Thursday, April 12, 2018
11:00-11:50 am, Room SH 306
Vladimir Druskin (Schlumberger-Doll Research
Center, Boston)
Reduced order models, networks and applications to modeling and imaging with waves
Abstract. Geophysical seismic exploration, as well as radar and sonar imaging
require the solution of large scale forward and inverse problems for
hyperbolic systems of equations. In this talk I will show how model
order reduction can be used to address some intrinsic difficulties of
these problems. In model order reduction, one approximates the
response (transfer function) of a large scale dynamical system using a
smaller system, called the reduced order model (ROM). We consider
ROMs that capture properties of the large problem that are essential
for imaging and that can be realized via sparse graph-Laplacian
networks. The ROMs are data-driven, i.e., they learn the underlying
PDE problem from the transfer function. One of the better known
applications of our ROMs is the efficient discretization of PDE problems in
unbounded domains. Here I will focus on two recent applications: (i)
Multiscale modeling of elastic wave propagation via network
approximations, with low communication and computational cost; (ii) A
direct, nonlinear acoustic imaging algorithm in strongly heterogeneous
media, where the ROM is used to manipulate the data in such a way that
multiply scattered waves are separated from the single scattered ones.
Thursday, April 5, 2018
Title. Fractional partial differential equations: modeling,
numerical method, and analysis
Abstract. Fractional partial differential equations
(FPDEs) provide an accurate description of transport
processes from many applications, which exhibit anomalous
diffusion and long-range spatial interaction and time
memory. However, FPDEs raise mathematical and numerical
difficulties that have not been encountered in the context
of integer-order PDEs.
Computationally, because of the nonlocal property of
fractional differential operators, the numerical methods for
FPDEs often generate dense coefficient matrices for which
traditional direct solvers were used that have a
computational complexity of O(N^3) per time step and memory
requirement of O(N^2) where N is the number of unknowns.
This makes numerical simulation of three-dimensional FPDE
modeling computationally very expensive. Mathematically,
FPDEs exhibit mathematical properties that have fundamental
differences from those of integer-order PDEs.
We will go over the development of fast numerical methods for
FPDEs, by exploring the structure of the coefficient
matrices. These methods have approximately linear
computational complexity per time step and optimal memory
requirement.
We will discuss mathematical issues on FPDEs such as
wellposedness and regularity of the problems and their
impact on the convergence behavior of numerical methods.
Friday, March 30, 2018
Mitigation of self-force effects in particle-in-cell codes
using meshfree data transfer
Abstract. Particle-in-cell (PIC) codes are a key tool for
obtaining computationally tractable simulations of plasma.
The approach may be summarized as using Lagrangian particles
to track ions, performing a data transfer of charge to a
mesh, solving the governing equations to obtain
electromagnetic fields, and performing a second data
transfer of these field variables back to particles to
evaluate particle accelerations. While this process may be
shown to conserve momentum and energy for uniform grids, on
unstructured grids the error during data transfer manifests
itself as an artificial self force in which a single
particle in isolation may accelerate itself under the action
of its induced electric field. In this work we develop a
meshfree approach to minimize this effect. We begin with a
review of the approximation theory underpinning generalized
moving least squares (GMLS) approximation, which we have
used extensively in the past to develop meshfree
discretizations of PDE with notions of conservation and
compatibility. GMLS is an optimization based approach to
locally approximate linear functionals from scattered sets
of sampling functionals. A key property of GMLS is the
ability to enforce exact reproduction of the approximation
over a class of functions, and we exploit this to enrich our
reconstruction with singular functions to better reproduce
the singular Greens function solution associated with the
electrostatic case in an infinite domain. We demonstrate
that this approach is equally applicable to any mesh-based
solution of the underlying field by investigating both
traditional nodal finite elements and the
Raviart-Thomas/piecewise constant mixed finite element pair.
Thursday, March 22, 2018
Vertex models of epithelial tissue mechanics - application in
notum morphogenesis
Abstract. Cell-based models provide powerful computational
tools for studying the mechanisms underlying the
morphological dynamics of biological tissues. An increasing
amount of quantitative data with cellular resolution has
paved the way for the quantitative parameterization and
validation of such models. Out of the cell-based models,
vertex models have been used to study a variety of processes
in epithelial tissues (sheets of cells). Recently we
implement the vertex model in the quantitative analyses of
live imaging of the Drosophila dorsal thorax during
metamorphosis and suggest a novel mechanism that can
generate contractile forces within the plane of epithelia
– via cell proliferation in the absence of growth. In
this talk, I will talk about both the results of this study
and the state of the art of the vertex models and their
numerical implementations.
Thursday, March 15, 2018
A domain-decomposition model reduction method for linear
convection-diffusion equations with random coefficients
Abstract.
We will focus on linear steady-state convection-diffusion
equations with random-field coefficients. Our particular
interest to this effort are two types of partial
differential equations (PDEs), i.e., diffusion equations
with random diffusivities, and convection-dominated
transport equations with random velocity fields. For each of
them, we investigate two types of random fields, i.e., the
colored noise and the discrete white noise. We developed a
new domain-decomposition-based model reduction (DDMR)
method, which can exploit the low-dimensional structure of
local solutions from various perspectives. We divide the
physical domain into a set of non-overlapping sub-domains,
generate local random fields and establish the correlation
structure among local fields. We generate a set of reduced
bases for the PDE solution within sub-domains and on
interfaces, then define reduced local stiffness matrices by
multiplying each reduced basis by the corresponding blocks
of the local stiffness matrix. After that, we establish
sparse approximations of the entries of the reduced local
stiffness matrices in low-dimensional subspaces, which
finishes the offline procedure. In the online phase, when a
new realization of the global random field is generated, we
map the global random variables to local random variables,
evaluate the sparse approximations of the reduced local
stiffness matrices, assemble the reduced global Schur
complement matrix and solve the coefficients of the reduced
bases on interfaces, and then assemble the reduced local
Schur complement matrices and solve the coefficients of the
reduced bases in the interior of the sub-domains. The
advantages and contributions of our method lie in the
following three aspects. First, the DDMR method has the
online-offline decomposition feature, i.e., the online
computational cost is independent of the finite element mesh
size. Second, the DDMR method can handle the PDEs of
interest with non-affine high-dimensional random
coefficients. The challenge caused by non-affine
coefficients is resolved by approximating the entries of the
reduced stiffness matrices. The high-dimensionality is
handled by the DD strategy. Third, the DDMR method can avoid
building polynomial sparse approximations to local PDE
solutions. This feature is useful in solving the
convection-dominated PDE, whose solution has a sharp
transition caused by the boundary condition. We will
demonstrate the performance of our method based on the
diffusion equation and convection-dominated equation with
colored noises and discrete white noises.
Thursday, Feb 22, 2018
Nonlinear Filtering Methods for Time-Varying Parameter
Estimation
Abstract. Many real-world applications involve unknown
system parameters that must be estimated using little to no
prior information. In addition, these parameters may be
time-varying and possibly subject to structural
characteristics such as periodicity. We show how nonlinear
Bayesian filtering techniques can be employed to estimate
periodic, time-varying parameters, while naturally providing
a measure of uncertainty in the estimation. Results are
demonstrated using data from several applications from the
life sciences.
Thursday, November 16, 2017
On a spatiotemporal population dynamics model and its applications to fishery science
Abstract. Structured population models (SPM) have been
used to model density, age and mass of
individuals over time. Traditional ecological models do not
introduce a separate partial dif-
ferential equation for mass; rather they model the population
as being subdivided into classes
parameterized by mass and then number density is written as a
function of spatial location,
time and mass. In this talk, a new approach of modeling mass
as dependent variable will
be discussed. The resultant model yields a coupled
reaction-diffusion and hyperbolic partial
differential equations. To show the applicability of the
model, I will discuss the following
issues from fishery science:
(1) how the mobility of species affects the yield with
multiple fishing zones and network of
marine protected areas (MPAs),
(2) the efficacy of MPAs under multiple fishing zones, and
(3) how mass dependent mortality influences density and mass
of a population.
This is a joint work with Dr. Jay R. Walton
(Department of Mathematics) and Dr.
Masami Fujiwara (Department of Wildlife and Fisheries), Texas
A&M University.
Thursday, November 09, 2017
Deterministic and Stochastic Inverse Scattering Problems
Using Fast Direct Solvers
Abstract. Inverse scattering problems arise in many areas of
science and engineering, including medical imaging, remote
sensing, ocean acoustics, nondestructive testing, geophysics
and radar. In this talk, we focus specifically on the
deterministic and stochastic inverse acoustic scattering
medium problem in two dimensions. There are a multitude of
challenges that should be addressed for the solution of
those problems, among them the fact that those problems are
nonlinear, ill-posed and computationally expensive.
The first two problems were previously addressed extensively
in the literature using diverse techniques, however, those
techniques require the solution of a large number of forward
scattering problems that, until recently, were extremely
computational expensive. The last decade has seen a lot of
progress in the development of fast direct solvers that use
the idea that distant interactions have a low rank
approximation. Those solvers are perfect fast accurate tools
to speed-up the solution of a large number of forward
scattering problems.
I will describe a fast, stable algorithm that can be applied
as a framework for the solution of both the discrete and the
stochastic inverse scattering problems. Given full aperture
far field measurements of the scattered field for multiple
angles of incidence, we use the Gauss-Newton method together
with the recursive linearization algorithm (RLA) to
reconstruct a band-limited approximation of the domain
solving a sequence of linear least square problems of
successively high frequencies using fast direct solvers.
Using this framework, we can obtain the solution of the
deterministic inverse scattering problem up to high
resolution, as never seen before, as well as the solution of
the stochastic inverse scattering problem using a large
number of samples.
Thursday, November 02, 2017
Polynomial Preserving Recovery for
Gradient and Hessian
Abstract. Post-processing techniques are important in
scientific and engineering computation. One of such
technique, Superconvergent Patch Recovery (SPR) proposed by
Zienkiewicz-Zhu in 1992, has been widely used in finite
element commercial software packages such as Abaqus, ANSYS,
Diffpack, etc.; another one, Polynomial Preserving Recovery
(PPR) has been adopted by COMSOL Multiphysics since 2008. In
this talk, I will give a survey for the PPR method and
discuss its resent development to obtain the Hessian matrix
(second derivatives) from the computed data.
Thursday, September 28, 2017
Numerical collocation method for simulating 1-D PDE model of
human tear film dynamics
Abstract. The one dimensional case for a single
lubrication-type equation modeling the dynamics of
the human tear film in a blink cycle has been proposed by
Braun et al. The equations must be
solved on time varying domain and they are of nonlinear,
fourth-order in space, and
first order in time. We describe the implementation of
pseudospectral collocation methods to
compute its numerical solutions on a transformed domain and
to impose the third-order
boundary conditions arising from flux conditions at both end
points. At the end, we present
some numerical results for full and partial blink cases and
compare them with experimental
data. This talk is also suitable for undergraduate students
who have taken or are currently
taking numerical PDE course.
Thursday, September 14, 2017
The Lubricated Immersed Boundary Method
Abstract.
Many real-world examples of fluid-structure interaction,
including the transit of red blood cells through the narrow
slits in the spleen, involve the near-contact of elastic
structures separated by thin layers of fluid. Motivated by
such problems, we introduce an immersed boundary method that
uses elements of lubrication theory to resolve thin fluid
layers between immersed boundaries. We apply this method to
two-dimensional flows of increasing complexity, including
eccentric rotating cylinders and elastic vesicles near walls
in shear flow, to show its increased accuracy compared to
the classical immersed boundary method.
11:00-11:50 am, Room SH 306
Hong Wang
(Department of Mathematics, University of South Carolina)
2:00-2:50 pm, Room HL 202 (Joint seminar with AE colloquium)
Nathaniel Trask (Sandia National Laboratory)
11:00-11:50 am, Room SH 306
Min Wu
(WPI)
11:00-11:50 am, Room SH 306
Guannan Zhang (Oak Ridge National Laboratory)
11:00-11:50 am, Room SH 203
Andrea Arnold (WPI)
Schedule Fall 2017
11:00-11:50 am, Room SL 406
Shankara Narayana Rao, Bheemaiah V
(WPI)
11:00-11:50 am, Room SL406
Carlos Borges
(University of Texas-Austin)
11:00-11:50 am, Room SL406
Zhimin
Zhang (Wayne State University)
11:00-11:50 am, Room SL406
Alfa R.H.
Heryudono (University of Massachusetts Dartmouth)
11:00-11:50 am, Room SL406
Thomas
Fai (Harvard University)
Past talks
List of Seminars at Department of mathematical Sciences