and a Prelude to Algorithms By Design
Abstract:
A new unified theory underlying the theoretical design of linear
computational algorithms in the context of time dependent linear first
order systems is presented. Providing new perspectives, and unlike various
formulations existing in the literature, it involves the following
considerations: i)it provides new avenues for designing new computational
algorithms to foster the notion of algorithms by design and recovering
existing algorithms in the literature, ii) describes a theory for the
evolution of time operators via a unified mathematical framework, and iii)
places into context and explains/contrasts future new developments
including existing algorithm designs and the various relationships among
the different classes of algorithms in the literature such as linear
multistep methods, sub-stepping methods, Runge-Kutta type methods, higher
order time accurate methods, etc. Subsequently, it provides design
criteria and guidelines for contrasting and evaluating time dependent
computational algorithms.
The notion of algorithms by design concept capitalizes upon :i) the
recently developed unfied theory mentioned above, and ii) newly established
algorithmic measures. The linear computaional algorithms in the context of
first order systems are classified as distictly pertaining to Type 1, Type
2 and Type 3 classifications of time discretized operators. A generalized
stability and accuracy limitation barrier theorem underlies the generic
designs of computational algorithms with arbitrary order of accuracy(the
Dahlquist theorem is a paricular case limited to second order accuracy),
and establishes guidelines which cannot be circumvented.
Title: On the Rate of Convergence of Finite-difference Approximations for Bellman Equations with Constant Coefficients
Abstract:
We consider elliptic Bellman equations with coefficients independent
of the variable x. Error bounds for certain types of finite-difference
schemes are obtained. These estimates are sharper than those of earlier results.
Title: The Non-Overlapping Domain Decomposition Method For Scattering Problems
Abstract:
In this talk we shall present some developments on the
non-overlapping domain decomposition method in the context
of acoustic scattering problems. After a brief introduction
of the method, we will discuss the central issue of convergence
of the iterative procedure that underlies the approach.
More precisely, we shall show that the wave nature of solutions
imposes subtle requirements on the choice of inter-domain (Robin)
boundary conditions to optimize the convergence properties.
If time permits, we shall also describe how to resolve the
issue of "cross points" within domain decomposition methods,
as well as how to use these schemes in BEM/FEM formulations
of the scattering problem.
Title: A multiscale model for the dynamics of solids
Abstract:
We present a multiscale method for coupling atomistic and continuum models
of solids. Both models are formulated in the form of physical conservation
laws, and the coupling is achieved through balancing the fluxes. Error
estimate is provided, which is of great help in choosing the size of the
macro grid and microscale system. Finally we shall show some applications
including phase transformation, and dynamic fracture mechanics.
Title: Linear Stability of Stratified Fluids and the Associated Nonlinear Eigenvalue Problem
Abstract:
It's a problem that has long puzzled fluid dynamicists:
How long does it take the waves in a container of fluid to settle?
To date there is no complete mathematical analysis; the
air/liquid/wall contact line and surface tension complicate things.
But for "supercritical" fluids at high pressure (important in
several industrial processes, such as decaffeination of coffee)
modeled by the incompressible Navier-Stokes equations, the
sharp distinction between liquid and vapor disappears. Viscosity
can be expected to damp internal waves with a characteristic
exponential relaxation time associated with the slowest decaying
mode of the system. This work proves that, surprisingly, there is
no slowest decaying mode in such stratified fluids. (This is a
joint work with R. L. Pego and K. F. Gurski.)
Title: An efficient integral equation method for electromagnetic and acoustic scattering simulations: convergence of multiple scattering iterations
Abstract:
One of the main difficulties in high-frequency electromagnetic and
acoustic scattering simulations is that any numerical scheme based
on the full-wave model entails the resolution of wavelength. It is
due to this challange that simulations involving even very simple
geometries are beyond the reach of classical numerical schemes.
In this talk, we shall present an analysis of a recently proposed
integral equation method that bypasses the need for the resolution
of wavelength, and thereby delivers solutions in frequency-independent
computational times. Within single scattering configurations, the
method is based on the use of an appropriate ansatz for the unknown
surface densities and on suitable extensions of the method of stationary
phase. The extension to multiple-scattering configurations, in turn,
is attained through consideration of an iterative (Neumann) series
that successively accounts for multiple reflections. We show that
the convergence properties of this series in the high-frequency regime
depends solely on geometrical characteristics. Moreover,for periodic
orbits, we determine the convergence rate with explicit error bounds.
Finally, we show that this insight suggests the use of alternative
summation mechanisms that can greatly accelerate the convergence of
the series.
Title: Variational PDE Models in Imaging and Vision
Abstract:
Traditional signal and image processings have mainly been
built upon spetral (Fourier) analysis. In the past two
decades, variational and PDE methods have gradually gained
their momentum in the modeling and analysis of imaging and
vision, mainly due to their flexibilities in modeling and
computation. In this talk, I will explain this novel approach
based on some of my own recent works. I would like to thank
Professors Tony Chan and Stan Osher for introducing to me
this approach during my postdoctor training at UCLA several
years ago.
Title: Optimal Routing of a Sailboat in Steady Winds
Abstract:
A family of novel, insightful, but yet tractable problems regarding
the optimal routing of a sailboat in the plane, such as a racing yacht,
in steady (i.e. time-invariant) winds is considered. Typically, the
optimization criterion is to minimize the route travel time from a
start point to a finish point. The solution to the constant wind
case is well-known and provides a solid basis for how the problem
generalizes and becomes even more realistic. In particular, the
mathematical properties of the boat polar are studied in detail
and the significance of the luff and downwind dip zones noted.
Limited corroborating experimental work was conducted on an
instrumented C & C 38 yacht with regard to: (1) boat polar
characterization and (2) optimal routing in constant winds. For
the case of non-constant, but smooth wind fields, the minimum-time
problem takes on the form of a Zermelo problem, resulting in a
2-point boundary-value problem generated from application of the
calculus of variations and requires a single iteration of the
initial boat heading angle. However, a major subtlety concerns
the continuity level of the optimal route and how it is affected
by the mathematical properties of the boat polar. Specifically, if
the entire optimal route is sufficiently continuous (e.g. C2),
the calculus of variations applies directly. If not, it may be
applied in piece-wise fashion -- this provides motivation for
introducing the notion of a discontinuity map associated with the
set of finish points in the plane and their minimal discontinuity
based optimal routes. Various formulations of increasing spatial
dimensionality are identified and checked with Mathematica, as
characterized by the type of wind field and the location of the
finish point. Major cases investigated include: (1) 1-dimensional
(1D) wind direction field with constant wind speed, (2) 2D wind
direction field with constant wind speed, (3) constant direction
wind field with speed variation, and (4) 2D wind speed and direction
field. Other optimization criteria and associated solutions are
also studied -- several velocity made good (VMG) techniques in
particular, one of which is equivalent to the minimum-time criterion.
Numerical simulation studies using MATLAB are provided for an
interesting scenario involving 1D spatially oscillating winds that
illustrate how the minimum-time route is generated and also how
its continuity is affected by the boat polar's properties. Other
optimization criteria and corresponding solutions are compared and
contrasted as well (i.e. two VMG approaches, minimum-distance).
In summary, this talk summarizes attempts to make a contribution
in the area of sailing mechanics and optimal routing problems,
with a special emphasis on, and appreciation of, continuity issues.
Title: Signal recovery from partial information
Abstract:
This lecture will demonstrate theoretically and empirically that a
greedy algorithm called Orthogonal
Matching Pursuit (OMP) can reliably recover a signal with $m$
nonzero entries in dimension $d$ given $O(m \ln d)$ random linear
measurements of that signal. This is a massive improvement over
previous results for OMP, which require $O(m^2)$ measurements. The
new results for OMP are comparable with recent results for another
algorithm called Basis Pursuit (BP). The OMP algorithm is much
faster and much easier to implement, which makes it an attractive
alternative to BP for signal recovery problems. This work is in
collaboration with Anna Gilbert.