Abstracts (As of 5/4/2011)
[PDF version]
Plenary Talks
Optimization-Based Computational Modeling, or How to
Achieve Better
Predictiveness with Less Complexity
Pavel Bochev, Applied Mathematics and Applications, Sandia National Lab
Discretization converts infinite dimensional mathematical
models into finite dimensional algebraic equations that can be solved on a
computer. This process is accompanied by unavoidable information losses
which can degrade the predictiveness of the discrete equations. Compatible
and regularized discretizations control these losses directly by using
suitable field representations and/or by modifications of the variational
forms. Such methods excel in controlling “structural” information losses
responsible for the stability and well-posedness of the discrete equations.
However, direct approaches become increasingly complex and restrictive for
multi-physics problems comprising of fundamentally different mathematical
models, and when used to control losses of “qualitative” properties such as
maximum principles, positivity, monotonicity and local bounds preservation.
In this talk we show how optimization ideas can be used to control
externally, and with greater flexibility, information losses which are
difficult (or impractical) to manage directly in the discretization process.
This allows us to improve predictiveness of computational models, increase
robustness and accuracy of solvers, and enable efficient reuse of code. Two
examples will be presented: an optimization-based framework for
multi-physics coupling, and an optimization-based algorithm for constrained
interpolation (remap). In the first case, our approach allows to synthesize a
robust and efficient solver for a coupled multi-physics problem from simpler
solvers for its constituent components. To illustrate the scope of the
approach we derive such a solver for nearly hyperbolic PDEs from standard,
off-the-shelf algebraic multigrid solvers, which by themselves cannot solve
the original equations. The second example demonstrates how optimization
ideas enable design of high-order conservative, monotone, bounds preserving
remap and transport schemes which are linearity preserving on arbitrary
unstructured grids, including grids with polyhedral and polygonal cells.
This is a joint work with D. Ridzal , G. Scovazzi (SNL) and M. Shashkov
(LANL)
Gravitational Wave Physics: Challenges and Opportunities
for the Numerical
Analyst
Jan Hesthaven, Division of Applied Mathematics, Brown University
During the last decade there has been a surge of activity
aimed at modeling strong gravitational wave sources such as binary black
hole systems or black hole - neutron star dynamics. A central driver for
this can be found in the international efforts to develop large experimental
facilities aimed at detecting gravity waves. The detection of gravity waves
is considered the final confirmation of the validity of Einstein's general
theory of relativity in the nonlinear regime and, as such, is one of the
places where Nobel prices grow.
However, even with the use of state-of-the-art technology, the detection of
gravity waves is at the very limit of what is physically possible and the
use of advanced computational models are seen as critical component to
enable successful detection. While this has lead to the development of
reliable and efficient modeling tools, many challenges, both of a
theoretical, computational, and practical nature, remains.
In this presentation we will, after having given a very brief introduction
to gravitational wave physics, discuss some of the numerical and
computational challenges in this area. By discussing some of the dominating
models, we identity some of the many challenges and illustrate how the use
of contemporary computational techniques, in this case discontinuous
Galerkin methods and reduced basis methods, may offer some interesting
opportunities and improvements over dominating existing methods used in this
area.
Throughout the talk we will try to identity open questions and challenges,
illustrating the richness of problems in this application area - problems
than span the entire range from fundamental theoretical problems of interest
to the analyst to the need for very large scale computational science to
accurately model large astrophysical phenomena and the generation of
gravitational waves.
FASP Methods for Solving Large Scale
Discretized PDEs
Jinchao Xu, Department of Mathematics, Pennsylvania
State University
In many practical applications, the solution of large
scale linear algebraic systems resulted from the discretization of various
partial differential equations (PDEs) are still often solved by traditional
methods such as Gaussian elimination (and variants) . Mathematically optimal
methods, such as multigrid methods, have been developed for decades but they
are still not that much used in practice. In this talk, I will report some
recent advances in the development of optimal iterative methods that can be
applied to various practical problems in a user-friendly fashion. Starting
from some basic ideas and theories on multiscale methods such as multigrid
and domain decomposition methods, I will give a description of a general
framework known as the Fast Auxiliary Space Preconditioning (FASP) Methods
and report some applications in various problems including Newtonian and
non-Newtonian models, Maxwell equations, Magnetohydrodymics and reservoir
(porous media) simulations.
Contributed Talks
Stability and Error Estimates for an Equation Error Method
for Elliptic
Equations
Mohammad Al-Jamal, Department of Mathematical
Sciences, Michigan Technological University
There are a growing number of applications which call for
estimating a coefficient (or parameter) in an elliptic boundary value
problem from measurements of the forward solution. In this talk, I will
present the equation error approach for estimating a spatially varying
parameter in an elliptic PDE with Neumann boundary condition. I will show
stability and convergence results and derive error estimates. Also, I will
present some numerical examples illustrating the method.
Self Similar Growth of Single
Precipitate in
Inhomogeneous Elastic
Medium
Amlan Barua, Department of
Applied Mathematics, Illinois Institute of
Technology
In this talk results from linear analysis will be used to
present a new formulation for the self similar diffusional evolution of a
single precipitate growing in presence of elastic fields. The precipitate is
bounded by matrix having infinite extent. The elasticity problem is
isotropic but inhomogeneous. Numerical support for the formulation will be
given by performing non linear simulations using boundary integral methods.
The non linear method is spectrally accurate in space and in time it is
second order.
A Superconvergent Local
Discontinuous Galerkin Method
for Elliptic Problems
Mahboub Baccouch, Department
of Mathematics, University of Nebraska at Omaha
Slimane Adjerid, Department of Mathematics, Virginia Tech
In this talk, we
develop, analyze and test a superconvergent local discontinuous Galerkin(LDG)
method for two-dimensional diffusion and convection-diffusion problems and
investigate its convergence properties. Numerical computations suggest that
the proposed method yield $O(h^{p+1})$ optimal $\mathcal{L}^2$ convergence
rates and $O(h^{p+2})$ superconvergent solutions at Radau points. More
precisely, a local error analysis reveals that the leading term of the LDG
error for a $p$-degree discontinuous finite element solution is spanned by
two $(p+1)$-degree right-Radau polynomials in the $x$ and $y$ directions.
Thus, $p$-degree LDG solutions are superconvergent at right-Radau points
obtained as a tensor product of the shifted roots of the $(p+1)$-degree
right-Radau polynomial. For $p=1$, we discover that the first component of
the solution's gradient is $O(h^3)$ superconvergent at tensor product of
the roots of the quadratic left-Radau polynomial in $x$ and right-Radau
polynomial in $y$ while the second component is superconvergent at the
tensor product of the roots of the quadratic right-Radau polynomial in $x$
and left-Radau polynomial in $y$. We use these results to construct simple,
efficient, and asymptotically correct {\it a posteriori} error estimates and
present several computational results to validate the theory.
Hybridizable Discontinuous Galerkin Methods
for
Elliptic Problems
Fatih Celiker,
Mathematics, Wayne State University
We introduce a family of discontinuous Galerkin methods
called {\em hybridizable} discontinuous Galerkin (HDG) methods. The
distinctive feature of the methods in this framework is that the only
globally coupled degrees of freedom are those of an approximation of the
solution defined only on the boundaries of the elements. Since the
associated matrix is sparse, symmetric, and positive definite, these methods
can be efficiently implemented. We begin with introducing these methods in
the simple framework of second order elliptic problem $\Delta u = f$. We
then show how to generalize these methods to fourth order elliptic problems,
in particular to the biharmonic equation $\Delta^2u = f$. We rewrite the
biharmonic
problem as a first order system for separate unknowns $u$, $\nabla u$,
$\Delta u$, and $\nabla\Delta u$, then we introduce the HDG method for which
the only globally coupled
degrees of freedom are those of the approximation to $u$ and $\Delta u$ on
the faces of the elements. We show that a suitable choice of the numerical
traces results in optimal convergence for all the unknowns except for the
approximation to $\nabla\Delta u$ which converges with order $k+1/2$ when
polynomials of degree at most $k$ are used. This is joint work with Bernardo
Cockburn and Ke Shi at the University of Minnesota.
Efficient Spectral Methods for Systems of Coupled Elliptic
Equations with
Applications to Cahn-Hilliard Type Equations
Feng Chen,
Department of Mathematics, Purdue University
I will talk about how our newly developed spectral method
for systems of arbitrary number of coupled elliptic equations in both two
and three dimensions. The method is suboptimal in the sense that its
complexity is O($N^{d+1}$) with $N$ the cutoff in unidirection and $d$ the
dimension of the problem. It can be applied to solve highly nonlinear and
high-order evolution equations in various situations, including the
isotropic Cahn-Hilliard equation from microstructure evolution, the
anisotropic Cahn-Hilliard equation from crystal growth, and some other Cahn-Hilliard
type equations from PEM full cell modelling. Furthermore, it serves as a
potential solver for phase-field-crystal equations, rotating Navier-Stokes
equations, and Schrodinger equations.
Numerical Linear Algebra Challenges in
Very Large Scale Data
Analysis
Jie Chen, Mathematics and Computer Science
Division, Argonne National Laboratory
With the ever increasing computing power of
supercomputers, nowadays computational sciences and engineering demand
numerical solutions for problems in a larger and larger scale. One important
problem that finds a wide range of applications in such as physical
simulations and machine learning, is in a stochastic process the generation
of random data from a prescribed covariance rule, and the inverse question
of fitting the covariance rule given experimented data. This problem gives
rise to a number of numerical linear algebra challenges, where one needs to
deal with dense and irregularly structured covariance matrices of mega-,
giga- or even much larger sizes. In this talk, I will illustrate specific
encountered challenges, including computing the square root of the matrix,
estimating the diagonal, solving linear systems, and preconditioning the
matrix. For some of these challenges, we have developed efficient and
scalable methods that are capable of dealing with matrices of size at least
in the mega-scale, on a single desktop machine. As a natural extension, high
performance codes run on supercompters are being developed; however, there
remain other unsolved challenging tasks along the line, which call for
innovative algorithms as well as theory.
Parallel Computation of the Mixed Volume
Tianran Chen,
Tsung-Lin Lee, and Tien-Yien Li, Department of Mathematics, Michigan State
University
Calculating mixed cells which produces mixed volume as a
by-product is the vital step in solving systems of polynomial equations by
the polyhedral homotopy methods. Our original algorithm for this purpose,
implemented in MixedVol-2.0, is highly serial. In this talk, we propose a
reformulation of our algorithm, making it much more fine-grained and
scalable. It can be readily adapted to both distributed and shared memory
computing systems. Remarkably, very high speed-ups were achieved in our
numerical results, and we are now able to compute mixed cells of polynomial
systems of very large scale, such as VortexAC6 system with mixed-volume
27,298,952 and total degree $2^{30}$ (around 1 billion).
Towards Lightweight Projection
Methods for Systems
with Multiple
Right-hand Sides
Efstratios Gallopoulos,
Department of Computer Engineering &
Informatics, University of Patras, Greece
An Efficient and Stable Spectral Method for
Electromagnetic
Scattering from a Layered
Periodic Structure
Ying He, Department of Mathematics,
Purdue University
The scattering of acoustic and electromagnetic waves by periodic structures
plays an important role in a wide range of problems of scientific and
technological interest. This contribution focuses upon the stable and
high--order numerical simulation of the interaction of time--harmonic
electromagnetic waves incident upon a periodic doubly layered dielectric
media with sharp, irregular interface. We describe a Boundary Perturbation
Method for this problem which avoids not only the need for specialized
quadrature rules but also the dense linear systems characteristic of
Boundary Integral/Element Methods. Additionally, it is a provably stable
algorithm as opposed to other Boundary Perturbation approaches such as Bruno
\& Reitich's ``Method of Field Expansions'' or Milder's ``Method of Operator
Expansions.'' Our spectrally accurate approach is a natural extension of the
``Method of Transformed Field Expansions'' originally described by Nicholls
\& Reitich (and later refined to other geometries by the authors) in the
single--layer case.
A Finite Element Solver for the Kohn-Sham
Equation with
a Mesh Redistribution
Technique
Guanghui Hu, Department of
Mathematics, Michigan State University
A finite element method is presented with an adaptive mesh
redistribution technique for solving the Kohn-Sham equation. The solver
consists of two independent iterations. The first one is a self-consistent
field (SCF) iteration which generates the self-consistent electron density,
while the second one is an iteration which optimizes the distribution of
mesh grids in terms of the self-consistent electron density. In the SCF
iteration, the Kohn-Sham equation is discretized by the standard finite
element method. The electrostatic potential is obtained by solving the
Poisson equation, with the algebraic multi-grid (AMG) method as the Poisson
solver. The local density approximation (LDA) is adopted to approximate the
exchange-correlation potential. Both the all-electron and the Evanescent
Core pseudopotential are considered for the external potential. To stabilize
the SCF iteration, the linear mixing scheme is introduced for updating the
electron density. After the SCF iteration, the distribution of mesh grids is
optimized by an adaptive technique, which is based on the harmonic mapping.
A monitor function which depends on the gradient of the electron density is
proposed to partially control the movement of mesh grids. To further improve
the mesh quality, a smoothing strategy which is derived from diffusive
mechanism is presented. From the numerical experiments, it can be observed
that important regions in the domain such as the vicinity of a nucleus, and
between atoms of chemical bonds are resolved successfully with our mesh
redistribution technique. Higher numerical accuracy and efficiency of our
solver are demonstrated in the numerical experiments.
Instant System Availability
Kai Huang,
Department of Mathematics, Florida International University
In this talk, I will present our recent work on the instant
availability A(t) of a repairable system through integral equation. We will
prove some properties of the instant system availability, Numerical
algorithm for computing A(t) is proposed. Examples show high accuracy and
efficiency of this algorithm. This is a joint work with J. Mi.
Efficient Computation of Failure
Probability
Jing Li, Department of Mathematics, Purdue University
Evaluation of failure probability of a given
system requires sampling of the system response and can be computationally
expensive. Therefore it is desirable to construct an accurate surrogate
model for the system response and subsequently to sample the surrogate
model. In this talk we demonstrate that the straightforward sampling of a
surrogate model can lead to erroneous results, no matter how accurate the
surrogate model is. We then propose a hybrid algorithm combines the
surrogate and sampling approaches and address the robust problem described
above. The resulting algorithm is significantly more efficient than the
traditional sampling method, and is more accurate and robust than the
straightforward surrogate model approach and numerical examples will be
presented.
The
Chebyshev Integral Formulation for Performing
High Spatial
Resolution Collocation Simulations
Benson Muite, Department of
Mathematics, University of Michigan
We describe an efficient implementation of the Chebyshev integration
formulation. The implementation allows for spatially accurate simulations
with $O(n\log n)$ computational costs and for the accurate recovery of
derivatives. High spatial resolution simulations using the implementation
will be demonstrated and factors limiting higher resolution simulations from
being done discussed.
An Algebraic Multigrid Preconditioner
with
Guaranteed Condition Number
Artem Napov, Lawrence
Berkeley National Lab
Yvan Notay, Universit Libre de Bruxelles, Belgium
We present an algebraic
multigrid preconditioner that has a guaranteed condition number for the
class of nonsingular symmetric M-matrices with nonnegative row sum. Our main
ingredient is a new algorithm for algebraic grid generation (coarsening) by
aggregation of unknowns which insures that the two-grid condition number
remains below a prescribed (user defined) parameter. This is achieved by
using a bound based on the worst aggregates' ``quality''. For a sensible
choice of this parameter, it is shown that the recursive use of the two-grid
procedure yields a condition number independent of the number of levels,
providing that one uses a proper AMLI-cycle. On the other hand, the cost of
the preconditioner is of optimal order if the mean aggregates' size is large
enough. This point is addressed analytically for the model Poisson problem
and, further, numerically through a wide range of numerical experiments,
demonstrating the robustness of the method. The experiments are performed on
low order discretizations of second order elliptic PDEs in two and three
dimensions, with both structured and unstructured grids, some of them with
local refinement and/or reentering corner, and possible jumps or
anisotropies in the PDE coefficients. This is joint work with Yvan Notay (Universit
Libre de Bruxelles, Belgium).
Polynomial Interpolation on Arbitrary Nodal Sets
in High Dimensions
Akil Narayan, Department of
Mathematics, Purdue University
Motivated by the problem of stochastic collocation in
high-dimensional spaces, we present a generalized algorithm for the `least
interpolant' method of Carl de Boor and Amos Ron for polynomial
interpolation on arbitrary data nodes in multiple dimensions. Our variation
on the least interpolant produces an interpolant that can be tailored for
various probability distributions. We present properties of this polynomial
interpolant and empirically analyze conditioning of the associated
Vandermonde-like matrices. We also present a few examples illustrating
utility of the method for interpolation on arbitrary nodal sets in
high-dimensional spaces.
Galerkin-type Approximation for
Stochastic PDE of
Nonlinear Beam with
Additive Noise
Henri Schurz, Department of Mathematics, Southern Illinois
University
The Galerkin-type approximation of strong solutions of
some quasi-linear stochastic PDEs with cubic nonlinearity and Q-regular
random space-time perturbations is discussed. The SPDE relates to the
nonlinear beam problem in engineering and physics. The existence of unique
solutions with homogeneous boundary and square integrable initial conditions
is shown by truncated Galerkin techniques. For our analysis, we exploit the
techniques of Fourier series solutions, Lyapunov-functions and monotone
operators. The related Fourier coefficients are computed by nonstandard
numerical methods to control the qualitative behavior of associated mean
energy functional in a consistent manner. If time permits, we sketch how we
prove consistency rates, convergence and stability along total mean energy
functional. This presentation is connected to a joint work with Boris
Belinskiy (UTC).
A Parallel Implementation of an ODE Solver
for
a River Basin Model
Scott Small, Department of
Mathematics, University of Iowa
Laurent O. Jay, Department of Mathematics, University
of Iowa
Many factors influence the flow of rivers, including
rainfall, properties of the soil, vegetation, and melting snow. We consider
a model for the discharge of water from an entire river basin. The model
consists of a large-scale system of ODEs defined on a sparse tree structure.
We consider using standard Runge-Kutta methods. However, solving the entire
system with the same of ODEs stepsize for all equations creates
inefficiencies. As a remedy, we propose a parallel asynchronous integration
approach to improve efficiency. Numerical results on basins in Iowa will be
presented.
Nonnegative Sparse Blind Source Separation
for NMR
Spectroscopy by Data
Clustering, Model Reduction,
and $l_1$ Minimization
Yuanchang Sun, Department of Mathematics,
University of California at Irvine
A novel blind source separation (BSS) approach is introduced to deal with
the nonnegative and correlated signals arising in NMR spectroscopy of
bio-fluids (urine and blood serum). BSS problem arises when one attempts to
recover a set of source signals from a set of mixture signals without
knowing the mixing process. Various approaches have been developed to solve
BSS problems relying on the assumption of statistical independence of the
source signals. However, signal independence is not guaranteed in many
real-world data like the NMR spectra of chemical compounds. To work with the
nonnegative and correlated data, we replace the statistical independence by
a constraint which requires dominant interval(s) from each source signal
over some of the other source signals in a hierarchical manner. This
condition is applicable for many real-world signals such as NMR spectra of
urine and blood serum for metabolic fingerprinting and disease diagnosis.
Exploiting the hierarchically dominant intervals from the source signals,
the method reduces the BSS problem into a series of sub-BSS problems by a
combination of data clustering, linear programming, and successive
elimination of variables. Then in each sub-BSS problem, an $l_1$
minimization problem is formulated for recovering the source signals in a
sparse transformed domain. The method is substantiated by examples from NMR
spectroscopy data and is promising towards separation and detection in
complex chemical spectra without the expensive multi-dimensional NMR data.
This is joint work with Prof. Jack Xin.
Error Bound for Numerical Methods for the
Rudin-Osher-Fatemi Image Smoothing Model
Jingyue Wang, Math, University of Georgia
The Rudin-Osher-Fatemi variational model has been extensively
studied and used in image analysis. There have been several very successful
numerical algorithms developed to compute the numerical solutions. We study
the convergence of the numerical solutions to various finite-difference
approximation
to this model. We bound the difference between the solution to the
continuous ROF model and the numerical solutions. These bounds apply to
``typical'' images, i.e.,
images with edges or with fractal structure.
Phase Field Modeling for Mesoscale Materials by
Differential Variational Inequality
Lei Wang, Argonne National Laboratory
The phase field method has recently emerged as a powerful computational
approach to modeling the defect and the microstructure dynamics in mesoscale
materials. We employ coupled Cahn-Hilliard and Allen-Cahn systems with a
double-obstacle free energy potential to simulate the physics. Differential
Variational Inequality(DVI) is employed in order to guarantee that the
discrete solutions satisfy appropriate constraints. We reformulated the DVIs
to a complementarity problem, which allows us to use parallel matrix-free
solvers such as PETSc and TAO. Several numerical test cases will be shown.
A Superfast
and Stable Solver for Toeplitz Linear Systems
via Randomized Sampling
Yuanzhe Xi, Department of
Mathematics, Purdue University
Jianlin Xia, Department of Mathematics, Purdue University
Toeplitz linear systems have been widely used in scientific computing and
engineering. Many stable and fast Toeplitz solvers (roughly $O(n^2)$ cost)
have been developed, and there are also many superfast (roughly $O(n)$ cost)
solvers which, however, are usually not stable. Here, we propose a new
stable and superfast Toeplitz solver. With a displacement structure, a
Toeplitz matrix is transformed into a Cauchy-like matrix by FFT. These
Cauchy-like matrix have a low-rank property. That is, its off-diagonal
blocks have small numerical ranks. By exploiting this property, the
Cauchy-like matrix can be further approximated by a rank structured form
called hierarchically semiseparable (HSS) matrix. The HSS Cauchy-like system
can be quickly solved in $O(n)$ complexity with $O(n)$ storage. In order to
construct such an HSS approximation quickly, we use randomized sampling
techniques together with fast Toeplitz-vector multiplications. In this way,
we only need to compress some much smaller matrices after the
multiplications of the Cauchy-like matrix with some Gaussian random
matrices. Further structured operations are also used during the HSS matrix
construction and factorization. All the steps are conducted qukcly and
stably. Numerical results demonstrate the efficiency of our solver, and show
that, when $n=2\times 10^4$, it is already about $10$ times faster than an
existing stable and superfast structured solver by Chandrasekaran, Gu, Xia,
et al. [SIAM J. Matrix Anal. Appl., 2007].
Fast Finite Element
Solver Development for
a Nonlocal Dielectric Continuum Model
Dexuan Xie, Department of Mathematical Sciences,
University of Wisconsin-Milwaukee
The nonlocal continuum dielectric model is an important extension of the
classical Poisson dielectric model but much more expensive to be solved
numerically. In this talk, I will introduce one commonly-used nonlocal
dielectric model and demonstrate its great promise in the calculation of
free energies with a much higher accuracy than the Poisson model. I then
will report our recent work on the development of fast finite element
solvers for this model. This project is a joined work with Prof. Ridgeway
Scott at the University of Chicago, and partially supported by by NSF grant
\#DMS-0921004.
WENO Divergence-Free
Reconstruction-Based Finite
Volume Scheme for Solving Ideal MHD
Equations
on Triangular Meshes
Zhiliang Xu, Applied and
Computational Mathematics and Statistics Department, University of Notre
Dame
In this talk, I will present our recent work on developing a 3rd order
accurate finite volume schemes for solving MHD equations on triangular
meshes. We advance the magnetic field with a constrained transport scheme
to preserve the divergence-free condition of the magnetic field. A high
order divergence-free reconstruction method is proposed for the magnetic
field that use the cell edge values. This reconstruction as well as the
reconstruction for flow variables are based on WENO finite volume method.
Numerical examples are given to demonstrate efficacy of the new schemes.
Fast Fourier--Jacobi
Methods for the Fokker--Planck
Equation of FENE Dumbbell Fluids
Haijun Yu, Department of Mathematics,
Purdue University
FENE Dumbbell model is one of the most simple mathematical models that
predict basic properties of Non-Newtonian Fluids. However the dynamics of
FENE dumbbell fluids are described by a high-dimensional Fokker--Planck
equation which needs very fast computer to simulate. Most of the existing
numerical algorithms involve factorization of a non-sparse matrix thus are
not suitable for discretizations with large degree of freedom. In this talk,
we will present a fast spectral Galerkin method using real Fourier series
and Jacobi polynomials as bases. This new algorithm has several virtues: 1.
The Galerkin approximation bases on a proper weighted weak formulation, in
which the numerical moments have spectral accuracy; 2. The numerical
approximation leads to linear sparse (banded indeed) system, thus can be
solved with linear computational cost; 3. The numerical algorithm can be
easily extended to solve the Fokker--Planck equation arising in
non-homogeneous systems, or the Navier--Stokes Fokker--Planck coupled
system.
Sequential Monte Carlo Sampling in Hidden
Markov Models of Nonlinear Dynamical
Systems
Xiaoyan Zeng, Mathematics and
Computer Science Division, Argonne National Laboratory
We investigate the issue of which state functionals can have their
uncertainty estimated efficiently in dynamical systems with uncertainty.
Because of the high dimensionality and complexity of the problem, sequential
Monte Carlo (SMC) methods are used. We prove that the variance of the SMC
method is bounded linearly in the number of time steps when the proposal
distribution is truncated normal distribution. We also show that for a
moderate large number of steps the error produced by approximation of
dynamical systems linearly accumulates on the condition that the logarithm
of the density function of noise is Lipschitz continuous. This finding is
significant because the uncertainty in many dynamical systems, in
particular, in chemical engineering systems that can be assumed to have
this nature.
A Lowest Order Divergence-free Finite
Element
on Rectangular Grids
Shangyou Zhang, University of
Delaware
Yunqing Huang, Xiangtan University
It is shown that the conforming $Q_{2,1;1,2}$-$Q_1'$ mixed element is
stable, and provides optimal order of approximation for the Stokes
equations on rectangular grids. Here $Q_{2,1;1,2}=Q_{2,1}\times Q_{1,2}$ and
$Q_{2,1}$ denotes the space of continuous piecewise-polynomials of degree
$2$ or less in the $x$ direction but of degree $1$ in the $y$ direction.
$Q_1'$ is the space of discontinuous bilinear polynomials, with spurious
modes filtered. To be precise, $Q_1'$ is the divergence of the discrete
velocity space $Q_{2,1;1,2}$. Therefore, the resulting finite element
solution for the velocity is divergence-free pointwise, when solving the
Stokes equations.
A Piecewise Constant Enriched Continuous Galerkin
Method for Problems with
Discontinuous Solutions
Shun Zhang, Division of Applied Mathematics, Brown
University
For problems with discontinuous solutions, the solutions are usually very
smooth in most regions except locations with shocks/contact
discontinuities. We propose a new approximation space, a space of
"continuous (which can be high-order/spectral) elements and piecewise
constants", to approximate those solutions. The continuous part of the
space can approximate the smooth part of the solution well, and the
piecewise constant enrichment can be used to approximate the discontinuous
phenomena. We hope the method can combine the good properties of high-order
methods and discontinuous Galerkin methods.
Eigen-based High Order Basis for Spectral Elements
Xiaoning Zheng,
Department of Mathematics, Purdue University
Steven Dong, Department of Mathematics, Purdue University
We present an efficient high-order expansion basis for the spectral element
approach. This belongs to the category of modal basis, but it is not
hierarchical. The interior modes are constructed by solving a small
generalized eigenvalue problem, while the boundary modes are constructed
based on such eigen functions in lower dimensions. We compare this expansion
basis with the commonly-used Jacobi polynomial-based expansion basis, and
demonstrate the considerably superior numerical efficieny of the new basis
in terms of conditioning and the number of iterations to convergence for
iterative solvers.
Contacts
Please contact the conference organizers for more information:
Jie Shen
(Chair),
Peijun Li, and
Jianlin Xia
at mwna@math.purdue.edu.
