Seminar: Numerical Methods for Partial Differential Equations

Math Department Calendar
List of Seminars

Organizers: Jean-Christophe Nave, Rodolfo Ruben Rosales

Day: Wednesdays
Time: 4pm-5pm
Location: Building 2, room 132

If you are interested in receiving announcements by email, please write to jcnave(at)mit.edu

Wednesday, September 23, 2009

4-5pm in 2-132
Gabriel Peyr� (CNRS and Universit� Paris-Dauphine)

Sparse Processing of Images

In this talk, I will review recent work on the sparse representations of natural images. I will focus on the application of these emerging models for the resolution of various imaging problems, which include compression, denoising and super-resolution of images, as well as compressive sensing and compressive wave computations. Natural images exhibit a wide range of geometric regularities, such as curvilinear edges and oscillating textures. Adaptive image representations select bases from a dictionary of orthogonal or redundant frames that are parameterized by the geometry of the image. If the geometry is well estimated, the image is sparsely represented by only a few atoms in this dictionary. The resolution of ill-posed inverse problems in image processing is then regularized using sparsity constraints in these adapted representations.

SPECIAL DAY / TIME / LOCATION: Monday, October 6th, 2009

Jointly with the Applied Math Colloquium
4:30-5:30pm in 4-370
Kai Schneider (Universite de Provence, Marseille, France)

Adaptive Space-Time Multiresolution Techniques for Nonlinear PDEs

We present efficient fully adaptive numerical schemes for evolutionary partial differential equations based on a finite volume (FV) discretization with explicit time discretization. A multiresolution strategy allows local grid refinement while controlling the approximation error in space. The costly fluxes are evaluated on the adaptive grid only. For time discretization we use an explicit Runge-Kutta scheme of second-order with a scale-dependent time step. On the finest scale the size of the time step is imposed by the stability condition of the explicit scheme. On larger scales, the time step can be increased without violating the stability requirement of the explicit scheme. Embedded Runge-Kutta methods of second and third order are then used to choose automatically the new time step while controlling the approximation error in time. Non-admissible choices of the time step are avoided by limiting its variation.
The implementation of the multiresolution representation uses a dynamic tree data structure, which allows memory compression and CPU time reduction. This new numerical scheme is validated using different classical test problems in one, two and three space dimensions. The gain in memory and CPU time with respect to the finite volume scheme on a regular grid is reported, which demonstrates the efficiency of the new method.

This work is joint work with M. Domingues, S. Gomes and O. Roussel.

Wednesday, October 14, 2009

4:30-5:30pm in 2-132
Markus Schmuck (MIT - Chem. Engr.)

Modeling, Analysis, and Numerics Of The Navier-Stokes-Nernst- Planck-Poisson System

We introduce a macroscopic model which allows to describe the essential electrokinetic phe- nomena as electrophoresis and -osmosis. Then, we present the basic analytical results for the Navier-Stokes-Nernst-Planck-Poisson system. Next, we propose and analyze two convergent nite element discretizations which preserve all char- acteristic properties of weak solutions in the discrete setting. We begin with a scheme based on perturbation and then show how we can improve the properties and consistency of the scheme by using a suitable truncation. In the last part of the talk, we derive eective macroscopic properties of a porous solid-electrolyte composite. This leads to a huge dimensional reduction by upscaling the microstructure. The results are gained by the formal multiple-scale method in the context of homogenization.

Wednesday, November 4, 2009

4:30-5:30pm in 2-132
Qi Qi Wang (MIT - Aero./Astro.)

Solving adjoint equations for unsteady fluid flows

Methods based on solving adjoint equations are widely used in trajectory optimization, shape design, inverse methods, optimal control and uncertainty quantification. The adjoint equation solves the sensitivity of an objective function with respect to the governing equations and its parameters. The adjoint sensitivity gradient can be used in approximating the objective function and in gradient based optimization. The adjoint equations for unsteady incompressible Navier-Stokes equations are derived and solve. Efficient backwards time integration of the adjoint equation is performed using the dynamic checkpointing scheme. The adjoint solution reveals the sensitivity of the objective function with respect to both the geometry and velocity of solid bodies in the flow field. Applications of the adjoint solution in optimization and uncertainty quantification are discussed.

Wednesday, November 18, 2009

4:30-5:30pm in 2-132
Leslie Greengard (NYU - CIMS)

The Nonuniform FFT, Heat Flow, and Magnetic Resonance Imaging Reconstruction

The nonuniform FFT arises is a variety of applications, from medical imaging to radio astronomy to the numerical solution of partial differential equations. In a typical problem, one is given an irregular sampling of N data points in the frequency domain with the goal of reconstructing the corresponding function at N points in the physical domain. When the sampling is uniform, the fast Fourier transform (FFT) allows this calculation to be carried out in O(N log N) operations. Unfortunately, when the data is nonuniform, the FFT does not apply. In the last few years, a number of algorithms have been developed which overcome this limitation and are often referred to as nonuniform FFTs. In this talk, we describe the basic algorithm and some of its applications.

Poster Announcement Spring 2009

Wednesday, February 18, 2009

4-5pm in 2-146
Martin Frank (Department of Mathematics, University of Kaiserslautern)

Optimal treatment planning in radiotherapy based on Boltzmann transport equations

Treatment with high energy ionizing radiation is one of the main methods in modern cancer therapy that is in clinical use. During the last decades, two main approaches to dose calculation were used, Monte Carlo simulations and semi-empirical models based on Fermi-Eyges theory. A third way to dose calculation has only recently attracted attention in the medical physics community. This approach is based on the deterministic kinetic equations of radiative transfer.
In this work, we present a Boltzmann transport model for dose calculation in radiation therapy. We discuss simplifications of this model and additionally formulate an optimal control problem for the desired dose. Based on this formulation, we derive optimality conditions. Numerical results in one and two dimensions are presented.

Wednesday, February 25, 2009

4-5pm in 2-146
Shing Yu Leung (Department of Mathematics, UC Irvine)

A grid based particle method for the evolution of open curves and surfaces

We present in this talk a new numerical method for modeling motion of open curves in two dimensions and open surfaces in three dimensions. Following the grid based particle method we have recently proposed, we represent the open curve or the open surface by meshless Lagrangian particles sampled according to an underlying fixed Eulerian mesh. The underlying grid is used to provide a quasi-uniform sampling and neighboring information for meshless particles. The key idea in the talk is to represent and to track end-points of the open curve and boundary points of the open surface explicitly and consistently with interior particles. We apply our algorithms to several applications including spiral crystal growth modeling and image segmentation using active contours.

Wednesday, March 11, 2009

4-5pm in 2-146
Roland Bouffanais (Department of Mechanical Engineering, MIT)

Simulation of unsteady transitional swirling flow with a moving free surface using a spectral element method

Unsteady incompressible viscous flows of a fluid enclosed in a cylindrical container with an open top surface are discussed. These moving free-surface flows are generated by the steady rotation of the solid bottom end-wall. Such type of flows belongs to a group of recirculating lid-driven cavity flows with geometrical axisymmetry. The top surface of the cylindrical cavity is left open so that the free surface can freely deformed. The Reynolds regime corresponds to unsteady transitional flows with some incursions in the fully laminar regime. The approach taken here revealed new nonaxisymmetric flow states that are investigated based on a fully three-dimensional solution of the Navier-Stokes equations for the free-surface cylindrical swirling flow, without resorting to any symmetry property unlike all other results available in the literature. Theses solutions are obtained through direct numerical simulations based on a highly-accurate Legendre spectral element method combined with a moving-grid technique.

Monday, April 6, 2009

Joint seminar with Applied Mathematics Colloquium
4:30-5:30pm in 4-237
John Lowengrub (Department of Mathematics, UC Irvine)

Multi-scale models of solid tumor growth and angiogenesis

We present and investigate models for solid tumor growth that incorporate features of the tumor microenvironment including tumor-induced angiogenesis. Using analysis and nonlinear numerical simulations, we explore the effects of the interaction between the genetic characteristics of the tumor and the tumor microenvironment on the resulting tumor progression and morphology. We account for variable cell-cell/cell-matrix adhesion in response to micro-environmental conditions (e.g. hypoxia) and to the presence of multiple tumor cell species. We focus on glioblastoma and quantify the interdependence of the tumor mass on the microenvironment and on the cellular phenotypes. The model provides resolution at various tissue physical scales, including the microvasculature, and quantifies functional links of molecular factors to phenotype that for the most part can only be tentatively established through laboratory or clinical observation. This allows observable properties of a tumor (e.g. morphology) to be used to both understand the underlying cellular physiology and to predict subsequent growth or treatment outcome, thereby providing a bridge between observable, morphologic properties of the tumor and its prognosis.

Wednesday, April 8, 2009

4-5pm in 2-146
Jan Hesthaven (Division of Applied Mathematics, Brown University)

Discontinuous Galerkin methods for the modeling of free surface flows using high-order Boussinesq approximations

We shall discuss the modeling of free surface flows and fluid-structure interactions using high-order Boussinesq approximations. These sets of equations are characterized by being purely dispersive and strongly non-linear, with additional complications introduced by high-order spatial derivatives and cross-derivatives. The key elements of the formulation and some of the properties of the Boussinesq system will be discussed in some detail. This shall be used to argue why discontinuous Galerkin methods may be a suitable approach for the solution of these equations. We shall develop the basic elements required for solving this system, discuss a number of subtleties and address practical concerns of performance and efficient solvers. The computational approach will be extensively validated with both benchmark tests and experimental data.
This is work done in collaboration with A.P. Engsig-Karup (DTU, Denmark), P. Madsen (DTU, Denmark), H. Bingham (DTU, Denmark) and T. Warburton (Rice).

Wednesday, April 22, 2009

4-5pm in 2-146
Pavel Grinfeld (Department of Mathematics, Drexel University)

Hamiltonian dynamic equations for fluid films

Two dimensional models for hydrodynamic systems, such as soap films, have been studied for hundreds of years. Yet there has not existed a fully nonlinear system of dynamic equations analogous to the classical Euler equations. We propose an exact nonlinear system for the dynamics of a fluid film. The system is derived in the classical Hamiltonian framework and neither the velocities nor the deviation from the equilibrium are assumed small. We discuss the properties of the proposed equations and results of numerical simulations.

Wednesday, April 29, 2009

4-5pm in 2-146
Frederic Gibou (Department of Mechanical Engineering, UC Santa Barbara)

Sharp interface methods for moving boundary problems

In this talk, we will describe recent (and less recent) advances in the field of free boundary problems, with an emphasis on level-set and ghost-fluid methods. We will discuss their applications to two-phase flows, the Stefan problem and image guided surgery. Novel methods for adaptive mesh refinement will be discussed as well.

Wednesday, May 6, 2009

4-5pm in 2-146
Jacob White (Department of Electrical Engineering and Computer Science, MIT)

Subtleties associated with biochemical oscillator sensitivity analysis in the presence of conservation laws

Sensitivity analysis is often dismissed as a trivial application of Taylor series, even when implicitly defined operators such as differential equations are involved. The subject maybe a bit more subtle when the Jacobian of the implicitly defined operator is singular, and the resolution of this subtlety can be quite problem dependent. In this talk we consider oscillators described by mass action kinetics, review the known material on sensitivity analysis for oscillators posed as two-point boundary value problems, and present the additional subtlety associated with the presence of conservation laws in the oscillator. In particular, we show that generalized eigenvectors play a surprising (at least to the authors) and essential role. This is joint work with Jared Toettcher, Anya Castillo, Paul Barton and Bruce Tidor.

Wednesday, May 13, 2009

~~4-5pm in 2-146~~
New time: 2:30-3:30pm in 2-136
Alexander Vladimirsky (Department of Mathematics, Cornell University)

Causality and efficiency: Non-iterative numerical methods

Our knowledge of the direction of information flow is fundamental for many efficient numerical methods (e.g., time-marching for evolutionary PDEs). However, for many problems (including first-order static nonlinear PDEs) the direction of information flow might be a priori unknown even if it is otherwise well-defined. This leads to a common use of iterative methods, which can be unnecessarily inefficient. For certain systems of nonlinear equations, the "causality" present in the problem can be used to uncover the direction of information flow at runtime. Exploiting causality to effectively de-couple nonlinear systems is the fundamental idea behind Dijkstra's classical method for finding shortest paths on graphs. We will use a continuous analogue of this principle to build efficient methods for a wide class of causal problems. We will illustrate this approach using examples from:

continuous and hybrid optimal control (e.g., optimal traveling on foot and using the buses);
multi-objective optimal control (e.g., quickest paths constrained by maximum pathlength & shortest paths constrained by maximum travel time);
anisotropic front propagation (e.g., first-arrivals and multiple-arrivals in seismic imaging);
optimal control under uncertainty (e.g., optimal traveling when the map or the terminal time is not quite known);
Markov decision processes (e.g., stochastic shortest paths on graphs);
dynamical systems (e.g., approximation of "geometrically stiff" invariant manifolds of vector fields).

Monday, September 28, 2008

Joint seminar with Applied Mathematics Colloquium
4:30-5:30pm in 2-231
Andrea L. Bertozzi (Department of Mathematics, University of California Los Angeles)

Swarming by nature and by design

The cohesive movement of a biological population is a commonly observed natural phenomenon. With the advent of platforms of unmanned vehicles, this occurrence is attracting renewed interest from the engineering community. This talk will review recent research results on both modeling and analysis of biological swarms and also design ideas for efficient algorithms to control groups of autonomous agents. For biological models we consider two kinds of systems: driven particle systems based on force laws and continuum models based on kinematic rules. Both models involve long-range social attraction and short range dispersal and yield patterns involving clumping, mill vortices, and surface-tension-like effects. For artificial platforms we consider the problem of boundary tracking of an environmental material and consider both computer models and demonstrations on real platforms of robotic vehicles. We also consider the motion of vehicles using artificial potentials.

Wednesday, October 1, 2008

4-5pm in 2-151
Raúl A. Radovitzky (MIT Aeronautics and Astronautics, MIT Institute for Soldier Nanotechnologies)

Discontinuous Galerkin methods applied to fracture mechanics

In this talk, I will present our progress in exploring the numerical formulation of problems in the mechanics of solid materials within the framework of discontinuous Galerkin (DG) methods. The interest in DG methods stems from its potential to address some limitations in conventional finite element formulations of problems involving complex phenomena including fracture and nonlocal material response.
Included in this talk will be the weak formulation of the boundary value problem of finite deformation elasticity, the discretized problem and its numerical properties, and the numerical implementation of the method within a conventional finite element framework. I will also discuss the extension to problems involving dynamic plastic deformations and the scalability in the case of explicit time integration.
Finally, I will discuss the application to problems involving fracture. In this respect, we discuss the connections of the method to the well-established Cohesive Zone Model (CZM) approach to crack nucleation and propagation. We show that the latter method describes the elasticity in the cohesive zone in an intrinsically inconsistent way, whereas the DG approach is inherently consistent.
Time permitting I will describe the application of DG methods to higher order problems, including shell theory and nonlocal isotropic elasticity.

Wednesday, October 15, 2008

4-5pm in 2-151
Chi-Wang Shu (Division of Applied Mathematics, Brown University)

High order methods for convection dominated PDEs - An overview

In this talk we will give an overview of algorithm development and application, with an emphasis on recent progress, on high order methods for convection dominated partial differential equations. We will mainly discuss the finite difference weighted essentially non-oscillatory (WENO) schemes, finite volume WENO schemes, and discontinuous Galerkin (DG) finite element methods. A comparison of their relevant advantages and disadvantages will be given.

Wednesday, October 29, 2008

4-5pm in 2-151
Benjamin Seibold (MIT, Applied Mathematics)

Particle methods - Sparsity and exact conservation

In many numerical approaches that operate on a fixed grid, a proper treatment of convection poses the largest challenge. Particle methods are a way out. Particles are moved with the flow, and thus take care of convection automatically. This advantage comes at a price: The governing equations have to be discretized on point cloud, and particle management (merging upon particle collision, and insertion of particles into holes) is required.
On the aspect of discretization, we consider meshfree finite difference approximations of the Laplace operator. A fundamental problem is how to select a small number of neighbors to a point, but guarantee the stability of the arising stencil. We present an approach, based on linear optimization, that selects optimally sparse finite difference stencils, while guaranteeing stability.
On the aspect of particle management, we present a new approach that approximates scalar 1D conservation laws only by particle motion and particle merging and insertion. The method conserved area exactly, and in addition conserves entropy exactly when no shocks are present. We outline an application that is relying heavily on such exact conservation properties.

Wednesday, November 12, 2008

4-5pm in 2-151
Sigal Gottlieb (UMass Dartmouth, Mathematics)

Time stepping methods for numerical solution of hyperbolic PDEs with shocks

When numerically solving a hyperbolic conservation law it is important to consider the properties of the spatial discretization combined with the time discretization. If the problem is smooth, it is sufficient to linearize the problem and analyze the L2 stability properties of the resulting discretization. However, if the solution is nonsmooth, stability in the L2 norm is not sufficient. This is because for PDEs with discontinuous solutions, the presence of oscillations prevents the approximation from converging uniformly. To ensure that the method does not allow oscillations to form, we require stability of the nonlinear system in the maximum norm or in the TV semi-norm. Strong stability preserving (SSP) high order time discretizations were developed to ensure these types of nonlinear stability properties SSP methods preserve the strong stability properties -- in any norm, seminorm or convex functional -- of the spatial discretization coupled with first order Euler time stepping. I will describe the development of SSP methods and the current state-of-the-art of these methods.

Wednesday, November 19, 2008

4-5pm in 2-151
John E. Dolbow (Duke University, Civil and Environmental Engineering)

An embedded interface finite element method: Application to modeling stimulus-responsive hydrogels

Stimulus-responsive hydrogels (SRHs) are macromolecular polymer networks immersed in a solvent, synthesized to exhibit large volumetric swelling in response to small changes in environmental stimuli. For example, SRHs have been designed to actuate in response to changes in temperature, solvent concentration, pH, and light. The unique properties of these ``soft-wet" materials make them appealing for a large range of applications. They have been used for sensors to detect trace contaminants in fuel lines and autonomous control in microfluidic systems, just to name a few. However, a lack of understanding into the relationships between gel composition, kinetics, and mechanical response has hindered designs based on SRHs and delayed the technological transfer from the laboratory to the marketplace. This presentation will focus on our recent efforts to characterize the unique behavior of SRHs and develop robust finite element models of the same. We have developed continuum-based models for chemically and thermally-induced volume transitions in hydrogels (Dolbow et al., 2004, 2005). Consistent with experimental observations, the models allow for a sharp interface separating swelled and collapsed phases in SRHs. The models predict characteristic swelling times that are proportional to the square of the characteristic linear dimension of the specimen. Our results have also suggested several synthetic pathways that might be pursued to engineer hydrogels with optimal response times. We discretize our models using an embedded interface finite element method, wherein the interface geometry is allowed to be independent of the underlying mesh (Dolbow and Franca, 2008; Dolbow and Harari, 2008). The method enhances the approximation basis in the vicinity of the interface to capture discontinuities in both primary and secondary fields, and is applicable to fully unstructured quad and tet meshes. Interfacial constraints are enforced weakly using a modification of a classical, variationally consistent, interior penalization method. Numerical tests indicate the accuracy of the method to be competitive with widely used finite-difference methods for elliptic interface problems.

Wednesday, December 3, 2008

4-5pm in 2-151
Alain Karma (Northeastern University, Physics Department and Center for Interdisciplinary Research on Complex Systems)

Phase-field modeling of fracture

The phase-field method has emerged as a powerful computational tool to describe the complex evolution of interfaces in a wide range of materials science problems ranging from alloy solidification to solid-state precipitation to thin-film patterning and dislocation dynamics. The chief advantage of this method is to avoid front-tracking by the introduction of a scalar order parameter that varies smoothly in space, thereby making the interface between two phases or two crystal grains spatially diffuse. Evolution equations for these order parameters are derived variationally from a Lyapounov functional that represents the total free-energy of the system.
This talk will give an overview of the more recent application of this method to fracture where the phase-field order parameter is used as a local measure of damage. The method allows to describe both the short-scale physics of failure inside the microscopic process zone around the crack tip and macroscopic linear elasticity within a self-consistent set of partial differential equations that can be readily simulated. In addition, those equations can be analyzed in certain limits to derive, rather than to guess, laws of crack motion. Results shed light on the origin and validity of the several decade old principle of local symmetry used to determine crack paths under various loading conditions and on its extension to anisotropic materials.