Numerical Analysis Archives for Fall 2016 to Spring 2017
Iterative solution methods for reduced-order models of parameterized PDEs
When: Tue, September 8, 2015 - 3:30pmWhere: Math 3206
Speaker: Virginia Forstall (Department of Mathematics, University of Maryland) -
Abstract: This talk will address efficient computational algorithms for solving
parameterized PDEs using reduced-order modeling. In some settings, it
is necessary to compute discrete solutions of the PDEs at many
parameter values. Accuracy considerations often lead to algebraic
systems with many unknowns whose solution via traditional methods can
be expensive. Reduced-order models use a reduced space to approximate
the parameterized PDE, where the reduced space is of a significantly
smaller dimension. Solving an approximation of the problem on the
reduced space leads to reduction in cost, often with little loss of
accuracy. In the traditional offline-online approach to these
problems, iterative methods can be used to improve efficiency in the
solution of reduced-order models using preconditioners constructed in
the offline step. In the case of nonlinear operators, these
techniques are combined with the discrete empirical interpolation
method, which reduces the cost of assembling the nonlinear component
of the model, to produce an efficient reduced solution. In cases that
do not follow the offline-online approach, like Krylov subspace
recycling methods, the reduced models are constructed to be well
conditioned.
Interpolatory Matrix Approximations
When: Tue, September 15, 2015 - 3:30pmWhere: Math 3206
Speaker: Mark Embree (Department of Mathematics, Virginia Tech) - http://www.math.vt.edu/people/embree/
Abstract: Interpolatory matrix factorizations provide alternatives to the singular value decomposition for obtaining low-rank approximations;
this class includes the CUR factorization, where the C and R matrices are formed from a subset of columns and rows of the target matrix.
While interpolatory approximations lack the SVD's optimality, their ingredients are easier to interpret than singular vectors: since they come directly from the matrix itself, they inherit the data's key properties (e.g., nonnegative/integer values, sparsity, etc.).
We shall provide an overview of these approximate factorizations, describe how they can be analyzed using interpolatory projectors, and introduce a new method for their construction based on the Discrete Empirical Interpolation Method (DEIM). This talk describes joint work with Dan Sorensen (Rice).
Nonequilibrium models for active matter systems, their numerical treatment and applications to complex biological systems
When: Tue, September 22, 2015 - 3:30pmWhere: Math 3206
Speaker: Prof. Qi Wang (Department of Mathematics, University of South Carolina) -
http://people.math.sc.edu/wangq/
Abstract: I will present a general approach to the derivation of nonequilibrium models for active matter systems, in particular, hydrodynamic models for active liquid crystal flows, explaining the way to deal with activities due to chemical or biological energy input at the molecular level to the otherwise dissipative system. Then, I will discuss an energy stable scheme for numerically solving the hydrodyamical model consisting of partial differential equations. Then, I will show how this set of moedling and computational tools can be employed to simulate cytokinesis of animal cells and traveling waves in swirling school of fish.
Minimal geodesics along volume preserving maps, through semi-discrete optimal transport
When: Tue, September 29, 2015 - 3:30pmWhere: Math 3206
Speaker: Jean-Marie Mirebeau (CNRS, University Paris-Sud, Department of Mathematics, UMR 8628, Orsay, France. ) - https://www.ceremade.dauphine.fr/~mirebeau/Page_de_Jean-Marie_Mirebeau/Main_page.html
Abstract: Geodesics along the group of volume preserving maps are known to solve Euler's equations of inviscid incompressible fluids, as observed by Arnold. On the other hand, the projection onto volume preserving maps amounts to an optimal transport problem, as follows from the generalized polar decomposition of Brenier.
In a joint work with Q. Merigot, we introduce a numerical scheme for Euler's Equations relying semi-discrete optimal transport. It is robust enough to extract non-classical, multi-valued solutions of Euler's equations, for which the flow dimension is higher than the domain dimension, a striking and unavoidable consequence of this model.
Gauges, sparsity, and spectral optimization
When: Wed, September 30, 2015 - 11:00amWhere: 4172 AV Williams
Speaker: Prof. Michael P. Friedlander (University of California, Davis) - https://www.math.ucdavis.edu/~mpf/
Abstract: Gauge optimization is the class of problems for finding the element
of
a convex set that is minimal with respect to a gauge (e.g., the
least-norm solution of a linear system). These conceptually simple
problems appear in a remarkable array of applications of sparse
optimization. Their structure allows for a special kind of duality
framework that can lead to new algorithmic approaches to challenging
problems. Low-rank spectral optimization problems that arise in two
signal-recovery application, phase retrieval and blind deconvolution,
illustrate the benefits of the approach.
A new approach to numerical computation of Hausdorff dimension of invariant sets associated to iterated function systems.
When: Tue, October 6, 2015 - 3:30pmWhere: Math 3206
Speaker: Prof. Richard Falk (Department of Mathematics, Rutgers, The State University Of New Jersey) - https://www.math.rutgers.edu/~falk/
Abstract: We develop a new approach to the computation of the Hausdorff dimension of
the invariant set associated to an iterated function system. The key idea
is to associate a family of Perron-Frobenius operators $L_s$ to the mappings
of the iterated function system. The Hausdorff dimension of the invariant
set is then the value $s=s_*$ for which the largest eigenvalue $\lambda_s$
of $L_s$ is equal to one. This eigenvalue problem is then approximated by a
collocation method using continuous piecewise linear functions (in one
dimension) or bilinear functions (in two dimensions). Using the theory of
positive operators and explicit a priori bounds on the derivatives of the
(strictly positive) eigenvector corresponding to $\lambda_{s_*}$, rigorous
upper and lower bounds are given for the Hausdorff dimension, which converge
to the value $s_*$ as the mesh size approaches zero.
Inference as optimal transportation
When: Tue, October 13, 2015 - 3:30pmWhere: Math 3206
Speaker: Youssef M. Marzouk (aerospace computational design lab, MIT) - http://aeroastro.mit.edu/faculty-research/faculty-list/youssef-m-marzouk
Abstract: Characterizing the high-dimensional and non-Gaussian posterior distributions that arise in Bayesian inference is a challenging task. A recent variational approach to this problem seeks to construct a deterministic transport map from a reference distribution to the posterior. Independent and unweighted posterior samples can then be obtained by pushing forward reference samples through the map. Constructing such maps efficiently, however, requires the identification and exploitation of low-dimensional structure.
Formally, the transport map can be obtained as the solution of an optimization problem over a function space. We will present this framework and discuss sources of low dimensionality therein. First is the conditional independence structure of the target distribution: we show that this structure can be revealed through the analysis of certain average derivative functionals, and that conditional independence in turn implies a particular sparsity of the transport maps. This structure can be represented with undirected graphical models, a connection that yields many useful algorithms for efficient ordering, decomposition, and inference---here, generalized to the continuous and non-Gaussian setting. Second, we analyze situations where conditional independence may not be present in the original parameterization of the problem, but can be induced through a suitable change of basis; these rotations correspond to low-rank approximations of the prior-to-posterior update in Bayesian inverse problems. The resulting algorithms compose a sequence of transport maps, interleaved with rotations, that adapts to the structure of high-dimensional target or posterior distribution. We demonstrate our approach on Bayesian inference problems arising in spatial statistics and PDEs.
This is joint work with Alessio Spantini (MIT).
$C^0$ DG methods for elliptic problems in non-divergence form
When: Tue, October 27, 2015 - 3:30pmWhere: Math 3206
Speaker: Michael Neilan (Department of Mathematics, University of Pittsburgh) - http://www.pitt.edu/~neilan/Neilan/index.html
Abstract:
In this talk, we describe a class of finite element methods for $W^{2,p}$ strong solutions of second-order linear elliptic PDEs in non-divergence form. The main novelty of the method is the inclusion of an interior penalty term, which penalizes the jump of the flux across the interior element edges/faces, to augment a nonsymmetric piecewise defined and PDE-induced bilinear form. Existence, uniqueness and error estimate in a discrete $W^{2,p}$ energy norm are proved for the proposed finite element method. This is achieved by establishing a discrete Calderon-Zygmund-type estimate and mimicking strong solution PDE techniques at the discrete level.
Incorporating local boundary conditions into nonlocal theories
When: Tue, November 3, 2015 - 3:30pmWhere: Math 3206
Speaker: Prof. Burak Aksoylu (Wayne State University, Department of Mathematics TOBB University of Economics and Technology, Department of Mathematics ) - http://www.math.lsu.edu/~burak
Abstract: We study nonlocal wave equations on bounded domains related to
peridynamics. We generalize the standard integral based convolution to an
abstract convolution operator defined by a Hilbert basis. This operator is
a function of the classical operator which allows us to incorporate local
boundary conditions into nonlocal theories. We present a numerical study of
the solutions. For discretization, we employ a weak formulation based on a
Galerkin projection which allows discontinuites on element boundaries.
Algebraic multigrid for the de Rham complex
When: Tue, November 10, 2015 - 3:30pmWhere: Math 3206
Speaker: Dr. Andrew Barker (Livermore National Labs) - http://people.llnl.gov/barker29
Abstract: Fast, scalable solution of very large problems in scientific computing can be performed effectively with algebraic multigrid. However, most existing algebraic multigrid algorithms target only the H1 space and are tailored to bilinear forms involving the gradient operator. In this talk we discuss the development of element-based algebraic multigrid methods for the whole de Rham sequence of H1, H(curl), H(div) and L2 spaces. The resulting method uses geometric information on the fine mesh, but generates coarse meshes in a purely algebraic way while maintaining the exactness properties of the de Rham complex, allowing for the development of scalable solvers for the entire complex.
Greedy algorithms and thresholding for dictionaries
When: Tue, November 17, 2015 - 3:30pmWhere: Math 3206
Speaker: Wolfgang Dahmen (Institut fur Geometrie und Praktische Mathematik, RWTH-Aachen University) - https://www.igpm.rwth-aachen.de/personen/dahmen
Abstract: Redundant systems like frames or even more general dictionaries offer advantages with respect to reducing the effect of transmission errors and, in principle, more flexibility with respect to best n-term approximation. On the other hand, the redundancy poses severe obstructions to actually computing best n-term approximations. In this talk we analyze and discuss the performance of concrete schemes like thresholding or greedy methods, which are known to realize best n-term approximations for orthonormal bases, in a more general dictionary setting.
Tensor Sparsity - a Regularity Notion for High Dimensional PDEs [Aziz Lecture]
When: Wed, November 18, 2015 - 3:15pmWhere: Math 3206
Speaker: Prof. Wolfgang Dahmen (Institut fur Geometrie und Praktische Mathematik, RWTH-Aachen University) - https://www.igpm.rwth-aachen.de/personen/dahmen
Abstract: The numerical solution of PDEs in a spatially
high-dimensional regime (such as the electronic
Schrodinger or Fokker-Planck equations) is severely
hampered by the "curse of dimensionality": the
computational cost required for achieving a desired
target accuracy increases exponentially with respect to
the spatial dimension.
We explore a possible remedy by exploiting a typically
hidden sparsity of the solution to such problems with
respect to a problem dependent basis or dictionary. Here
sparsity means that relatively few terms from such a
dictionary suffice to realize a given target accuracy.
Specifically, sparsity with respect to dictionaries
comprised of separable functions -- rank-one tensors --
would significantly mitigate the curse of dimensionality.
The main result establishes such tensor-sparsity for
elliptic problems over product domains when the data are
tensor-sparse, which can be viewed as a structural
regularity theorem.
Analysis of an augmented mixed-FEM for the Navier-Stokes problem
When: Tue, November 24, 2015 - 3:30pmWhere: Math 3206
Speaker: Dr. Jessica Camano (Departamento de Matematica y Fisica Aplicadas, Universidad Catolica de la Santisima Concepcion) -
Abstract: In this work we propose and analyze a new augmented mixed finite element method for the Navier-Stokes problem. Our approach is based on the introduction of a nonlinear-pseudostress tensor linking the pseudostress tensor with the convective term, which leads to a mixed formulation with the nonlinear-pseudostress tensor and the velocity as the main unknowns of the system.
Blended atomistic-to-continuum hybrid methods for modeling crystalline materials [Special Date]
When: Thu, December 3, 2015 - 3:30pmWhere: Math 3206
Speaker: Prof. Helen Li (Department of Mathematics and Statistics, University of North Carolina at Charlotte) - https://sites.google.com/a/brown.edu/xingjie-helen-li/
Abstract: The development of consistent and stable atomistic-to-continuum coupling models for multi-dimensional crystalline solids remains a challenge. In this talk, we consider two prototypical atomistic-to-continuum coupling methods of blending type: the energy-based and the force-based quasicontinuum methods, with a comprehensive error analysis that is valid in two and three dimensions, for simple crystals, and in the presence of lattice defects (point defects and dislocations). Recently, we extend our analysis to multi-lattices crystals, for example, graphene type structures.
Based on a precise choice of blending mechanism, the error estimates are considered in terms of degrees of freedom. The numerical experiments for simple crystals confirm the theoretical predictions.
Numerical analysis for Neumann boundary control problems governed by semilinear elliptic equations in polygonal domains
When: Tue, December 8, 2015 - 3:30pmWhere: Math 3206
Speaker: Johannes Pfefferer (Universitat der Bundeswehr Munchen) - https://www.unibw.de/bauv1/forschung/personen/pfefferer/
Abstract: This talk is concerned with the discretization error analysis for semilinear elliptic Neumann boundary control problems in polygonal domains where the control has to fulfil pointwise inequality constraints. In order to solve this problem the state and the adjoint state are discretized by linear finite elements, whereas the control is approximated by piecewise constant functions. In a postprocessing step approximations of the continuous optimal control are constructed which possess superconvergence properties, i.e., by imposing second order sufficient optimality conditions it is possible to prove nearly second order convergence for the postprocessed control on quasi-uniform meshes in domains with largest interior angle smaller than $2\pi/3$. However, for larger interior angles the presence of corner singularities lowers the convergence rates in general. In that case mesh grading techniques are used to compensate this negative influence. Finally, the quality of the approximations is demonstrated by a numerical example.
TBA
When: Mon, January 11, 2016 - 3:30pmWhere: 3258 AV Williams
Speaker: Sven Leyffer (Argonne National Laboratory) - https://wiki.mcs.anl.gov/leyffer/index.php/Sven_Leyffer
A new approach to finite element simulation of general relativity
When: Tue, February 2, 2016 - 3:30pmWhere: Math 3206
Speaker: Dr. Vincent Quenneville Belair (Columbia University) - http://www.columbia.edu/~vq2111/
Abstract: In order to study gravitational waves, I introduce a new approach to finite element simulation of general relativity. This approach is based on approximating the Weyl curvature directly through new stable mixed finite elements for the Einstein-Bianchi system. I design and analyze these novel finite elements by adapting the recently developed Finite Element Exterior Calculus (FEEC) framework to abstract Hodge wave equations. This framework enables me to borrow key ideas from Reissner-Mindlin plate bending and elasticity with weakly imposed symmetries to maintain stability of the method. The stability of a discretization often relies on deep connections between fundamental branches of mathematics: the FEEC mimics these connections for the numerical method to achieve similar stability to that of the original equations. The recent development of FEEC has had a transformative impact on electromagnetism and related computational problems, and I am now expanding it to general relativity.
The Optimal Transportation Problem: applications and numerical methods (joint PDE/NA seminar)
When: Thu, February 4, 2016 - 3:30pmWhere: Math 3206
Speaker: Prof. Adam Oberman (McGill University) - http://www.adamoberman.net/
Abstract: The Optimal Transportation problem has been the subject of a great
deal of attention by theoreticians in last couple of decades. The
Wasserstein (or Earth Mover) distance allows for the metrization of
the space of probability measures. However computation of these
distances (and the associated maps) has been intractable, except for
very small problems.
Current applications of Optimal Transportation include: Freeform
Illumination Optics for shaping light or laser beams, Shape
Interpolation (in computer graphics), Machine learning (comparing
histograms), discretization of nonlinear PDEs (using the gradient
flow in the Wasserstein metric), parameter estimation in geophysics,
matching problems in mathematical economics, and Density Functional
Theory in physical chemistry.
Recent advances have allowed for more efficient computation of
solutions of the Monge-Kantorovich problem of optimal transportation.
In the special, but important case of quadratic costs, the map can
be obtained from the solution of the elliptic Monge-Ampere partial
differential equation with nonstandard boundary conditions. For more
general costs, the Kantorovich plan can be approximated by a finite
dimensional linear program. In this talk we will compare the cost
and quality of the solutions obtained by two different methods.
I will also discuss some nonlinear PDE problems (curvature flows,
2-Hessian equation) which can be solved using similar techniques to
those applied to the Monge-Ampere PDE.
Optimal control of fractional order PDEs
When: Tue, February 9, 2016 - 3:30pmWhere: Math 3206
Speaker: Prof. Harbir Antil (Department of Mathematical Sciences George Mason University) - http://math.gmu.edu/~hantil/
Abstract: Diffusion is the tendency of a substance to evenly spread into an available space, and is one of the most common physical processes. The classical models of diffusion lead to well-known equations. However, in recent times, it has become evident that many of the assumptions involved in these models are not always satisfactory or not even realistic at all. Consequently, different models of diffusion have been proposed, fractional diffusion being one of them. The latter has received a great deal of attention recently, mostly fueled by applications in diverse areas such as finance, turbulence and quasi-geostrophic flow models, image processing, peridynamics, biophysics, and many others.
This talk will serve as an introduction to fractional diffusion equation - fractional derivative in both space and time. A novel PDE result by Caffarelli and Silvestre '07 has led to innovative schemes to realize the fractional order operators. We will discuss these numerical methods and their application to PDE constrained optimization problems.
On the error control for fully discrete approximations of evolution PDEs
When: Tue, February 16, 2016 - 3:30pmWhere: Math 3206
Speaker: Dr. H. Karakatzani (University of Chester, England) - http://www.chester.ac.uk/departments/mathematics/staff/dr-f-karakatsani
Abstract: We consider fully discrete schemes for linear parabolic problems. For the discretization in space we use the finite element method and for the discretization in time we consider the Crank-Nicolson and the fractional step \theta-scheme. We study the effect of mesh modification on the stability of fully discrete approximations as well as its influence on residual-based aposteriori error estimators. The aposteriori estimates are derived by using the reconstruction technique.
Duality-based model adaptation in context of multiscale methods
When: Tue, February 23, 2016 - 3:30pmWhere: Math 3206
Speaker: Dr. Matthias Maier (University of Minnesota) - http://math.umn.edu/~msmaier/cv.shtml
Abstract: A large class of modeling problems in Physics and Engineering is of multiscale character, meaning, that relevant physical processes act on highly different length scales. This usually implies high computational cost for a full resolution of the problem. One way to avoid such a full resolution are multiscale schemes, where, generally speaking, an effective model is solved on a coarse scale with upscaled, effective parameters. Those paramaters are determined with the help of localized sampling problems on a fine scale.
Multiscale schemes introduce significant complexity with respect to sources of error, not only are there discretization errors on a coarse and fine scale, but also a model error introduced by the modeling assumption. This makes suitable a posteriori strategies highly necessary.
In this talk different model adaptation strategies for the Variational Multiscale Method (VMM) and the Heterogeneous Multiscale Method (HMM) are examined and a general framework for model adaptation(based on the HMM) is introduced. The framework is derived within the setting of ``goal-oriented'' adaptivity given by the so-called Dual Weighted Residual (DWR) method.
Based on the framework a sampling-adaptation strategy is proposed that allows for simultaneous control of discretization and model errors with the help of classical refinement strategies for mesh and sampling regions. Further, a model-adaptation approach is derived that interprets model adaptivity as a minimization problem of a local model-error indicator.
Toward Multiscale Material Modeling with Peridynamics
When: Tue, March 1, 2016 - 3:30pmWhere: Math 3206
Speaker: Dr. Pablo Seleson (Oak Ridge National Laboratory) -
Abstract: Peridynamics is a nonlocal reformulation of the classical theory of continuum mechanics, based on integral equations, suitable for material failure and damage simulation. Unlike the classical (local) theory, based on partial differential equations, constitutive models in peridynamics do not require a spatial differentiability assumption on displacement fields. As a nonlocal model, peridynamics possesses a length scale which
can be controlled for multiscale modeling. For instance, classical elasticity has been presented as a limiting case of a peridynamic model. In addition, certain discretizations of peridynamic models possess the same computational structure as molecular dynamics equations, motivating the study of peridynamics as a coarse-graining of molecular dynamics. In this talk, I will present analytical and numerical connections of peridynamics to molecular dynamics and classical continuum mechanics, and I will discuss various capabilities of peridynamics toward multiscale material modeling.
Solution paths of convex regularizations
When: Tue, March 8, 2016 - 3:30pmWhere: Math 3206
Speaker: Bjorn Bringmann (Technische Universitat Munchen , Germany) -
Abstract: Energy functionals with quadratic data delity terms and `1- regularizations are frequently used in image and signal analysis to recover a signal from linear measurements. An approximate reconstruction is obtained by solving variational problems. In order to compute a reconstruction close to the original signal, the regularization parameter t, which determines the trade-off
between fitting the measured data and obtaining regularized signals, has to be chosen wisely. Instead of solving the problem just for a fixed t, I will explain how to compute a solution for every nonnegative regularization parameter in finite time.
Under a condition on the solution path, called "one-at-a-time", the well-known Homotopy method is able to compute a piecewise linear and continuous solution path by solving a linear system at every kink. To illustrate that this condition is necessary, I will discuss a toy example in which the standard Homotopy method fails.
By replacing the linear system with a nonnegative least squares problem, our method works for arbitrary measurement matrices and input data. I will give a full characterization of the set of possible directions and discuss the finite termination property of our algorithm. Finally, I will describe some open problems.
This is joint work with Felix Krahmer, Michael Moeller, and Daniel Cremers
Meshfree finite difference methods for fully nonlinear elliptic equations
When: Tue, March 22, 2016 - 3:30pmWhere: Math 3206
Speaker: Prof. Brittany Froese (Department of Mathematical Sciences, New Jersey Institute of Technology) - https://web.njit.edu/~bdfroese/
Abstract: The relatively recent introduction of viscosity solutions and the Barles-Souganidis convergence framework have allowed for considerable progress in the numerical solution of fully nonlinear elliptic equations. Convergent, wide-stencil finite difference methods now exist for a range of problems that satisfy a comparison principle.
However, these schemes are defined only on uniform Cartesian meshes over a rectangular domain. We describe a framework for constructing convergent meshfree finite difference approximations for a class of nonlinear elliptic operators. These approximations are defined on unstructured point clouds, which allows for computation on
non-uniform meshes and complicated geometries. Because the schemes are monotone, they fit within the Barles-Souganidis convergence framework and can serve as a foundation for higher-order filtered methods. We present computational results for several examples including problems posed on random point clouds, computation of convex envelopes, obstacles problems, and Monge-Ampere equations. In some settings, even a simple Dirichlet boundary condition needs to be interpreted in a weak sense, the resulting viscosity solutions can be discontinuous, and the usual comparison principle is not valid. However, it is still possible to produce meshfree methods that converge in the interior of the domain, as we demonstrate for a Monge-Ampere equation used to generate surfaces of prescribed Gaussian curvature.
Reduced basis approximation and a posteriori error estimates for parametrized elliptic eigenvalue problems
When: Tue, March 29, 2016 - 3:30pmWhere: Math 3206
Speaker: Dr. Ivan Fumagalli (Politecnico di Milano, Italy) -
Abstract: We develop a new reduced basis (RB) method for the rapid and reliable approximation of parametrized elliptic eigenvalue problems. The method is based upon dual weighted residual type a posteriori error indicators which estimate, for any value of the parameters, the error between the high-fidelity finite element approximation of the first eigenpair and the corresponding reduced basis approximation. The proposed error estimators are exploited both to certify the RB approximation with respect to the high-fidelity one, and to set up a greedy algorithm for the offline construction of a reduced basis space. Several numerical experiments show the overall validity of the proposed RB approach, comparing it also with the standard POD-RB approximation. (Joint work with A. Manzoni, N. Parolini, M. Verani)
A Structure Preserving Discretization For Liquid Crystals With External Fields (joint PDE and Numerical Analysis Seminar)
When: Thu, April 14, 2016 - 3:30pmWhere: Math 3206
Speaker: Prof. Shawn Walker (Department of Mathematics and center for computation and technology, Louisiana State University) - https://www.math.lsu.edu/~walker/
Abstract: We present a finite element method (FEM) for computing equilibrium configurations of liquid crystals with variable degree of orientation. The model consists of a Frank-like energy with an additional "s" parameter that allows for line defects with finite energy, but leads to a degenerate elliptic equation for the director field. Our FEM uses a special discrete form of the energy that does not require regularization, and allows us to obtain a stable (gradient flow) scheme for computing minimizers of the energy. We also include external fields to model the so-called "Freedricksz Transition". Simulations in 2-D and 3-D are presented to illustrate the method.
Numerical Approximation of Variational Problem on Bounded Domain involving the Fractional Laplacian
When: Tue, April 19, 2016 - 3:30pmWhere: Math 3206
Speaker: Wenyu Lei (Texas A&M University) -
Abstract: The mathematical theory and numerical analysis of non-local operators has been the topic of intensive research in recent years. One class of applications come from replacing Brownian motion diffusion by diffusion defined by a symmetric $\alpha$-stable Levy?s process, i.e., where the Laplace operator is replaced by a fractional Laplacian.
In this talk, we propose a numerical approximation of equations with these types of diffusion terms. We focus on the simplest example of an elliptic variational problem coming from this fractional Laplacian on a bounded domain with homogeneous Dirichlet conditions on the complement. Although it is conceptually feasible to study the Galerkin approximation based on a standard finite element space, such a direct approach is not viable as the exact computation of the resulting stiffness matrix entries is not possible (at least in two or more spatial dimensions).
Instead, we will develop a non-conforming method by approximating the action of the stiffness matrix on a vector (sometimes referred to as a matrix free approach). We first derive an improper integral representation of the bilinear form corresponding to the above variational problem involving solutions of elliptic problems defined on $\mathbb{R}^d$. The numerical approximation of the action of the corresponding stiffness matrix consists of three stages: (i) Apply a SINC quadrature scheme to approximate the integral representation by a finite sum where each term involves the solution of an elliptic partial differential equation defined on $\mathbb{R}^d$; (ii) Reduce each term to a truncated problem on a bounded domain; (iii) Use the finite element method to approximate the solution of the truncated problem. The consistency error analysis for the first two steps is discussed together with the numerical implementation of the entire algorithm. The results of computations illustrate the error behavior in terms of the mesh size restricted to the bounded domain, the domain truncation parameter and the quadrature spacing parameter.
This is joint work with Andrea Bonito and Joseph E. Pasciak
A Meshless Galerkin Method For Non-Local Diffusion Using Localized Kernel Bases [Special Room]
When: Tue, April 26, 2016 - 3:30pmWhere: MTH 0407
Speaker: Dr. Richard Lehoucq (Sandia National Laboratories ) - http://www.sandia.gov/~rblehou/
Abstract: We introduce a meshless method for solving both continuous and discrete variational formulations of a volume constrained, nonlocal diffusion problem. Our method is nonconforming and uses a localized Lagrange basis that is constructed out of radial basis functions. By verifying that certain inf-sup conditions hold, we demonstrate that both the continuous and discrete problems are well-posed, and also present numerical and theoretical results for the convergence behavior of the method. The stiffness matrix is assembled by a special quadrature routine unique to the localized basis. Combining the quadrature method with the localized basis produces a well-conditioned, symmetric matrix.
This is joint work with Francis J. Narcowich (Texas A&M), Stephen T. Rowe (Sandia National Laboratories), Joseph D. Ward (Texas A&M).
Improving Numerical Weather Prediction and Assessing Observing System Impact Using Data Assimilation
When: Tue, May 3, 2016 - 3:30pmWhere: Math 3206
Speaker: Prof. Ide Kayo (Department of Atmospheric and Oceanic Science, Center for Scientific Computation And Mathematical Modeling, Institute for Physical Science and Technology ) -
Abstract: Data assimilation is an iterative method for scientific estimation and prediction. It attempts to combine the two sources of information in a statistically consistent manner: one source is dynamic forecast of the state by the computational models and the other is the real-time observations of the physical system. Numerical weather prediction (NWP) is a form of data assimilation and provides daily weather services. This talk will cover the overview, with focus on practice and challenges for the global and regional NWP.
[Aziz Lecture special time] Quantum Dots and Dislocations: Dynamics of Materials Defects
When: Wed, May 4, 2016 - 3:15pmWhere: Math 3206
Speaker: Prof. Irene Fonseca (Department of Mathematical Sciences, Carnegie Mellon University ) - http://www.math.cmu.edu/math/faculty/Fonseca
Abstract: The formation and assembly patterns of quantum dots have a significant impact on the optoelectronic properties of semiconductors. We will address short time existence for a surface diffusion evolution equation with curvature regularization in the context of epitaxially strained three-dimensional films. Further, the nucleation of misfit dislocations will be analyzed. This is joint work with Nicola Fusco, Giovanni Leoni and Massimiliano Morini.
[Applied Math Seminar] Homogenization of Integral Energies Under Periodically Oscillating Differential Constraints
When: Thu, May 5, 2016 - 3:30pmWhere: Math 3206
Speaker: Prof. Irene Fonseca (Department of Mathematical Science, Carnegie Mellon University) - http://www.math.cmu.edu/math/faculty/Fonseca
Abstract: A homogenization result for a family of integral energies is presented, where the fields are subjected to periodic first order oscillating differential constraints in divergence form. We will give an example that illustrates that, in general, when the operators differential operators have non constant coefficients then the homogenized functional maybe be nonlocal, even when the energy density is convex. This is joint work with Elisa Davoli, and is based on the theory of A-quasiconvexity with variable coefficients and on two-scale convergence techniques.