Universality and chaos in clustering and branching processes
Prof. Robert Pego
Carnegie-Mellon University

Scaling limits of Smoluchowski's coagulation equation are related to probability theory in numerous remarkable ways. E.g., such an equation governs the merging of ancestral trees in critical branching processes, as observed by Bertoin and Le Gall. A simple explanation of this relies on how Bernstein functions relate to a weak topology for Levy triples. From the same theory, we find the existence of 'universal' branching mechanisms which generate complicated dynamics that contain arbitrary renormalized limits. I also plan to describe a remarkable application of Bernstein function theory to a coagulation-fragmentation model introduced in fisheries science to explain animal group size.

Maximum Norm Stability and Error Estimates for Stokes and Navier-Stokes Problems
Prof. Vivette Girault
Université Pierre et Marie Curie, Paris, France

Energy norm stability estimates for the finite element discretization of the Stokes problem follow easily from the variational formulation provided the discrete pressure and velocity satisfy a uniform inf-sup condition. But deriving uniform stability estimates in L^∞ is much more complex because variational formulations do not lend themselves to maximum norms. I shall present here the main ideas of a proof that relies on weighted L² estimates for regularized Green's functions associated with the Stokes problem and on a weighted inf-sup condition. The domain is a convex polygon or polyhedron. The triangulation is shape-regular and quasi-uniform. The finite element spaces satisfy a super-approximation property, which is shown to be valid for most commonly used stable finite-element spaces. Extending this result to error estimates and to the solution of the steady incompressible Navier-Stokes problem is straightforward.

Semismooth Newton Methods: Theory, Numerics and Applications
Prof. Michael Hintermüller
Department of Mathematics
Humboldt-Universität, Berlin, Germany

Many mathematical models of processes or problems in engineering sciences, mathematical imaging, biomedical sciences or mathematical finance rely on non-smooth structures, either directly through non-differentiable associated energy models, due to (quasi)variational inequality formulations or the presence of inequality constraints in pertinent energy minimization tasks. Based on non-smooth operator equation based (re)formulations of the above problem classes, in this talk a generalized Newton framework in function space is discussed. For this purpose the concept of semismoothness in function space is addressed. Relying on the latter concept, locally superlinear convergence of the associated semismooth Newton iteration is established and its mesh independent convergence behavior upon discretization is shown. In a second part of the talk, the efficiency and wide applicability of the above semismooth Newton framework is highlighted by considering constrained optimal control problems for fluid flow, contact problems with or without adhesion forces, phase separation phenomena relying on non-smooth homogeneous free energy densities and restoration tasks in mathematical image processing.

Optimal and Practical Algebraic Solvers for Discretized PDEs
Prof. Jinchao Xu
Center for Computational Mathematics and Applications
Penn State University

An overview of fast solution techniques (such as multi-grid, two-grid, one-grid and nil-grid methods) will be given in this talk on solving large scale systems of equations that arise from the discretization of partial differential equations (such as Poisson, elasticity, Stokes, Navier-Stokes, Maxwell, MHD, and black-oil models). Mathematical optimality, practical applicability and parallel (CPU/GPU) scalability will be addressed for these algorithms and applications.

Complex Fluids
Prof. Peter Constantin
Department of Mathematics
University of Chicago

The talk will be about some of the models used to describe fluids with particulate matter suspended in them. Some of these models are very complicated. After a bit of history and a review of known results, I will try to point out some open problems, isolate some of the mathematical difficulties, and illustrate some of the phenomena on simpler didactic models.

Discontinuous Galerkin Finite Element Methods for High Order Nonlinear Partial Differential Equations
Prof. Chi-Wang Shu
Brown University

Discontinuous Galerkin (DG) finite element methods were first designed to solve hyperbolic conservation laws utilizing successful high resolution finite difference and finite volume schemes such as approximate Riemann solvers and nonlinear limiters. More recently the DG methods have been generalized to solve convection dominated convection-diffusion equations (e.g. high Reynolds number Navier-Stokes equations), convection-dispersion (e.g. KdV equations) and other high order nonlinear wave equations or diffusion equations. In this talk we will first give an introduction to the DG method, emphasizing several key ingredients which made the method popular, and then we will move on to introduce a class of DG methods for solving high order PDEs, termed local DG (LDG) methods. We will highlight the important ingredient of the design of LDG schemes, namely the adequate choice of numerical fluxes, and emphasize the stability of the fully nonlinear DG approximations. Numerical examples will be shown to demonstrate the performance of the DG methods.

A Taste of Compressed Sensing
Prof. Ronald DeVore
Texas A&M University

Compressed Sensing is a new paradigm in signal and image processing. It seeks to faithfully capture a signal/image with the fewest number of measurements. Rather than model a signal as a bandlimited function or an image as a pixel array, it models both of these as a sparse vector in some representation system. This model fits well real world signals and images. For example, images are well approximated by a sparse wavelet decomposition. Given this model, how should we design a sensor to capture the signal with the fewest number of measurements. We shall introduce ways of measuring the effectiveness of compressed sensing algorithms and then show which of these are optimal.

Isogeometric Analysis
Prof. Thomas J. R. Hughes
Institute for Computational Engineering and Sciences 
University of Texas at Austin

Geometry is the foundation of analysis yet modern methods of computational geometry have until recently had very little impact on computational mechanics. The reason may be that the Finite Element Analysis (FEA), as we know it today, was developed in the 1950's and 1960's, before the advent and widespread use of Computer Aided Design (CAD) programs, which occurred in the 1970's and 1980's. Many difficulties encountered with FEA emanate from its approximate, polynomial based geometry, such as, for example, mesh generation, mesh refinement, sliding contact, flows about aerodynamic shapes, buckling of thin shells, etc., and it s disconnect with CAD. It would seem that it is time to look at more powerful descriptions of geometry to provide a new basis for computational mechanics.

The purpose of this talk is to describe the new generation of computational mechanics procedures based on modern developments in computational geometry. The emphasis will be on Isogeometric Analysis in which basis functions generated from NURBS (Non-Uniform Rational B-Splines) and T-Splines are employed to construct an exact geometric model. For purposes of analysis, the basis is refined and/or its order elevated without changing the geometry or its parameterization. Analogues of finite element h- and p-refinement schemes are presented and a new, more efficient, higher-order concept, k-refinement, is described. Refinements are easily implemented and exact geometry is maintained at all levels without the necessity of subsequent communication with a CAD description.

In the context of structural mechanics, it is established that the basis functions are complete with respect to affine transformations, meaning that all rigid body motions and constant strain states are exactly represented. Standard patch tests are likewise satisfied. Numerical examples exhibit optimal rates of convergence for linear elasticity problems and convergence to thin elastic shell solutions. Extraordinary accuracy is noted for k-refinement in structural vibrations and wave propagation calculations. Surprising robustness is also noted in fluid and non-linear solid mechanics problems. It is argued that Isogeometric Analysis is a viable alternative to standard, polynomial-based, finite element analysis and possesses many advantages. In particular, k-refinement seems to offer a unique combination of attributes, that is, robustness and accuracy, not possessed by classical p-methods, and is applicable to models requiring smoother basis functions, such as, thin bending elements, and strain-gradient and various phase-field theories.

A modelling paradigm for patient-specific simulation of cardiovascular fluid-structure interaction is reviewed, and a précis of the status of current mathematical understanding is presented.

The Fast Multipole Method and its Applications
Leslie Greengard
Courant Institute of Mathematical Sciences, New York University

In this lecture, we will describe the analytic and computational foundations of fast multipole methods (FMMs), as well as some of their applications. They are most easily understood, perhaps, in the case of particle simulations, where they reduce the cost of computing all pairwise interactions in a system of N particles from O(N2) to O(N) or O(N log N) operations. FMMs are equally useful, however, in solving partial differential equations by first recasting them as integral equations. We will present examples from electromagnetics, elasticity, and fluid mechanics.

Topology optimization of structures
Prof. Gregoire Allaire
Ecole Polytechnique

The typical problem of structural optimization is to find the "best" structure which is, at the same time, of minimal weight and of maximum strength or which performs a desired deformation. In this context I will present the combination of the classical shape derivative and of the level-set methods for front propagation. This approach has been implemented in two and three space dimensions for models of linear or non-linear elasticity and for various objective functions and constraints on the perimeter. It has also been coupled with the bubble or topological gradient method which is designed for introducing new holes in the optimization process. Since the level set method is known to easily handle boundary propagation with topological changes, the resulting numerical algorithm is very efficient for topology optimization. It can escape from local minima in a given topological class of shapes and the resulting optimal design is largely independent of the initial guess.

New materials from mathematics: real and imagined
Prof. Richard D. James
University of Minnesota

In this talk I will give two examples where mathematics played an important role for the discovery of new materials, and a third example where mathematics suggests a systematic way of searching for broad classes of yet undiscovered materials.

Adaptive Approximation by Greedy Algorithms
Prof. Albert Cohen
Laboratoire Jacques-Louis Lions, Université Pierre et Marie Curie, 
Paris, France

This talk will discuss computational algorithms that deal with the following general task: given a function f and a dictionary of functions D in a Hilbert space, extract a linear combination of N functions of D which approximates f at best. We shall review the convergence properties of existing algorithms. This work is motivated by applications as various as data compression, adaptive numerical simulation of PDE's, statistical learning theory.

Compressive Sampling
Prof. Emmanuel J. Candes
California Institute of Technology

One of the central tenets of signal processing is the Shannon/Nyquist sampling theory: the number of samples needed to reconstruct a signal without error is dictated by its bandwidth-the length of the shortest interval which contains the support of the spectrum of the signal under study. Very recently, an alternative sampling or sensing theory has emerged which goes against this conventional wisdom. This theory allows the faithful recovery of signals and images from what appear to be highly incomplete sets of data, i.e. from far fewer data bits than traditional methods use. Underlying this metholdology is a concrete protocol for sensing and compressing data simultaneously.

This talk will present the key mathematical ideas underlying this new sampling or sensing theory, and will survey some of the most important results. We will argue that this is a robust mathematical theory; not only is it possible to recover signals accurately from just an incomplete set of measurements, but it is also possible to do so when the measurements are unreliable and corrupted by noise. We will see that the reconstruction algorithms are very concrete, stable (in the sense that they degrade smoothly as the noise level increases) and practical; in fact, they only involve solving very simple convex optimization programs.

An interesting aspect of this theory is that it has bearings on some fields in the applied sciences and engineering such as statistics, information theory, coding theory, theoretical computer science, and others as well. If time allows, we will try to explain these connections via a few selected examples.

Imaging in random media
Prof. George C. Papanicolaou
Mathematics Department 
Stanford University

I will present an overview of some recently developed methods for imaging with array and distributed sensors when the environment between the objects to be imaged and the sensors is complex and only partially known to the imager. This brings in modeling and analysis in random media, and the need for statistical algorithms that increase the computational complexity of imaging, which is done by backpropagating local correlations rather than traces (interferometry). I will illustrate the theory with applications from non-destructive testing and from other areas.

String integration of some MHD equations
Prof. Yann Brenier
Laboratoire Alexandre Dieudonné 
Université de Nice-Sophia-Antipolis, France

We first review the link between strings and some Magnetohydrodynamics equations. Typical examples are the Born-Infeld system, the Chaplygin gas equations and the shallow water MHD model. They arise in Physics at very different (from subatomic to cosmologic) scales. These models can be exactly integrated in one space dimension by solving the 1D wave equation and using the d'Alembert formula. We show how an elementary "string integrator" can be used to solve these MHD equations through dimensional splitting. A good control of the energy conservation is needed due to the repeated use of Lagrangian to Eulerian grid projections. Numerical simulations in 1 and 2 dimensions will be shown.

Multiscale Analysis in Micromagnetics
Prof. Felix Otto
Institute for Applied Mathematics
University of Bonn, Germany

From the point of view of mathematics, micromagnetics is an ideal playground for a pattern forming system in m aterials science: There are abundant experiments on a wealth of visually attractive phenomena and there is a well-accepted continuum model.

In this talk, I will focus on two specific experimental pattern for thin film ferromagnetic elements. One pattern is a ground state, the other pattern is a metastable state. Starting point for our analysis is the micromagnetic model which has three length scales and thus many parameter regimes. For both pattern, we identify the appropriate paramater regime and rigorously derive a reduced model via G-convergence. We numerically simulate the reduced model and compare to experimental data.

This is joint work with A. DeSimone, R. V. Kohn, and S. Müller for the first part and with R. Cantero-Alvarez and J. Steiner for the second part.

Multiscale Modeling in Biosciences: Ion Transport through Membranes
Prof. Willi Jäger
Institute for Applied Mathematics
University of Heidelberg, Germany

Electromagnetic imaging for small inhomogeneities
Prof. Michael Vogelius
Department of Mathematics, Rutgers University

Mathematical models for cell motion
Prof. Benoît Perthame
École Normale Supérieure, Paris

Multiscale Modeling and Computation of Flow in Heterogeneous Media
Prof. Tom Hou
Caltech

Mathematical and Numerical Modeling of the Cardiovascular System
Prof. Alfio Quarteroni
Politecnico di Milano, Milan, Italy, and 
EPFL, Lausanne, Switzerland

The regularity of minimizers in elasticity
Prof. John Ball
Department of Mathematics, Oxford

Multigrid: From Fourier to Gauss
Prof. Randolph E. Bank
Department of Mathematics, University of California at San Diego

Mathematical Problems in Meteorology and Oceanography
Prof. Roger Temam
Institute for Scientific Computing and Applied Mathematics, Indiana University

Recent Approaches in the Treatment of Subgrid Scales
Prof. Franco Brezzi
Istituto di Analisi Numerica del CNR and Dipartimento di Matematica, Universita di Pavia, Italy

Time Stepping in Parabolic Problems - Approximation of Analytic Semigroups
Prof. Vidar Thomée
Dept. of Mathematics, Chalmers University of Technology and Göteborg University 

Colliding Black Holes and Gravity Waves: A New Computational Challenge
Prof. Douglas N. Arnold
Dept. of Mathematics, Pennsylvania State University

A Priori and A Posteriori Error Estimates in Finite Element Approximation
Prof. Lars B. Wahlbin
Dept. of Mathematics, Cornell University

Mathematical Problems Related to the Reliability of Finite Element Analysis in Practice: When Can We Trust the Computational Results for Engineering Decisions
Prof. Ivo Babuska
University of Texas, Austin, Emeritus Professor at University of Maryland