Numerical Analysis Archives for Fall 2014 to Spring 2015
High Order FEM for Wave Propagation: Like it or lump it!?
When: Tue, September 10, 2013 - 3:30pmWhere: Math 3206
Speaker: Mark Ainsworth (Division of Applied Mathematics, Brown University) -
Abstract: High order finite element methods have long been used for computational wave propagation for both first order and second order equations alike. A key issue with computational wave propagation is the phase accuracy of the methods: getting the waves to propagate with the right speed is often at least as, if not more, important than the convergence rate. The main variants are the finite element method (FEM) and spectral element method (SEM) with each technique having its band of disciples.
SEM can be viewed as a higher order mass-lumped FEM. Despite the predominance of these methods, there is comparatively little by way of hard analysis on what each variant offers in terms of phase accuracy, and there is considerable misinformation and confusion in the literature on this topic. In this presentation, we shall attempt to shed some light on the matter, and also briefly
mention new methods that improve on both FEM and SEM.
A decomposition technique for integrable functions with applications to the divergence problem on bad domains
When: Tue, September 17, 2013 - 3:30pmWhere: Math 3206
Speaker: Fernando Lopez Garcia (Woschester Polytechnic Institute) -
Abstract: Let U be a bounded domain that can be written as the union of a countable collection of subdomains with certain properties, for example, the union of Whitney cubes. In this talk, we shall show a technique to decompose a function f, with vanishing mean value, into the sum of a collection of functions, with the same property, and subordinated to the collection of subdomains of U. This result generalizes an idea published by Bogovskii when the collection of subdomains is finite. Moreover, we apply this decomposition to show the existence of a solution in weighted Sobolev spaces of the divergence problem div u= f, and the well-posedness of the Stokes equations on bad domains, for example, domains with external cusps.
Centroidal Voronoi Tessellations: from Lloyd method to multigrid and beyond
When: Tue, September 24, 2013 - 3:30pmWhere: Math 3206
Speaker: Maria Emelianenko (Department of Mathematics, George Mason University) - http://math.gmu.edu/~memelian/
Abstract: This talk will focus on recent advances associated with fast and
reliable algorithms for computing centroidal Voronoi tessellations
(CVT) - special tessellations of the domain into Voronoi cells having
centers of mass as their generators. CVT concept is at the heart of
many modern applications, from image segmentation to signal analysis
and mesh generation. Although many methods for constructing CVTs have
been introduced in the last decades, their analysis is far from
complete. Only recently rigorous convergence results for the commonly
used Lloyd method have been established, and many conjectures remain
open. We will touch upon several classes of methods, from traditional
methods for which new convergence estimates are obtained, to novel
multilevel techniques which allow for uniform convergence with
respect to the problem size. Numerical demonstrations in various
applied settings will be provided and advantages/disadvantages of
each method will be discussed.
Hierarchical Krylov Methods and Nested Krylov Methods for Extreme-Scale Computing
When: Tue, October 1, 2013 - 3:30pmWhere: Math 3206
Speaker: Lois Curfman McInnes (Mathematics and Computer Science Division Argonne National Laboratory) - http://press.mcs.anl.gov/curfman/
Abstract: Preconditioned Krylov methods are widely used and offer many advantages for solving sparse linear systems that do not have highly convergent, geometric multigrid solvers or specialized fast solvers. Unfortunately, however, Krylov methods encounter well-known difficulties in scaling beyond 10,000 processor cores because each iteration requires at least one vector inner product, which in turn requires a global synchronization that scales poorly because of internode latency. To help overcome these difficulties, we have developed hierarchical and nested Krylov methods in the PETSc library that reduce the number of global inner products required across the entire system (where they are expensive), while freely allowing inner products across smaller subsets of the entire system (where they are inexpensive) or using inner iterations that do not invoke vector inner products at all. We demonstrate that these methods significantly reduce overall simulation time on the Cray XK6 and Blue Gene/P for the PFLOTRAN subsurface flow application when using 10,000 through 224,000 cores.
An energetic variational approach for Ionic Fluids and Ion Channels
When: Tue, October 8, 2013 - 3:30pmWhere: Math 3206
Speaker: Prof. Chun Liu (Department of Mathematics Penn State University) - http://www.personal.psu.edu/cxl41/
Abstract: The interactions of ions flowing through biological systems has been a central topic in biology for more than 100 years. Flows of ions produce signaling in the nervous system, initiation of contraction in muscle, coordinating the pumping of the heart and regulating the flow of water through kidney and intestine.
Ion concentrations inside cells are controlled by ion channel proteins through the lipid membranes. In this talk, a continuum model is derived from the energetic variational approach which include the coupling between the electrostatic forces, the hydrodynamics, diffusion and crowding (due to the finite size effects). The model provides some basic understanding of some important properties of proteins, such as the ion selectivity and sensor mechanism.
Transport of charged particles and ions in biological environments is by nature a multiscale and multiphysics problem. I will also discuss the roles of other important ingredients such as those of nonlocal diffusion and also the connection between kinetic description and continuum approaches.
Discrete Emprirical Interpolation for Nonlinear Model Reduction
When: Tue, October 15, 2013 - 3:30pmWhere: Math 3206
Speaker: Prof. Danny Sorensen (Department of Computational and Applied Mathematics Rice University) - http://www.caam.rice.edu/~sorensen/
Abstract: I will present a method for nonlinear model reduction that is based upon proper orthogonal decomposition (POD) combined with a discrete empirical interpolation method (DEIM). This POD-DEIM approach has provided spectacular dimension and complexity reduction for challenging systems of large scale ordinary differential equations(ODEs). Reductions from 15,000 variables to 40 variables in the reduced model with very little loss of
accuracy have been achieved. For parabolic problems, this accuracy can be established rigorously.
The DEIM is surprisingly simple and amounts to replacing orthogonal projection with an interpolatory projection of the nonlinear term that only requires the evaluation of a few selected components of the original nonlinear term.
Bounds on Eigenvalues of Matrices Arising from Interior-Point Methods
When: Tue, October 22, 2013 - 3:30pmWhere: Math 3206
Speaker: Prof.Chen Greif (Department of Computer Science The University of British Columbia) - http://www.cs.ubc.ca/~greif/
Abstract: Interior-point methods feature prominently among numerical methods for inequality-constrained optimization problems, and involve the need to solve a sequence of linear systems that typically become increasingly ill-conditioned with the iterations. To solve these systems, whose original form has a nonsymmetric 3-by-3 block structure, it is common practice to perform block elimination and either solve the resulting reduced saddle-point system, or further reduce the system to the Schur complement and apply a symmetric positive definite solver. In this talk we use energy estimates to obtain bounds on the eigenvalues of the matrices, and conclude that the original unreduced matrix has more favorable eigenvalue bounds than the alternative reduced versions. Our analysis includes regularized variants of those matrices that do not require typical regularity assumptions. We also discuss a few possible ways for iteratively solving the linear system using specialized block preconditioners.
This is join work with Dominique Orban and Erin Moulding.
A New Mixed Formulation For a Sharp Interface Model of Stokes Flow and Moving Contact Lines
When: Tue, October 29, 2013 - 3:30pmWhere: Math 3206
Speaker: Shawn Walker (Department of Mathematics, Louisiana State University) - http://www.math.lsu.edu/~walker/
Abstract: Two phase fluid flows on substrates (i.e. wetting phenomena) are important in many industrial processes, such as micro-fluidics and coating flows. These flows include additional physical effects that occur near moving (three-phase) contact lines. We present a new 2-D variational (saddle-point) formulation of a Stokesian fluid with surface tension that interacts with a rigid substrate. The model is derived by an Onsager type principle using shape differential calculus (at the sharp-interface, front-tracking level) and allows for moving contact lines and contact angle hysteresis through a variational inequality. We prove the
well-posedness of the time semi-discrete and fully discrete (finite element) model and discuss error estimates. Simulation movies will be presented to illustrate the method. We conclude with some discussion of a 3-D version of the problem as well as future work on optimal control of these types of flows.
[APPLIED MATH COLLOQUIUM ] Coagulation dynamics of random shocks and phase boundaries
When: Thu, November 7, 2013 - 2:30pmWhere: Math 3206
Speaker: Prof. Robert Pego (Department of Mathematics, Carnegie Mellon University ) - http://www.math.cmu.edu/~bobpego/
Abstract: Smoluchowski?s coagulation equation is a simple kinetic model for clustering, and its dynamics are still poorly understood except for a few cases `solvable' by Laplace transform. However, among these cases are some directly connected with classic problems in PDE. Two relate to random solutions of the inviscid Burgers equation and models of the coarsening of patterns of phase boundaries for the Allen-Cahn equation. I plan to review existing results on scaling dynamics for these particular models of ballistic aggregation and annihilation, indicate current progress and describe some of the numerous open problems in this subject.
[Aziz Lecture Unusual time] Universality and chaos in clustering and branching processes
When: Fri, November 8, 2013 - 3:15pmWhere: Math 3206
Speaker: Prof. Robert Pego (Department of Mathematics, Carnegie Mellon University ) - http://www.math.cmu.edu/~bobpego/
Abstract: Scaling limits of Smoluchowski's coagulation equation are related to probability theory in numerous remarkable ways. E.g., such an equation governs the merging of ancestral trees in critical branching processes, as observed by Bertoin and Le Gall. A simple explanation of this relies on how Bernstein functions relate to a weak topology for Levy triples. From the same theory, we find the existence of `universal' branching mechanisms which generate complicated dynamics that contain arbitrary renormalized limits. I also plan to describe a remarkable application of Bernstein function theory to a coagulation-fragmentation model introduced in fisheries science to explain animal group size.
On the Existence of Flows Driven by Body Couples and Couple Stresses in Magnetic Nanoparticle Suspensions
When: Tue, November 12, 2013 - 3:30pmWhere: Math 3206
Speaker: Prof. Carlos Rinaldi (J. Crayton Pruitt Family Department of Biomedical Engineering Department of Chemical Engineering University of Florida, Gainesville) - http://www.bme.ufl.edu/people/rinaldi_carlos
Abstract: Suspensions of magnetic nanoparticles, so-called ferrofluids, are a commercially and scientifically relevant example of fluids whose description requires consideration of the balance of internal angular momentum. For over 40 years, researchers in this field have sought to understand the mechanisms responsible for observations of flows of these fluids driven solely by the action of rotating uniform magnetic fields. One possible mechanism is that these flows are driven by the action of body couples, exerted by the magnetic field on the particle’s magnetic dipoles, and so-called couple stresses, which account for the short-range transport of internal angular momentum. However, other possible mechanisms have been advanced in the literature, such as flow driven by traveling wave magnetic field gradients or by surface phenomena, and experiments have failed to shed light on the matter due to limitations in past experimental approaches. In this talk I will briefly summarize our past theoretical and experimental work investigating this phenomenon, as well as present new experimental results using model ferrofluids consisting of suspensions of magnetic nanoparticles with thermally-blocked magnetic dipoles, for which quantitative agreement can be obtained between experiment and theoretical predictions taking into account the action of body couples and couple stresses.
Active Fluids on Surfaces
When: Tue, November 19, 2013 - 3:30pmWhere: Math 3206
Speaker: Prof. Michael Shelley (Courant Institute, New York) - http://math.nyu.edu/faculty/shelley/
Abstract: Active fluids are complex fluids with active microstructure that create non-thermodynamic stresses within the fluid even in the absence of external forcing. A typical example of such a out-of-equilibrium system is a bacterial bath where these stresses can create chaotic mixing flows. However, synthetic systems, which are typically better characterized and
controlled, are increasingly available to study. I'll talk about modeling and simulating two such systems, one a so-called "active nematic" arising from mixtures of biopolymers and motor-proteins, and the other involving chemically active particles and intimately related to the Keller-Segel model of chemotaxis. In both cases their dynamics is naturally constrained to immersed surfaces.
Modeling and simulation of non-Newtonian fluids: fluid-particle suspensions, mantle flow and ice dynamics
When: Tue, November 26, 2013 - 3:30pmWhere: Math 3206
Speaker: Georg Stadler ( Institute for Computational Engineering and Sciences, University of Texas at Austin) - http://users.ices.utexas.edu/~georgst/
Abstract: I will discuss viscous flow problems that require the solution of large-scale nonlinear incompressible Stokes equations with strong viscosity contrasts. The driving applications are a model for high particle-to-fluid volume ratio suspensions, the viscous flow in the earth's mantle and the dynamical behavior of polar ice sheets. After giving an overview of the main challenges, I will focus on two particular aspects:
(1) Analysis and numerical methods for a nonsmooth continuum mechanical model for dispersions with a high particle-to-fluid volume fraction will be presented. The proposed model is based on the Stokes equation with a discontinuous shear thickening constitutive relation, allows formulation as nonsmooth optimization problem and has the form of a free boundary value problem. I will present a primal-dual formulation of the problem, discuss theoretical properties of the solution, propose a semismooth Newton solution algorithm and show numerical simulations motivated by recent experimental studies of dispersions.
(2) I will discuss the efficient parallel solution of Stokes flow problems with high viscosity contrast, which are discretized using higher-order finite elements on adaptive nonconforming hexahedral meshes. The solver is based on a matrix-free iterative Krylov method with a pressure Schur complement block preconditioner, and uses (algebraic or geometric) multigrid to approximately invert blocks in the preconditioner. I will show results for challenging Stokes
problems with more than 10^9 unknowns.
Local Structure Analysis and Defect Characterization in Atomic-Resolution Images
When: Mon, January 13, 2014 - 10:00amWhere: Math 3206
Speaker: Matt Elsey (Courant Institute, NYU) - http://cims.nyu.edu/~melsey/
Abstract: Recent advances allow us to capture enormous atomic-resolution images, but techniques for efficiently analyzing these images are lacking. To address this shortcoming, we propose a variational method which yields a tensor field describing the local crystal strain at each point. Local values of this field describe the crystal orientation and elastic distortion, while the curl of the field locates and characterizes crystal defects and grain boundaries. The proposed energy functional has a simple L2-L1 structure which permits minimization via a split Bregman iteration, and GPU parallelization results in short computing times. Joint work with Benedikt Wirth.
Music of the Microspheres: Eigenvalue problems from micro-gyro design
When: Tue, January 28, 2014 - 3:30pmWhere: Math 3206
Speaker: Prof. David Bindel (Department of Computer Science, Cornell University) - http://www.cs.cornell.edu/~bindel/index.html
Abstract: In 1890, G. H. Bryan demonstrated that when a ringing wine glass rotates, the shape of the vibration pattern precesses, and this effect is the basis for a family of high-precision gyroscopes. Mathematically, the precession can be described in terms of a symmetry-breaking perturbation due to gyroscopic effects of a geometrically degenerate pair of vibration modes. Unfortunately, current attempts to miniaturize these gyroscope designs are subject to fabrication imperfections that also break the device symmetry. In this talk, we describe how these devices work and our approach to accurate and efficient simulations of both ideal device designs and designs subject to fabrication imperfections.
Transient fluid and solid dynamics on linear tetrahedral finite elements: An accurate and stable variational multi-scale approach
When: Tue, February 4, 2014 - 3:30pmWhere: Math 3206
Speaker: Prof. Guglielmo Scovazzi (Civil Engineering Duke University ) - http://www.cee.duke.edu/faculty/guglielmo-scovazzi
Abstract: A new tetrahedral finite element for transient dynamic computations of fluids [1] and solids [2] is presented. It utilizes the simplest possible finite element interpolations: Piece-wise linear continuous functions are used for displacements and pressures (P1/P1), while the deviatoric part of the stress tensor (if present, as in the case of solids) is evaluated with simple single-point quadrature formulas. The variational multiscale stabilization eliminates the pressure checkerboard instabilities affecting the numerical solution in the case of the Darcy-type operator related to compressible fluids computations, or the Stokes-type operator related to solid dynamics computations. The formulation is extended to strong shock computations in fluids and to elastic-plastic flow in solids. Extensive tests of shock flows in fluids, and of linear elasticity
and finite elastoplasticity (compressible as well as nearly incompressible) will be presented. Because of its simplicity, the proposed element could favorably impact complex geometry, fluid/structure interaction, and embedded discontinuity computations. Time permitting, a number of preliminary results on fluid-structure interaction problems will also be presented.
References
[1] G. Scovazzi, “Lagrangian shock hydrodynamics on tetrahedral meshes: A stable and accurate varia-
tional multiscale approach”, J. Comp. Phys., 231(24), pp. 8029-8069, 2012.
[2] G. Scovazzi and B. Carnes, “Accurate and stable transient solid dynamics computations on linear finite
elements: A variational multiscale approach”, Int. J. Num. Meth. Engr., (in preparation), 2013.
This work was supported by Sandia National Laboratories under various grants, among which the Com-
puter Science Research Foundations (2010-2013) and Laboratory Directed Research & Development (2013-
2015) programs.
Universal meshes: A tool for domain discretization in problems with evolving geometries
When: Tue, February 11, 2014 - 3:30pmWhere: Math 3206
Speaker: Dr. Ramsharan Rangarajan (Brown University ) - http://lavxm.stanford.edu/wiki/Ramsharan_Rangarajan
Abstract: Computational methods for simulating problems with evolving domains & discontinuities invariably require robust and automatic methods to discretize changing geometries. So is the case in shape optimization, where the geometry is part of the solution. We introduce and discuss a new method that recovers conforming triangulations for planar curved domains from background meshes. In the process, no new vertices are introduced and connectivities of triangles are maintained. We demonstrate its robustness while using only simple ideas from geometry and optimization. The method facilitates a straightforward construction of optimal high-order finite elements as well. We discuss its application to engineering problems including hydraulic fracturing and penetration of ballistic gels.
Near-Best hp-Adaptive Tree Approximation
When: Tue, February 18, 2014 - 3:30pmWhere: Math 3206
Speaker: Prof. Peter Binev (Department of Mathematics, South Carolina State University ) - http://www.math.sc.edu/~binev/
Abstract: One of the prominent numerical procedures for solving partial differential quations is the hp-adaptive finite element method. However, there are no theoretical results establishing the optimal rates of convergence for this method. A closely related approximation theory problem is the following: construct an efficient procedure that for a given function finds a piecewise polynomial fit on adaptive partitions, in which the order of the polynomials on different elements of the partition may vary but the total number of degrees of freedom is fixed.
In this talk we consider domain partitioning based on a fixed binary refinement scheme and a coarse-to-fine routine for making adaptive decisions about the elements to be split and the polynomial orders to be assigned. The problem of finding near-optimal results is managed by using greedy-type algorithms on binary trees and a modification of the local errors that take into account the local complexity of the adaptive approximation. We prove that the algorithm provides near-best approximation to any given function.
Adaptive Methods for Large-Scale Optimization
When: Wed, February 19, 2014 - 11:00amWhere: [special location] 4172 AV Williams
Speaker: Tom Goldstein (Department of Electrical & Computer Engineering, Rice University) - http://www.ece.rice.edu/~tag7/Tom_Goldstein/Welcome.html
Abstract: As massive datasets and distributed computation become commonplace, the field of optimization is quickly evolving to meet new challenges. As a response to the challenges posed by big data, first-order (splitting) methods have become the workhorse behind large-scale systems because of their low complexity. These methods have been well studied in the context of smooth, unconstrained optimization. For more sophisticated problems, important practical considerations such as adaptivity, acceleration, and stopping conditions are not well understood. Furthermore, incorporating these features into solvers creates challenges from both a theoretical and practical perspective. This talk will address these issues, and discuss applications of fast splitting methods in machine learning, imaging, and wireless communications.
Universal Meshes for Problems with Moving Boundaries
When: Tue, February 25, 2014 - 3:30pmWhere: Math 3206
Speaker: Evan Gawlik (Stanford University ) - http://www.stanford.edu/~egawlik/
Abstract: We introduce a framework for the design of finite element methods for moving boundary problems with prescribed boundary evolution that have arbitrarily high order of accuracy, both in space and in time. At the core of our approach is the use of a universal mesh: a stationary background mesh containing the domain of interest for all times that adapts to the geometry of the immersed domain by adjusting a small number of mesh elements in the neighborhood of the moving boundary. The resulting method maintains an exact representation of the (prescribed) moving boundary at the discrete level, yet is immune to large distortions of the mesh under large deformations of the domain. The framework is general, allowing one to achieve any desired order of accuracy in space and time by selecting a suitable finite-element space on the universal mesh and a suitable time integrator for ordinary differential equations. We illustrate our approach on several examples in one and two dimensions.
On Some Nonlinear Problems in Poroelasticity
When: Tue, March 4, 2014 - 3:30pmWhere: Math 3206
Speaker: A.J. Meir (Department of Mathematics and Statistics, Auburn University) - http://wp.auburn.edu/ajm/
Abstract: Poromechanics is the science of energy, motion, and forces, and their effect on porous material and in particular the swelling and shrinking of fluid-saturated porous media. Modeling and predicting the mechanical behavior of fluid-infiltrated porous media is significant since many natural substances, for example, rocks, soils, clays, shales, biological tissues, and bones, as well as man-made materials, such as, foams, gels, concrete, water-solute drug carriers, and ceramics are all elastic porous materials (hence poroelastic).
In this talk I will describe some nonlinear problems in poroelasticity and their mathematical analysis. I will also describe finite element based numerical methods for approximating solutions of (nonlinear) model problems in poroelasticity, and the available a-priori error estimates.
Adaptive High-Order Methods
When: Tue, March 11, 2014 - 3:30pmWhere: Math 3206
Speaker: Marco Verani (MOX, Department of Mathematics Politecnico di Milano) - http://mox.polimi.it/~verani/
Abstract: We study the performance of adaptive high-order Galerkin
methods for the solution of linear elliptic problems.
These methods offer unlimited approximation power only restricted
by solution and data
regularity. In particular, we consider adaptive Fourier-Galerkin
methods and adaptive Legendre-Galerkin
methods and examine their convergence and optimality properties.
This study is of intrinsic interest but is also a first step
towards understanding adaptivity for the $hp$-FEM. (Joint work with
C. Canuto and R.H. Nochetto)
Towards Domain Decomposition Methods for Stochastic Partial Differential Equations
When: Tue, March 25, 2014 - 3:30pmWhere: Math 3206
Speaker: Bedrich Sousedik (Department of Computer Science, University of Maryland at College Park) - http://www.cs.umd.edu/~sousedik/
Abstract: Our goal is to develop fast, parallel, iterative algorithms that would allow efficient solution of systems of linear equations obtained from stochastic finite element discretizations. Our strategy and this talk consist of two parts. In the first part of the talk I will present the Adaptive-Multilevel BDDC method, where BDDC stands for Balancing Domain Decomposition by Constraints.
As opposed to the original two-level BDDC, the multilevel approach preserves scalability as the number of subdomains increases, and the adaptivity enables detection of troublesome parts of the problem on each decomposition level. In the second part I will discuss the structure of the matrices obtained from stochastic Galerkin discretizations of stochastic elliptic boundary value problems, and outline preconditioners that take advantage of the recursive hierarchy in the matrix structure.
No seminar
When: Tue, April 1, 2014 - 3:30pmWhere: Math 3206
Speaker: () -
Numerical solution of transient nonlinear axisymmetric eddy current models with hysteresis.
When: Tue, April 8, 2014 - 3:30pmWhere: Math 3206
Speaker: Dr. Pablo Venegas (Department of Mathematics, University of Maryland) - http://www2.math.umd.edu/~pvenegas
Abstract: This work deals with the mathematical analysis and the computation of transient electromagnetic fields in nonlinear magnetic media with hysteresis. We complement previous results where the mathematical and numerical analysis of a 2D nonlinear axisymmetric eddy current model was performed under fairly general assumptions on the H–B curve but without considering hysteresis effects. In our case, the constitutive relation between H and B is given by a hysteresis operator, i.e., the values of the magnetic induction depend not only on the present values of the magnetic field but also on its past history. We assume axisymmetry of the fields and then we consider two kinds of boundary conditions. Firstly the magnetic field is given on the boundary (Dirichlet
boundary condition). Secondly, the magnetic flux through a meridional plane is given, leading to a non-standard boundary-value problem. For both problems, an existence result is achieved under suitable assumptions. For the numerical solution, we consider the Preisach model as hysteresis operator, a finite element discretization by piecewise linear functions, and the backward Euler time-discretization.We report a numerical test in order to assess the order of convergence of the proposed numerical method. Finally, we validate the numerical scheme with experimental results. With this aim, we consider
a physical application: the numerical computation of eddy current losses in laminated media as those used in transformers or electric machines.
Computing the asymptotic spectrum for networks representing energy landscapes.
When: Tue, April 15, 2014 - 3:30pmWhere: Math 3206
Speaker: Prof. Maria Cameron (Department of Mathematics, University of Maryland) - http://www2.math.umd.edu/~mariakc/
Abstract: The concept of metastability has caused a lot of interest in recent years. The spectral decomposition of the generator matrix of a stochastic network exposes all of the transition processes in the system while it evolves toward the equilibrium. I discuss an efficient way to compute the asymptotics for eigenvalues and eigenvectors starting from the low lying group for networks representing energy landscapes. I apply this algorithm to Wales's Lennard-Jones-38 stochastic network with 71887 states and 119853 edges whose underlying potential energy landscape has a double-funnel structure. The result turns out to be surprising at the first glance. The concept of metastability should be applied with care to this system.
Discretization of time-dependent quantum systems: prop- agation of the evolution operator
When: Tue, April 22, 2014 - 3:30pmWhere: Math 3206
Speaker: Prof. Joseph Jerome (Northwestern University and George Washington University) - http://www.math.northwestern.edu/~jwj/
Abstract: The talk represents joint work with Eric Polizzi and is based on
a paper of similar title recently published online in Applicable Analysis. We
discuss time-dependent quantum systems on bounded domains; these repre-
sent closed systems and are relevant for application to Carbon Nanotubes and
molecules. Included in our framework are linear iterations involved in time-
dependent density functional theory as well as the global nonlinear model which
includes the Hartree potential. A key aspect of the analysis of the algorithms is
the use of time-ordered evolution operators, which allow for both a well-posed
problem and its approximation. The approximatiom theorems we obtain are op-
erator extensions of classical quadrature theorems. The global existence theorem
uses the Leray-Schauder fixed point theorem, coupled to a modified conserva-
tion of energy principle. The simulations were performed by Eric Polizzi using
his algorithm FEAST. The evolution operators used in the talk are due to T.
Kato and their properties will be summarized. Application areas make signifi-
cant use of these operators, particularly chemical physics. In the mathematical
literature, the Euclidean space problem has been studied by T. Cazenave and
others, employing the Strichartz inequalities. These are ultimately based on
semi-groups. Our results appear to be complementary to results of this type.
The solutions we discuss are strong solutions. We are currently studying more
general potentials via weak solutions. This work is in-progress.
Nearly time-periodic water waves
When: Tue, April 29, 2014 - 3:30pmWhere: Math 3206
Speaker: Prof. Jon Wilkening (Department of Mathematics, University of California Berkeley) - http://math.berkeley.edu/~wilken/
Abstract: We compute new families of time-periodic and quasi-periodic solutions of the free-surface Euler equations involving extreme standing waves and collisions of traveling waves of various types. A Floquet analysis shows that many of the new solutions are linearly stable to
harmonic perturbations. Evolving such perturbations (nonlinearly) over tens of thousands of cycles suggests that the solutions remain nearly time-periodic forever. We also discuss resonance and re-visit a long-standing conjecture of Penney and Price that the standing water
wave of greatest height should form wave crests with sharp, 90 degree interior corner angles. We conclude with a geophysical application in which nearly-coherent standing waves at the ocean surface can lead to rapidly-moving pressure zones at the sea floor. These pressure zones
can generate resonant elastic waves believed to be partially responsible for microseisms, the background noise observed in earthquake seismographs.
A finite element method for the second order linear elliptic equation in non-divergence form.
When: Tue, May 6, 2014 - 3:30pmWhere: Math 3206
Speaker: Dr. Wujun Zhang (Department of Mathematics, University of Maryland) - http://www2.math.umd.edu/~wujun/
Abstract: Fully nonlinear elliptic PDEs, including Monge-Ampere equation and Isaac's equation, arise naturally from differential geometry, optimal mass transport, stochastic games and the other fields of science and engineering. In contrast to an extensive PDE literature, the numerical approximation reduces to a few papers. One major difficulty is the notion of viscosity solution which hinges on the maximum principle, instead of a variational principle.
In this talk, we consider linear uniformly elliptic equations in non-divergence form, which may be regarded as linearization of fully nonlinear equations. We discuss the design of convergent numerical methods for such equations. We introduce a novel finite element method which satisfies a discrete maximum principle. This property, together with operator consistency, guarantees convergence to the viscosity solution provided that the coefficient matrix and right-hand side are continuous. We also present a discrete version of the Alexandroff-Bakelman-Pucci estimate, and use it to derive convergence rates under suitable regularity assumptions of the coefficient matrix and viscosity solution.
A PDE approach to space-time fractional parabolic problems.
When: Tue, May 13, 2014 - 3:30pmWhere: Math 3206
Speaker: Dr. Enrique Otarola (Department of Mathematics, University of Maryland) - http://www2.math.umd.edu/~eotarol1/
Abstract: We study solution techniques for evolution equations with fractional diffusion and fractional time derivative in a polyhedral bounded domain. The fractional time derivative, in the sense of Caputo, is discretized by a first order scheme and analyzed in a general Hilbert space setting. We show discrete stability estimates which yield an energy estimate for evolution problems with fractional time derivative. The spatial fractional diffusion is realized as the Dirichlet-to-Neumann map for a nonuniformly elliptic problem posed on a semi-infinite cylinder in one more spatial dimension. We write our evolution problem as a quasi-stationary elliptic problem with a dynamic boundary condition, and we analyze it in the framework of weighted Sobolev spaces. The rapid decay of the solution to this problem suggests a truncation that is suitable for numerical approximation. We propose and analyze a first order semi-implicit fully-discrete scheme to discretize the truncation: first degree tensor product finite elements in space and a first order discretization in time. We prove stability and a near optimal a priori error estimate of the numerical scheme, in both order and regularity.
Sedimentation in viscous fluids
When: Tue, May 20, 2014 - 3:30pmWhere: Math 3206
Speaker: Prof. Willy Dorfler (Karlsruher Institut fuer Technologie (KIT)) - http://www.math.kit.edu/ianm2/~doerfler/
Abstract: Sedimentation of small rigid particles in viscus fluid is a very interesting issue in chemical engineering. Here, a set of particles (mostly modelled by balls or ellipsoids) is located in the whole space and the movement is driven by gravity. We present some models describing the situation and report on numerical results for small and large number of particles. Some of the numerical simulations also include electric forces.