Numerical Analysis Archives for Academic Year 2014

Maximum principle preserving continuous finite element schemes

When: Wed, August 13, 2014 - 3:30pm
Where: Math 3206
Speaker: Murtazo Nazarov (Department of Mathematics, Texas A&M University) -
Abstract: We will introduce a new stabilized continuous Lagrange finite element method for the explicit approximation of scalar conservation equations that does not require any a priori knowledge of quantities like local wave-speed, proportionality constant, or local mesh-size. Provided the lamped mass matrix is positive definite and the flux is Lipschitz, the method is proved to satisfy the local maximum principle under a usual CFL condition. The method is independent of the cell type; for instance, the mesh can be a combination of tetrahedra, hexahedra, and prisms in three space dimensions without any particular regularity assumption. Then, using the Boris-Book-Zalesak flux correction technique, we extend the method to (at least) a second-order accuracy that satisfy maximum principle. The method is tested on a series of linear and nonlinear benchmark problems.

Analysis of numerical method for the Monge-Ampere Equation

When: Tue, September 9, 2014 - 3:30pm
Where: Math 3206
Speaker: Prof. Gerard Awanou (Department of Mathematics, Statistics, and Computer Science, University of Illinois at Chicago) -
Abstract: The Monge-Amp`re equation is a nonlinear partial differential equation which appears
in a wide range of applications, e.g. optimal transportation and reflector design. Solutions of the Monge-Amp`re equation are in general not smooth, and hence difficult to compute with standard discretizations. I will review a large class of methods proposed so far, from the point of view of compatible discretizations. I will discuss how this new point of view leads to an analysis of the theoretical convergence properties of the methods.

Functional-analytic limitations and numerical aspects for variable splitting approaches in TV-regularization

When: Tue, September 16, 2014 - 3:30pm
Where: Math 3206
Speaker: Prof. Micheal Hintermuller (Department of Mathematics, Humbolt University, Berlin, Germany) -
Abstract: Variable splitting schemes for the function space version of the image
reconstruction problem with total variation regularization (TV-problem)
in its primal and pre-dual formulations are considered. For the primal
splitting formulation, while existence of a solution cannot be guaranteed,
it is shown that quasi-minimizers of the penalized problem are asymptotically
related to the solution of the original TV-problem. On the other hand, for
the predual formulation, a family of parametrized problems is introduced and
a parameter dependent contraction of an associated fixed point iteration
is established. Moreover, the theory is validated by numerical tests.
Additionally, the augmented Lagrangian approach is studied, details on an
implementation on a staggered grid are provided and numerical tests are shown.

Optimal boundary control of a semi-discrete Cahn-Hilliard/Navier-Stokes system

When: Thu, September 18, 2014 - 3:30pm
Where: Math 3206
Speaker: Prof. Donat Wegner (Department of Mathematics, Humbolt University, Berlin, Germany) -
Abstract: We consider a time discretization of a Cahn-Hilliard/Navier-Stokes type system known as model H. The special choice of the discretization allows to derive energy estimates which will be the key in the proof of existence of solution. This discussion in particular includes the case of the non-smooth double-obstacle potential. Thereafter, we study an optimal boundary control problem for this system, prove the existence of minimizers and derive an optimality system with the help of a mollified Yosida approximation. If time permits we discuss some
issues of the numerical implementation and difficulties.

A Study of the Agglomeration Equations for Agent-Countermeasure Particle Interactions

When: Tue, September 30, 2014 - 3:30pm
Where: Math 3206
Speaker: Dr. William G. Szymczak (Code 7131, Naval Research Laboratory Washington DC 20375) -
Abstract: The agglomeration equations first derived by Smolukowski in 1918 will be used to address a concept in which agent particles are neutralized through their interaction with countermeasure particles. A new generalized form of the discrete equations, valid for poly-sized particles, is first derived using a weak formulation of the continuous equations. This formulation is shown to preserve particle mass conservation provided the test functions form a partition of unity, and yields a class of algorithms having straightforward and efficient implementations. Verifications of the numerical algorithm and validations of agglomeration due to turbulent shear are included. A dimensional analysis of the equations is used to assess the effectiveness of agent-countermeasure particle systems using both turbulent shear and acoustic excitation agglomeration mechanisms. Extensions of the model for evaporating particles, higher order algorithms, and coupling to the Navier Stokes equations are briefly discussed.

Computational methods for large-scale matrix equations: recent advances [special time and location]

When: Thu, October 2, 2014 - 10:00am
Where: 4172 AV Williams
Speaker: Prof. Valeria Simoncini (Universita di Bologna, Italy) -
Abstract: Linear matrix equations such as the Lyapunov and Sylvester equations
and their generalizations play an important role in the analysis of
dynamical systems, in control theory, in eigenvalue computation,
and in other scientific and engineering application problems.

A variety of robust numerical methods exists for the
solution of small dimensional linear equations, whereas the large scale
case still provides a great challenge.

In this talk we review several well-established numerical
methods, and discuss new approaches specifically
designed for the solution of certain classes of linear matrix equations.

Massively Parallel, Adaptive Finite Element Computations in deal.II

When: Tue, October 7, 2014 - 3:30pm
Where: Math 3206
Speaker: Prof. Timo Heister (Clemson University, South Carolina) -
Abstract: We present an approach to allow for massively parallel finite element
computations on large computing clusters on adaptively refined meshes. Only
an algebraic partitioning gives the flexibility to solve many different kinds
of PDEs in a scalable manner. The proposed method is implemented in the open
source finite element library deal.II, where it is used successfully for many
different applications. We will show numerical results and discuss appropriate
linear solvers and preconditioners. Finally, future architectures and its
ramifications are discussed.

Approximation of elliptic equations with BMO coefficients

When: Fri, October 17, 2014 - 3:00pm
Where: Math 3206
Speaker: Prof. Abner J. Salgado (Department of Mathematics, University of Tennessee ) -
Abstract: We study solution techniques for elliptic equations in divergence form, where the coefficients are only of bounded mean oscillation. For p sufficiently close to 2 and a right hand side in $W^{-1}_p$ we show convergence of a finite element scheme, where the "closeness" depends on the oscillation of the coefficients.

Implicit solvers for incompressible flow problems

When: Tue, October 21, 2014 - 3:30pm
Where: Math 3206
Speaker: Prof. David Silvester (University of Manchester, England) -
Abstract: This survey talk reviews some recent developments in the design of robust solution methods for the Navier-Stokes equations modelling incompressible fluid flow. There are two building blocks in our solution strategy. First, an implicit time integrator that uses a stabilized trapezoid rule with an explicit Adams-Bashforth method for error control, and second, a robust Krylov subspace solver for the spatially discretized system. Numerical experiments are presented that illustrate the effectiveness of our generic approach. It is further shown that the basic solution strategy can be readily extended to more complicated models, including unsteady flow problems with coupled physics and steady flow problems that are stochastic in the sense that they have uncertain input data.

Simulations of DNA Binding Landscapes in Helical Coordinates

When: Tue, November 4, 2014 - 3:30pm
Where: Math 3206
Speaker: Prof. Garegin A. Papoian (Department of Chemistry and Biochemistry and Institute for Physical Science and Technology, UMD) -
Abstract: DNA molecules are highly charged semi-flexible polymers that are involved in a wide variety of dynamical processes such as transcription and replication. Characterizing the binding landscapes around DNA molecules is essential to understanding the energetics and kinetics of various biological processes. We present a curvilinear coordinate system that fully takes into account the helical symmetry of a DNA segment. The latter naturally allows to characterize the spatial organization and motions of ligands tracking the minor or major grooves, in a motion reminiscent of sliding. Using this approach, we performed umbrella sampling (US) molecular dynamics (MD) simulations to calculate the three-dimensional potentials of mean force (3D-PMFs) for a Na+ cation and for methyl guanidinium, an arginine analog. The computed PMFs show that, even for small ligands, the free energy landscapes are complex. We discuss how our findings have important implications for understanding how proteins !
and ligands associate and slide along DNA.

Isogeometric mortaring techniques [Appl. Math. Colloq.]

When: Tue, November 11, 2014 - 3:30pm
Where: Math 3206
Speaker: Prof. Annalisa Buffa (Istituto di Matematica Applicata e Tecnologie Informatiche (IMATI) Consiglio Nazionale delle Ricerche 27100 Pavia, Italy) -
Abstract: The treatment of interfaces with non compatible grids is one of the important issues in the numerical simulation of PDEs. We study this problem when the unknowns are described with splines or NURBS spaces.

I will present mortar techniques for spline-based methods, proposing suitable spaces of Lagrange multipliers. I will prove stability and also provide numerical experiments, which address the delicate issue of numerical quadrature for the interface integrals.

This work has been done in collaboration with E. Brivadis, B. Wohlmuth and L. Wunderlich.

The interplay between geometric modeling and simulation of partial differential equations [Aziz Lecture]

When: Wed, November 12, 2014 - 3:30pm
Where: Math 3206
Speaker: Prof. Annalisa Buffa (Istituto di Matematica Applicata e Tecnologie Informatiche (IMATI) Consiglio Nazionale delle Ricerche 27100 Pavia, Italy) -
Abstract: Computer-based simulation of partial differential equations involves approximation of the unknown fields and a description of geometrical entities such as the computational domain and the properties of the media. There are a variety of situations: in some cases this description is very complex, in some other the governing equations are very difficult to discretize. Starting with an historical perspective, I will describe the recent efforts to improve the interplay between the mathematical approaches characterizing these two aspects of the problem.

Numerical pricing of financial options with simple finite difference methods.

When: Tue, November 18, 2014 - 3:30pm
Where: Math 3206
Speaker: Prof. Jens Hugger (Department of Mathematical Sciences University of Copenhagen Denmark) -
Abstract: The talk is (almost) self-contained assuming only a bit of general knowledge of partial differential equations and finite difference methods.
First the subject area of financial options is introduced and an IBVP-model is presented together with some key features of the solution.
Then we look at convergence properties for a standard Crank-Nicolson method for the option problem and for simple enhancements of the method in the form of K?-optimization, Rannacher-timestepping, and mesh grading.

A Computational Study of Some Numerical Schemes for a Test case with Steep Boundary Layers

When: Tue, December 2, 2014 - 3:30pm
Where: Math 3206
Speaker: Dr. Rao Appadu (University of Pretoria, South Africa) -
Abstract: In this work, three numerical methods have been used to solve a 1-D Convection-Diffusion equation with specified initial and boundary conditions. The methods used are the third order upwind scheme (DEHGHAN, 2005), fourth order upwind scheme DEHGHAN2005 and a Non-Standard Finite Difference (NSFD) scheme (MICKENS, 1991). The problem we considered has steep boundary layers near $x=1$ (ZHENYU, 2011) and this is a challenging test case as many schemes are plagued by nonphysical oscillation near steep boundaries. We compute the $L_2$ and $L_\infty$ errors, dissipation and dispersion errors when the three numerical schemes are used and observe that the NSFD is much better than the other two schemes for both coarse and fine grids and also at low and high Reynolds numbers.

Turbulence and Transition in Energetic Reactive Flows

When: Tue, December 9, 2014 - 3:30pm
Where: Math 3206
Speaker: Prof. Elaine Oran (Department of Aerospace Engineering University of Maryland College Park) -
Abstract: The most complicated and difficult problems in fluid dynamics often involve transitions among what seem to be relatively stable steady states. In turn, one of the most complex and intriguing sequences of fluid transitions is the series of changes in the behavior of an energetic reactive flow that occurs as a small spark, often ignited quite accidentally, evolves into a powerful supersonic wave. These reactive-flow transitions are critical elements in systems ranging from engines for high-speed propulsion, to accidental fuel explosions in mines or fuel-storage facilities, to explosions of thermonuclear supernova.

This presentation will draw from a combination of numerical simulations of unsteady, multidimensional, compressible reactive flows, theoretical analyses, laboratory experiments, and large-scale accidental explosions in an attempt to determine when and how such transitions occur. In the course of describing the process from ignition to final explosion, we note the presence of a metastable complex of loosely coupled shocks and reaction fronts. The presence of these transition states, which occur in purely gas-phase reactive flows as well as in multiphase gas-granular reactive flows, might offer some isight into how to quench or hasten with the overall process.

Adaptivity and blowup detection for semilinear evolution convection-diffusion equations based on a posteriori error control

When: Tue, January 27, 2015 - 3:30pm
Where: Math 3206
Speaker: Dr. Irene Kyza (Department of Mathematics, University of Dundee) -
Abstract: We discuss recent results on the a posteriori error control and adaptivity for an evolution semilinear convection-diffusion model problem with possible blowup in finite time. This belongs to the broad class of partial differential equations describing e.g., tumor growth, chemotaxis and cell modelling. In particular, we derive a posteriori error estimates that are conditional (estimates which are valid under conditions of a posteriori–type) for an interior penalty discontinuous Galerkin (dG) implicit-explicit (IMEX) method using a continuation argument. Compared to a previous work, the obtained conditions are more localised and allow the efficient error control near the blowup time. Utilising the conditional a posteriori estimator we are able to propose an adaptive algorithm that appears to perform satisfactorily. In particular, it leads to good approximation of the blowup time and of the exact solution close to the blowup. Numerical experiments illustrate and complement our theoretical results. This is joint work with A. Cangiani, S. Metcalfe, and E.H. Georgoulis from the University of Leicester.

Modeling complex geometries applied to fire dynamics simulation

When: Tue, February 3, 2015 - 3:30pm
Where: Math 3206
Speaker: Marcos Vanella (Dept. of Mechanical and Aerospace Engineering, The George Washington University, Washington, DC) -
Abstract: The speaker will comment on ongoing work for the FDS (fire dynamics simulator) software,developed and maintained by the Fire Research
Division at NIST. This software is employed in numerical simulation
of fire scenarios with application to evaluation and design of fire protection systems, forensic work, and study of wild land fires, among
others. In order to model the gas phase hydrodynamics, the low-­‐Mach
approximation for the Navier-­‐Stokes equations is employed. Momentum, sensible enthalpy, and species transport equations are discretized using a staggered mesh finite volume scheme (FV). A method for discrete representation of complex geometry within FDS will be discussed, in particular for the treatment of scalar transport equations being evolved. Considering the conservation requirements of
FDS, a cut-­‐cell FV method is adopted towards this end. The small cell problem derived from the cut-­‐cell scheme is tackled using explicit-­‐implicit domain decomposition, within the time integration algorithm used in FDS. Details on the derivation and implementation of the numerical method being developed will be given. Also, a comment on the scheme for the velocity-­‐pressure coupling on the cut-­‐cell region will be provided.

Novel Stochastic Methods in Biochemical Electrostatics

When: Tue, February 10, 2015 - 3:30pm
Where: Math 3206
Speaker: Prof. Michael Mascagni (Department of Computer Science and Department of Mathematics and Department of Scientific Computing and Graduate Program in Molecular Biophysics Florida State University) -
Abstract: Electrostatic forces and the electrostatic properties of molecules in solution are among the most important issues in understanding the structure and function of large biomolecules. The use of implicit-solvent models, such as the Poisson-Boltzmann equation (PBE), have been used with great success as a way of computationally deriving electrostatics properties such molecules. We discuss how to solve an elliptic system of partial differential equations (PDEs) involving the Poisson and the PBEs using path-integral based probabilistic, Feynman-Kac, representations. This leads to a Monte Carlo method for the solution of this system which is specified with a stochastic process, and a score function. We use several techniques to simplify the Monte Carlo method and the stochastic process used in the simulation, such as the walk-on-spheres (WOS) algorithm, and an auxiliary sphere technique to handle internal boundary conditions. We then specify some optimizations using the error (bias) and variance to balance the CPU time. We show that our approach is as accurate as widely used deterministic codes, but has many desirable properties that these methods do not. In addition, the currently optimized codes consume comparable CPU times to the widely used deterministic codes. Thus, we have an very clear example where a Monte Carlo calculation of a low-dimensional PDE is as fast or faster than deterministic techniques at similar accuracy levels.

A finite element method for nematic liquid crystals with variable degree of orientation

When: Tue, February 17, 2015 - 3:30pm
Where: Math 3206
Speaker: Dr. Wujun Zhang (University of Maryland, College Park) -
Abstract: We consider the simplest one-constant model, put forward by J. Ericksen, for nematic liquid crystals with variable degree of orientation. The equilibrium state is described by a director field and its degree of orientation, which minimize a sum of Frank-like energies and a double well potential. In particular, the Euler-Lagrange equations for the minimizer contain a degenerate elliptic equation for the director field, which allows for line and plane defects to have finite energy.

We present a structure preserving discretization of the liquid crystal energy without regularization, and show that it is consistent. We prove convergence of the continuous piecewise linear finite solutions as the mesh size goes to zero. We develop a weighted gradient flow scheme for computing discrete equilibrium solutions and prove that it has a strict energy decrease property. We present simulations in two and three
dimensions that exhibit both line and plane defects and illustrate key features of the method.

The NIST Adaptive Mesh Refinement Benchmark Suite

When: Tue, February 24, 2015 - 3:30pm
Where: Math 3206
Speaker: Dr. William Mitchell (Applied and Computational Mathematics Division
National Institute of Standards and Technology) -
Abstract: The development of new algorithms and computer codes for the solution of partial differential equations (PDEs) usually involves the use of proof-of-concept test problems. Such test problems have a variety of uses such as demonstrating that a new algorithm is effective, verifying that a new code is correct in the sense of achieving the theoretical order of convergence, and comparing the performance of different algorithms and codes.

Self-adaptive methods to determine a quasi-optimal grid are a critical component of the improvements that have been made in PDE algorithms in recent years. This field is referred to as adaptive mesh refinement, or adaptive grid refinement. Although adaptive mesh refinement techniques are now in widespread use in applications, they remain an active field of research, particularly in the context of hp-adaptive techniques, non-elliptic problems, accurate error estimators, etc.

Nearly every paper on algorithms for solving PDEs contains a numerical results section with one or more test problems. Although there are a few commonly used problems for adaptive mesh refinement research, such as the L-domain problem, the test problems vary widely among papers. The purpose of the NIST Adaptive Mesh Refinement Benchmark Suite is to provide a standard set of problems suitable for benchmarking and testing adaptive mesh refinement algorithms and error estimators. The problems exhibit a variety of types of singularities, near singularities, and other difficulties.

This talk will describe this ongoing effort and the web resource at It will also discuss some related efforts including the use of a reference solution to assess solution accuracy and convergence, locating an element that contains a given point by using space filling curves, and an examination of the effectiveness of high order finite elements.

Statistical modeling and adaptive control of reduced-order-model error in uncertainty quantification

When: Tue, March 3, 2015 - 3:30pm
Where: Math 3206
Speaker: Kevin Carlberg (Sandia National Laboratories Livermore, CA ) -
Abstract: Many tasks in uncertainty quantification require hundreds or thousands of `forward' model simulations. In Bayesian inference for data assimilation, for example, each sample from the posterior distribution requires (at least) one forward simulation. Employing high-fidelity, large scale computational models for such tasks is infeasible, as a single simulation can consume days or weeks on a supercomputer.

To make such problems tractable, we 1) replace the large-scale model with a low-dimensional projection-based reduced-order model (ROM), 2) rigorously account for the additional (epistemic) uncertainty introduced by the ROM, and 3) control this uncertainty by adaptively refining the ROM as needed.

Two newly developed methods enable this strategy. For step 2 above, we propose the ROM error surrogate (ROMES) method [1], which employs stochastic processes to construct accurate statistical models of the ROM error. For step 3, we propose an adaptive h-refinement approach [2] that borrows many concepts from adaptive mesh refinement.

[1] M. Drohmann and K. Carlberg. “The ROMES method for statistical modeling of reduced-order-model error,” SIAM/ASA Journal on Uncertainty Quantification, in press (2014).
[2] K. Carlberg. “Adaptive h-refinement for reduced-order models,” International Journal for Numerical Methods in Engineering, Special Issue on Model Reduction, in press (2014). doi:10.1002/nme.4800

How do Bacterial Mechanosensitive Channels Function in Multiple Scales?

When: Tue, March 10, 2015 - 3:30pm
Where: Math 1313 (Note Room Change)
Speaker: Prof. Sergei I. Sukharev (Department of Biology University of Maryland) -
Abstract: In this presentation I will outline key experimental observations and related physical and structural models that may explain the function of bacterial mechanosensitive channels. These channels act as fast osmolyte release valves which protect cells from strong osmotic forces.

1. Under extreme osmotic conditions when pressure inside the cell is expected to exceed 20 atm, the channels open, increasing membrane permeability by orders magnitude and dissipate osmolyte gradients within 10-50 ms. This will be illustrated by our recent light scattering experiments suggesting the kinetic nature of the osmotic rescuing mechanism and the crucial role of cell envelope mechanics.

2. The crystal structures of the two major channels, MscS and MscL, provide excellent starting points for modeling their conformational transitions. The physical parameters of unitary conductance and lateral expansion guide the modeling process. The analysis of surface hydrophobicity and water behavior in the pore hint at the energetic scales of the transitions.

3. An inactivation process in MscS suggests the involvement of inter-helical slip-bonds in its structure, which permit the molecule to differentiate forces applied at different rates.

4. MscS senses mechanical inputs arriving from different sides: (i) using its transmembrane barrel it senses the lateral pressure profile inside the lipid bilayer and (ii) by means of its hollow cytoplasmic domain (cage) it senses crowding pressure in the cytoplasm. The hypothesis of why MscS-like proteins with their characteristic cage domains are present exclusively in organisms with cell walls will be discussed.

Optimal Control of Elliptic Variational Inequalities: Optimality Conditions and Numerical Methods

When: Tue, March 24, 2015 - 3:30pm
Where: Math 3206
Speaker: Prof. Thomas Surowiec (Humboldt University Berlin, Germany) -
Abstract: In this talk, several methodologies for the derivation of optimality conditions for the optimal control of elliptic variational inequalities will be presented. The lack of Fr\`echet differentiability of the control-to-state mapping and the degeneracy of the constraint set inhibit the usage of standard approaches of PDE-constrained optimization. Following this, the numerical solution of these control problems will be discussed. In particular, we present new methods in which the explicit formula for the directional derivative of the control-to-state map is used to provide robust descent directions of the reduced objective functional. The methods are illustrated by several examples.

Optimization-based models and algorithms for the feature-preserving solution of transport equations

When: Thu, April 2, 2015 - 3:30pm
Where: Math 3206
Speaker: Dr. Denis Ridzal (Sandia Laboratories) -
Abstract: A fundamental mathematical challenge in the numerical modeling and solution of transport equations is to simultaneously achieve high spatial accuracy and preserve essential physical solution features, such as positivity, monotonicity, and mass or energy balance. To reconcile the notion of accuracy with the preservation of physical features we employ global nonlinear optimization models at various stages of the discretization and solution process. Specifically, the optimization problems are designed to minimize the distance to a suitable target quantity, responsible for the solution accuracy, subject to a system of equalities and inequalities that maintain physical constraints.

This approach is significantly more flexible than traditional discretizations of transport equations, which rely on some form of flux limiting, and offers valuable computational and theoretical advantages. First, by design it is guaranteed to compute the most accurate solution representation that simultaneously satisfies physical constraints. Second, it is independent of the underlying spatial discretization scheme and the mesh representation, and can be thought of as a mathematically rigorous "post-processing" of the solution.

As the first example, motivated by Semi-Lagrangian and Arbitrary Lagrangian-Eulerian methods for transport, we use optimization ideas to perform conservative and monotone incremental remap, i.e., to transfer a scalar conserved quantity between a sequence of meshes subject to local solution bounds. Two formulations of remap are compared and contrasted -- the flux-variable flux-target (FVFT) formulation and the mass-variable mass-target (MVMT) formulation -- yielding optimization algorithms with distinct performance characteristics. We highlight the MVMT formulation, a singly linearly constrained quadratic program with simple solution bounds, for which we design an efficient and easily parallelizable optimization algorithm.

As the second example, motivated by Eulerian finite element methods, we design a monotone and mass-conserving scheme for finite element transport. The scheme shares the optimization formulation and the algorithm with the above MVMT formulation. It provably maintains linear relationships between simulated quantities in multi-variable systems, such as those arising in models for atmospheric transport. As a side benefit, our framework also enables robust failure recovery on future high-performance computing architectures.

Bio: Dr. Denis Ridzal is a Principal Member of Technical Staff at Sandia National Laboratories. Dr. Ridzal received a Ph.D. in Computational and Applied Mathematics from Rice University in 2006. His research includes the development and analysis of algorithms for large-scale optimization, with emphasis on the efficient treatment of simulation constraints, iterative linear solvers and preconditioners; optimization-based models and algorithms for transport equations and optimization-based physics coupling; and the development and implementation of high-order and compatible PDE discretizations. Dr. Ridzal is a lead developer of Sandia's Trilinos package Rapid Optimization Library (ROL) for matrix-free large-scale optimization and the Trilinos package Intrepid for compatible PDE discretizations.

Homotopy continuation methods for nonlinear PDEs and applications

When: Tue, April 7, 2015 - 3:30pm
Where: Math 3206
Speaker: Dr. Wenrui Hao (Mathematical Biosciences Institute, Ohio State University ) -
Abstract: This talk will cover some recent work on homotopy continuation methods to solve systems of nonlinear partial differential equations (PDEs) arising from biology and physics. This new approach, which is used to compute multiple solutions and bifurcation of nonlinear PDEs, makes use of polynomial systems (with thousands of variables) arising by discretization. Examples from hyperbolic systems, tumor growth models, and a blood clotting model will be used to demonstrate the ideas.

Waves in random media: the story of the phase (Aziz Lecture) - Colloquium

When: Wed, April 15, 2015 - 3:15pm
Where: Math 3206
Speaker: Lenya Ryzhik (Stanford University) -
Abstract: The macroscopic description of wave propagation in random media typically focuses on the scattering of the wave intensity, while the phase is discarded as a uselessly random object. At the same time, most of the beauty in wave scattering come from the phase correlations. I will describe some of the miracles, as well as some limit theorems for the wave phase.

(Applied Math Colloq) Radiative Transport or Homogenization?

When: Thu, April 16, 2015 - 3:30pm
Where: Math 3206
Speaker: Lenya Ryzhik (Stanford University) -
Abstract: Radiative transport equations and other macroscopic kinetic models are widely used to describe multiple scattering of wave energy in random media. They account for the incoherent wave energy, and their validity is associated with the randomness of the wave field. The opposite regime is homogenization - here, the wave field has a deterministic macroscopic limit. I will try to describe the transition from one regime to the other, and understand which one is more generic. This is a joint work with G. Bal, T. Chen and T. Komorowski.

Equilibrated A Posteriori Error Estimators for Interior Penalty Discontinuous Galerkin Approximations of Second and Fourth Order Elliptic Problems

When: Tue, April 21, 2015 - 3:30pm
Where: Math 3206
Speaker: Prof. Ronald Hoppe (Department of Mathematics University of Houston) -
Abstract: Interior Penalty Discontinuous Galerkin (IPDG) methods for second and fourth
order elliptic boundary value problems can be derived from a mixed formu-
lation of the problem involving properly specified numerical flux functions
across interior faces of the underlying triangulation of the computational do-
main. Equilibrated a posteriori error estimators can be derived by means of
a two-energies principle also referred to as a hypercircle method or Prager-
Synge theorem. The two-energies principle is based on the construction of
an equilibrated flux for second order problems and an equilibrated moment
tensor for fourth order problems. It results in a reliability estimate without
generic constants except for possible data oscillations. On the other hand,
the efficiency can be established by verifying that the estimators are bounded
from above by residual-type a posteriori error estimators which are known to
be efficient. The results are based on joint work with Dietrich Braess and
Thomas Fraunholz.

[Applied Math Colloquium] Numerical approximation of non-divergence-form PDEs

When: Tue, May 5, 2015 - 3:30pm
Where: Math 3206
Speaker: Prof. Endre Suli (Oxford University) -
Abstract: Non-divergence form partial differential equations with discontinuous coefficients do not generally posses a weak formulation, thus presenting an obstacle to their numerical solution by classical finite element methods. Such equations arise in many applications from areas such as probability and stochastic processes. These equations also arise as linearizations to fully nonlinear PDEs, as obtained for instance from the use of iterative solution algorithms. In such cases, it can rarely be expected that the coefficients of the operator be smooth or even continuous. For example, in applications to Hamilton--Jacobi--Bellman equations, the coefficients will usually be merely essentially bounded. In contrast to the study of divergence form equations, it is usually not possible to define a notion of weak solution when the coefficients are non-smooth. In the case of continuous but possibly non-differentiable coefficients, the Calderon--Zygmund theory of strong solutions establishes the well-posedness of the problem in sufficiently smooth domains. However, without additional hypotheses, well-posedness is generally lost in the case of discontinuous coefficients. The aim of the lecture is to survey recent developments concerning the numerical approximation of such problems by finite element methods.

[Aziz Lecture] Mathematical challenges in kinetic models of dilute polymers: analysis, approximation and computation

When: Wed, May 6, 2015 - 3:15pm
Where: Math 3206
Speaker: Prof. Endre Suli (Oxford University) -
Abstract: We survey recent analytical and computational results for coupled macroscopic-microscopic
bead-spring chain models that arise from the kinetic theory of dilute solutions of polymeric
fluids with noninteracting polymer chains, involving the unsteady Navier--Stokes system in a bounded
domain and a high-dimensional Fokker--Planck equation. The Fokker--Planck equation emerges from a system of
(It$\hat{\rm o}$) stochastic differential equations, which models the evolution of a vectorial stochastic process
comprised by the centre-of-mass position vector and the orientation (or configuration) vectors of the polymer chain.
We discuss the existence of large-data global-in-time weak solutions to the coupled Navier--Stokes--Fokker--Planck system.
The Fokker--Planck equation involved in the model is a high-dimensional partial differential equation, whose
numerical approximation is a formidable computational challenge, complicated by the fact that for practically
relevant spring potentials, such as finitely extensible nonlinear elastic potentials,
the drift coefficient in the Fokker--Planck equation is unbounded.

Numerical solution of bilayer bending problems

When: Tue, May 12, 2015 - 3:30pm
Where: Math 3206
Speaker: Prof. Soeren Bartels (Freiburg University Germany) -
Abstract: Thin elastic bilayer structures arise in various modern applications,
e.g., in the fabrication of nanotubes or microgrippers. The mechanical behavior is
characterized by large isometric deformations with large curvature in one direction. The
mathematical modeling leads to a nonlinear fourth order problem with nonlinear
pointwise constraint. The reliable and efficient numerical treatment is therefore challenging. We prove the convergence of a finite element discretization within the framework of $\Gamma$-convergence and discuss the convergence of an iterative
solution method. The work is based on results by Friesecke, James, and M\"uller (2002)
as well as Schmidt (2007,) and extends methods for the approximation of
large bending problems.

The talk represents joint research with Andrea Bonito (Unversity of Texas, A\& M)
and Ricardo H. Nochetto (University of Maryland, College Park).