Seminar: Numerical Methods for Partial Differential Equations

Math Department Calendar
List of Seminars

Facts

Organizers: Lyuba Chumakova, Rodolfo Ruben Rosales

Mailing List

If you are interested in receiving announcements by email, please write to daisymae@math.mit.edu

Schedule Spring 2012

 

Announcement: This semester the NumPDE seminar is being run within the Physical Math Seminar (PMS) on

Tuesdays, 2:30-3:30pm, Room 2-105, Refreshments will be served in Room 2-290 at 3:30pm.

For the up-to-date schedule please check the PMS website.

Thursday, February 2nd, 2012

SPECIAL TIME AND LOCATION: Room 2-143 from 2:30 PM - 3:30 PM, Thursday, Feb 2nd.
Cristina Turner University of Cordoba, FaMAF, Argentina

Shape optimization for tumor location

In non-invasive thermal diagnostics, accurate correlations between the thermal image at skin surface and interior human physiology are desired. In this work an estimation methodology to determine unknown geometrical parameters of an embedded tumor is proposed. We defined a functional that represents the mismatch between a measured experimental temperature profile, which may be obtained by infrared thermography on the skin surface, and the solution of an appropriate boundary problem. This functional is related to the geometrical parameters through the solution of the boundary problem, in such a way that finding the minimum of this functional form also means finding the unknown geometrical parameters of the embedded tumor. Sensitivity analysis techniques coupled with the adjoint method were considered to compute the shape derivative of the functional and a nonmonotone spectral projected gradient method was implemented to solve the optimization problem of finding the optimal geometric parameters.

Schedule Fall 2011

 

Announcement: This semester the NumPDE seminar is being run within the Physical Math Seminar (PMS) on

Tuesdays, 2:30-3:30pm, Room 2-105, Refreshments will be served in Room 2-290 at 3:30pm.

For the up-to-date schedule please check the PMS website.

Schedule Spring 2011

Wednesday, February 16th, 2011

4-5pm in 2-131
Enkeleida Lushi Courant, NYU

An efficient numerical method for multiple surfactant-covered interfaces in 2D Stokes Flow

I will present an efficient and highly accurate numerical method for computing the deformation of many surfactant-coated interfaces in 2D slow viscous flow. Surfactants locally alter the surface tension in interfaces and thus change the nature of their motion. We consider bubbles as well as drops covered with interface-bound or insoluble surfactant. The advection-diffusion equation for the surfactant concentration on the interface is coupled with the Stokes equations in the fluid domain through a Laplace-Young condition. The Stokes equations are first recast as an integral equation based on the complex variable theory of the biharmonic equation. The numerical method employed is spectrally accurate and uses a fast-multipole accelerated iterative procedure. The computational cost per time step is only O (N logN ) operations, with N being the number of discretization points. The interface is described by a spectral mesh and is advected according to the fluid velocity in such a manner so as to preserve equal arc length spacing of marker points. This equal arc length framework has the dual advantage of dynamically maintaining the spatial mesh and allowing efficient, implicit treatment of the stiffest terms in the dynamics. Several phenomenologically-different examples will be presented, e.g. tip streaming, bursting bubbles, interacting bubbles, etc. Last, I will talk about the numerical method extension to study an interface with soluble surfactants.

Wednesday, March 16, 2011

4-5pm in 2-131
Burt Tilley WPI

Macroscale field effects on flows between closely spaced, thin laminates.

Fluid flows through anisotropic media are found in a wide variety of geophysical and biological systems. The macroscale behavior of these systems depend on the microstructure, which in turn may depend on local and global physical processes. Classically, geometric restrictions are needed to model these systems on the largest length scale, and we are interested in developing effective models which relax these restrictions. To explore the development of these multiscale models, we present two fundamental problems.

In the first problem, we consider an array of closely spaced, purely dielectric rigid laminates with nonuniform thickness. The laminate thickness and spacing varies on a length scale much longer than the characteristic thickness of the laminates. In the spacing between the laminates, an electrically conducting fluid is driven by an applied electric field through electroosmosis and electrophoresis along with an applied pressure gradient. Debye layers occur at the laminate-fluid interface, which are assumed to be much smaller than the laminate thickness. From a modification of the classical homogenization approach that relies on a physical microscale constraint in place of a geometric constraint, we derive an effective set of equations that describe the fluid pressure, the anion and cation concentrations in the fluid and the electric potential. Anisotropic dispersion effects in the electric field are included, and electroneutrality in the fluid is not imposed. We find that gradients in the laminate spacing can lead to charge accumulation when electroosmosis and the electrophoresis induced from the anisotropic dispersion effects balance.

Our second problem centers on the flow of an incompressible fluid that saturates an array of deformable laminates with gravity acting in the spanwise direction. The aspect ratio of the characteristic spacing between the laminates is much smaller than the characteristic scale of the laminate length, and we use this aspect ratio to find effective equations for the components of the stress tensor of the effective material. Momentum conservation on the macroscale at leading order result in a coupled set of elliptic equations for the unknown laminate spacing, orientation and the local pore pressure. At next order, effective stress-strain relations are derived based on the local geometry and material properties of the laminates and fluid. This work is done in collaboration with B. Vernescu and J.D. Plummer at WPI.

Wednesday, March 30, 2011

4-5pm in 2-131
Hans Johnston UMASS Amherst

Numerical Study of Turbulent Thermal Convection Between Conditions of Constant Temperature and Constant Flux

We report the results of high resolution direct numerical simulations of two-dimensional Rayleigh-Benard convection for Rayleigh numbers up to Ra=10^10 in order to study the influence of temperature boundary conditions on turbulent heat transport. Specifically, we consider the extreme cases of fixed heat flux (where the top and bottom boundaries are poor thermal conductors) and fixed temperature (perfectly conducting boundaries). Both cases display identical heat transport at high Rayleigh numbers fitting a power law Nu~0.138xRa^(.285) with a scaling exponent indistinguishable from 2/7 = 0.2857... above Ra = 10^7. The overall flow dynamics for both scenarios, in particular the time averaged temperature profiles, are also indistinguishable at the highest Rayleigh numbers. The findings are compared and contrasted with results of recent three-dimensional simulations.

This is joint work with C.R. Doering (U. Michigan), Cheng Wang (UMass-Dartmouth), Jian-Guo Liu (Duke)

Wednesday, April 6, 2011

4-5pm in 2-131
Jian-Guo Liu Duke Universtiy

Dynamics of orientational alignment and phase transition

Phase transition of directional field appears in some physical and biological systems such as ferromagnetism near Curie temperature, flocking dynamics near critical mass of self propelled particles. Dynamics of orientational alignment associated with the phase transition can be effectively described by a mean field kinetic equation. The natural free energy of the kinetic equation is non-convex with a minimum level set consisting of a sphere at super-critical case, a typical spontaneous symmetry breaking behavior in physics. In this talk, I will present some analytical results on this dynamics equation of orientational alignment and exponential convergence rate to the equilibria for both supper and sub critical cases, as well at algebraic convergence rate at the critical case. A new entropy and spontaneous symmetry breaking analysis played an important role in our analysis.

Wednesday, April 13, 2011

4-5pm in 2-131
Michael Minion University of North Carolina at Chapel Hill

Novel Methods for Temporal Integration of PDEs

I will discuss ongoing research on the development of novel methods for the temporal integration of ODEs and PDEs. The strategy employed in the numerical methods is based on an iterative deferred corrections approach, within which one can utilize operator splitting, multirate timestepping, and parallelization in the temporal direction to achieve better computational efficiency. Much of the recent work is motivated by fluid-structure interaction problems in biological systems and I will discuss the difficulties and progress related to some target applications in this area. Recent results concerning parallelization in the temporal direction will also be presented.

Wednesday, May 4, 2011

4-5pm in 2-131
Hyoungsu Baek MIT

Accuracy and stability enhancements of semi-implicit schemes for the Navier–Stokes equations through sub-iteration

We present an iterative semi-implicit scheme for the incompressible Navier–Stokes equations, which is stable at CFL numbers well above the nominal limit. We have implemented this scheme in conjunction with spectral discretizations, which suffer from serious time step limitations at very high resolution. Specifically, at each time level, the nonlinear convective term and the pressure boundary condition – both of which are treated explicitly in time – are updated using fixed-point iteration and Aitken relaxation. Instabilities from pressure boundary condition and convective term are suppressed by sub-iteration and relaxation of flow field, respectively. Eigenvalue analysis of Stokes problem confirms that this scheme is unconditionally stable. For Navier-Stokes flows, the proper value of the relaxation parameter is obtained as a function of the flow parameters, which leads to second- and third-order temporal accuracy at CFL number 5–14. Moreover, stability of the scheme does not depend on the order of polynomial basis employed in the spectral element method. Systematic accuracy, stability, and cost comparisons are presented against the standard semi-implicit method and a fully-implicit scheme that does not require Newton’s iterations.

Schedule Fall 2010

Wednesday, November 3, 2010

4-5pm in 2-132
Paul Chesler (MIT - Dept. of Physics)

Applied string theory: from gravitational collapse to heavy ion collisions

A remarkable result from heavy ion collisions at the Relativistic Heavy Ion Collider is that shortly after a collision, the medium produced behaves as a nearly ideal liquid. The system is very dynamic and evolves from a state of two colliding nuclei to a liquid in a time roughly equivalent to the time it takes light to cross a proton. Understanding the mechanisms behind the rapid approach to a liquid state is a challenging task. In recent years string theory has emerged as a powerful tool to study non-equilibrium phenomena, mapping the (challenging) dynamics of quantum systems onto the dynamics of classical gravitational systems. The creation of a liquid in a quantum theory maps onto the classical process of gravitational collapse and black hole formation. I will describe how one can use techniques borrowed from numerical relativity in astrophysics to study processes which mimic the dynamics of heavy ion collisions.

SPECIAL DAY / TIME / LOCATION: Friday, November 5, 2010

4-5pm in 2-135
Snezhana I. Abarzhi (University of Chicago)

Turbulent mixing and beyond: problems, concepts, solutions

Turbulent mixing plays an important role in a broad variety of natural and artificial systems, spanning astrophysical to atomistic scales and low to high energy densities. Examples include inertial confinement fusion, supernovae, stellar convection, non-canonical boundary layers and optical free-space communications. Theoretical description of non-equilibrium mixing transports is a challenging problem due to singular aspects of the governing (Euler or Navier-Stokes) equations. Furthermore these processes are statistically unsteady and their fluctuating quantities are essentially time-dependent and non-Gaussian. We apply the new theoretical concept, the rate of momentum loss, to describe the transports of mass, momentum and energy in turbulent mixing flow and to capture its anisotropic and inhomogeneous character. It is shown that invariant, scaling and spectral properties of unsteady turbulent mixing differ substantially from those of isotropic and homogeneous turbulence. Time- and scale-invariance of the rate of momentum loss leads to non-dissipative momentum transfer, to 1/2 and 3/2 power-law scale-dependencies of the velocity and Reynolds number and to non-Kolmogorov spectra. Turbulent mixing exhibits more order compared to isotropic turbulence, and its viscous and dissipation scales are finite and set by flow acceleration. We suggest how to describe the random character of the unsteady turbulent flow and show that the rate of momentum loss is the statistic invariant and a robust diagnostic parameter for either sustained or time-dependent acceleration. Some criteria are outlined for the estimate of the fidelity and information capacity of the experimental and numerical data sets.

Wednesday, November 17, 2010

4-5pm in 2-132
Paul Hand (MIT - Mathematics)

An Adaptive Multiscale Model for Simulating Cardiac Conduction

We present an adaptive multi-scale numerical method for simulating cardiac action potential propagation along a single strand of heart muscle cells. This method combines macroscale cable partial differential equations posed over the tissue with different microscale equations posed over discrete cellular geometry. The microscale equations are applied near a wavefront, where electric potential changes rapidly, and the macroscale equations are applied everywhere else. One-dimensional numerical simulations reveal that the adaptive scheme accurately reproduces the action potential waveforms and wavespeeds of the full microscale model. In particular the adaptive simulations capture the ephaptic effect that has been demonstrated by several mathematical models the literature. Our study demonstrates that capturing the behavior of action potential propagation may sometimes require simulations that are adaptive and multi-scale because (1) fully resolving all cells in a whole-heart simulation is computationally infeasable; (2) one-dimensional simulations at coarseness levels that are affordable for whole-heart simulations lead to substantial errors in conduction speed; and (3) the macroscopic cable equations are invalid at low levels of gap junctional coupling. More abstractly, this work provides an example of when and how to selectively couple the dynamics of a periodic microstructure to those of its macroscale description.

Wednesday, December 1, 2010

4-5pm in 2-132
David Shirokoff (MIT - Mathematics)

An efficient method for the incompressible Navier-Stokes equations on irregular domains with no-slip boundary conditions, high order up to the boundary.

Common efficient schemes for the incompressible Navier-Stokes equations, such as projection or fractional step methods, have limited temporal accuracy as a result of matrix splitting errors, or introduce errors near the domain boundaries (resulting in weakly convergent solutions). In this paper we recast the incompressible Navier-Stokes equations (with the velocity prescribed at the boundary) as an equivalent system, for the primary variables velocity and pressure. We do this in the usual way away from the boundaries, by replacing the incompressibility condition on the velocity by a Poisson equation for the pressure. The key difference from the usual approaches occurs at the boundaries, where we use boundary conditions that unequivocally allow the pressure to be recovered from knowledge of the velocity at any fixed time --- avoiding the common difficulty of an, apparently, over-determined Poisson problem. Since in this alternative formulation the pressure can be accurately and efficiently recovered from the velocity, the recast equations are ideal for numerical marching methods. This new system can be discretized using a variety of methods, in principle to any desired order of accuracy. In this work we illustrate the approach with a 2-D second order finite difference scheme on a Cartesian grid, and devise an algorithm to solve the equations on domains with curved (non-conforming) boundaries, including a case with a non-trivial topology (a circular obstruction inside the domain). This algorithm achieves second order accuracy in the $L^{\infty}\/$ norm, for both the velocity and the pressure. The scheme has a natural extension to 3-D.

SPECIAL DAY / TIME / LOCATION: Friday, December 10, 2010

4-5pm in 2-103
Benjamin Seibold (Temple University)

A new perspective on the moment closure problem in radiative transfer

Radiative transfer can be modeled by a kinetic equation that describes the evolution of the particle density function in a phase space of time, position, and the angle of flight (possibly more). While the direct simulation of this high dimensional mesoscopic equation is possible (albeit very costly), in many applications it is desirable to have a description in terms of macroscopic equations. An expansion in the angular variable yields an equivalent system of infinitely many macroscopic moment equations. The fundamental question how to best truncate this system is the moment closure problem. Many types of closure strategies exist. These are typically based on an asymptotic arguments or assume higher moments be quasi-stationary. In this talk, we present a very different approach to derive moment closures, based on the Mori-Zwanzig formalism of irreversible statistical mechanics. Here, the influence of the truncated moments on the resolved moments is modeled by a memory term. Moment closures are then defined by a choice of a probability measure on the phase space and suitable approximations to the memory term. We demonstrate that existing closures, such as PN, SPN, and diffusion correction closures, can be re-derived with this formalism. In addition, new closures arise, such as the crescendo-diffusion closure.

Wednesday, December 15, 2010

4-5pm in 2-132
Alex Marques (MIT - Mathematics)

High-Order Solution of the Poisson Equation with Interface Jump Conditions – Towards a Generalized Ghost Fluid Method

The Poisson equation with jump discontinuities across an interface is of central importance in multiphase diffusion phenomena. In particular, we must solve this equation to obtain the pressure distribution in multiphase flows. The difficulty in numerically solving this equation arises due to the discontinuous nature of the solution around the interface. Over the last three decades we have seen the development of several methods to address this issue. However, obtaining high order of accuracy in a systematic fashion still poses great challenges to the numerical fluid mechanics community.

In this presentation we shall introduce a new method to obtain high- order solutions to the Poisson Equation with interface jump conditions. The method is based on local smooth extrapolations of the solution field across the interface. The extrapolation procedure uses a local functional basis that satisfies the interface conditions to high order of accuracy. The overall approach is compatible with standard discretizations of the Laplace operator, and leads to minimal modifications to the final linear system to be inverted. In particular, we apply the method with cubic Hermit polynomials to develop fourth-order accurate schemes. As a result, standard Poisson solvers can be used with only minimal modifications. Details of the method and applications will be presented.

Schedule Spring 2010

Wednesday, March 17, 2010

4-5pm in 2-105
Maria Cameron (Courant Institute of Mathematical Science)

Computing transition paths for rare events

The overdamped Langevin equation is often used as a model in molecular dynamics. At low temperatures, a system evolving according to such an SDE spends most of the time near the potential minima and performs rare transitions between them. A number of methods have been developed to study the most likely transition paths. I will focus on one of them: the MaxFlux functional. The MaxFlux functional has been around for almost thirty years but not widely used because it is challenging to minimize. Its minimizer provides a path along which the reactive flux is maximal at a given finite temperature. I will show two ways to derive it in the framework of transition path theory: the lower bound approach and the geometrical approach. I will present an efficient way to minimize the MaxFlux functional numerically. I will demonstrate its application to the problem of finding the most likely transition paths in the Lennard-Jones-38 cluster between the face-centered-cubic and icosahedral structures.

Thursday, April 1, 2010

4:00 PM in Building 2, Room 105
Daniela Tordella (Politecnico di Torino, Italy)

Transients of Three-dimensional Perturbations and the Role of Long Waves in the Plane Wake. Relationships with Turbulence

Results from an exploratory analysis of the transient and long-term behavior of small three-dimensional perturbations in the circular cylinder wake are presented. A few comments on the role of long waves in this stability problem and on the role of small scales in turbulence (anisotropy, possible long-term interactions) are also proposed.

Wednesday, April 21, 2010

4:00 PM in Building 2, Room 105
John F. Gibson (Georgia Institute of Technology)

Invariant Solutions and State-space Dynamics of Low-Re Turbulence

It has recently become possible to compute precise 3D, nonlinear solutions of Navier-Stokes equations at Reynolds numbers above the onset of turbulence, for simple geometries such as pipes and channels. These solutions capture the form and dynamics of "coherent structures" and provide a starting point for understanding low-Reynolds turbulence as a dynamical system. In this talk I will present a number of equilibrium, traveling wave, and periodic orbit solutions of plane Couette flow, emphasizing visualizations of their physical structure and state-space dynamics, and comparisons to turbulent flow. Certain spatially localized solutions exhibit homoclinic snaking remarkably similar to that observed in simpler 1D PDE systems. What emerges is a picture of low-Reynolds turbulence as a walk among a set of weakly unstable invariant solutions.

Wednesday, April 28, 2010

4:00 PM in Building 2, Room 105
Professor Anthony T Patera( Department of Mechanical Engineering, MIT)

Certified Reduced Basis Methods; Application to Continuum Mechanics and Transport

We discuss reduced basis approximation and associated a posteriori error estimation for reliable and rapid solution of parametrized partial differential equations.

The crucial ingredients are rapidly convergent Galerkin approximations over a space spanned by "snapshots" on the parametrically induced solution manifold; effective constructions for stability-constant lower bounds; rigorous and sharp a posteriori error bounds for the outputs/quantities of interest; efficient POD (in time)/Greedy (in parameter) selection of quasi-optimal samples; and Offline-Online computational procedures for very rapid response in the real-time and many-query contexts.

We consider the application of these techniques to illustrative problems from heat transfer, solid mechanics, acoustics, and fluid dynamics. We also briefly describe recent implementations on deployed platforms.

Wednesday, May 5, 2010

4:00 PM in Building 2, Room 105
Kenneth Kamrin(Harvard University)

A Novel Finite-Difference Method for Large-Deformation Solid Mechanics

A classic problem in computational continuum mechanics is to simulate the deformation of a solid-like material to the point of very large strain. Standard approaches such as the finite-element method can lose accuracy when elements become overly distorted, and while there have been a number of mesh-repairing methods considered in the past, fully Eulerian algorithms provide a natural way to handle this issue. This talk develops and tests a fixed-grid approach to solid deformation based on a kinematic variable we call the "reference map". The method can be used in a variety of common circumstance cared about in solid mechanics: statics, quasi-statics, smooth dynamics, and shock propagation. Moreover, by the similarity of this method to finite-difference CFD algorithms, the approach also leads to a basic method for computing fluid/structure interactions on a single fixed grid.

 

 

Schedule Fall 2009

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.

Schedule Spring 2009

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).

Schedule Fall 2008

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.