(September 12) Professor Walter Gander: Generating Numerical Algorithms Using Computer Algebra - We show how numerical algorithms can be derived in a simple way using computer algebra. Examples are numerical differentiation, quadrature and multi-step methods for ODE. It is also shown how the discretization error of a method can be computed automatically. This approach not only makes formularies obsolete (in fact some errors were found in Abramowitz/Stegun) but is also useful in teaching since principles and fundamentals are emphasized and we can leave the sometimes tedious derivation of the specific algorithm to the machine.

(September 19) Dr. Milan Mihajlovic: Efficient Preconditioning of the Navier-Stokes Equations with the Free Surface Condition - A numerical model is presented for the simulation of viscoelastic flows (Oldroyd-B) with complex free surfaces in three space dimensions. The mathematical formulation of the model is similar to that of the volume of fluid (VOF) method, but the numerical procedures are different. The time discretization is based on a splitting scheme. The first step consists in solving advection equations whilst the second step corresponds to a simplified Oldroyd-B problem in a fixed domain. Numerical results will be presented for the buckling of a jet and for the stretching of a filament.

Extension to the stochastic Hookean dumbbells model is then discussed and existence on a fixed time interval is proved provided the data are small enough. This result is based on the maximal regularity property of the Stokes operator and on the analycity behavior of the corresponding semi-group. A finite element discretization in space is then proposed. Existence of the numerical solution is proved for small data, as well as a priori error estimates, using an implicit function theorem framework.

(September 26) Dr. Andrea Bonito: Three-Dimensional Viscoelastic Flows with Complex Free Surfaces: Numerical Simulations and Mathematical Analysis of Simplified Problems - A numerical model is presented for the simulation of viscoelastic flows (Oldroyd-B) with complex free surfaces in three space dimensions. The mathematical formulation of the model is similar to that of the volume of fluid (VOF) method, but the numerical procedures are different. The time discretization is based on a splitting scheme. The first step consists in solving advection equations whilst the second step corresponds to a simplified Oldroyd-B problem in a fixed domain. Numerical results will be presented for the buckling of a jet and for the stretching of a filament.

Extension to the stochastic Hookean dumbbells model is then discussed and existence on a fixed time interval is proved provided the data are small enough. This result is based on the maximal regularity property of the Stokes operator and on the analycity behavior of the corresponding semi-group. A finite element discretization in space is then proposed. Existence of the numerical solution is proved for small data, as well as a priori error estimates, using an implicit function theorem framework.

(October 10) Professor Ivo Babuska: Are the Computational Results Reliable? - The notion of the reliability is complex. It will be understood as the reliability of the prediction of the phenomena based on the computation. This prediction is then the basis for the decision. Few major engineering accidents representing different reasons for the non reliability will be presented. a.Modeling. The incorrect mathematical problem was solved. b.Discretization. The error in the numerical solution (of the correct problem). c.Computer science. The round off, underflow. d.Human error. Category a. belongs to the field of Validation, Categories b.c.d. to the Verification. Notions of the Mathematical problem, Verification, Validation, Quantification of the uncertainty and the regulatory assessment will be presented. The Validation pyramid of the Airbus A380 and the failure of the wing test (February 2006) will be discussed. A specific example of the frame where the regulatory assessment is to compute the probability that the displacement in a point is not larger than 3mm and the estimate of the confidence in the computed result. For disposition are the experimental data for the calibration, validation and the accreditation (which create the validation pyramid). Three groups of a different number of available data are given. The purpose was to show the influence of the amount of experimental data on the reliability of the results. This was one of the problems of the Sandia workshop. (May 2006). Mathematics of the Validation is very broad and new. Validation has many aspects, physics, experimentation, sensitivity, philosophy of science. Finally some very basic literature will be mentioned.

(October 17) Dr. Manuel Cascon: Convergence and Optimality of an AFEM for General Second Order Linear Elliptic PDE - Despite the overwhelming computational evidence that AFEM lead to optimal meshes, the mathematical theory started very recently with Binev et al [1]and Stevenson [3]. Paper [1] discusses geometric properties of bisection which turn out to be crucial for optimality; The AFEM in[1] includes a coarsening step which is not necessary in practise for linear elliptic PDE. Stevenson presents in [3] an algorithm with optimal complexity without coarsening, but it does not work with general linear elliptic operator and is unpractical.

In this paper, we modify the Morin-Nochetto-Siebert algorithm [2] with a new procedure for oscillation reduction relative to error estimator. We prove that the resulting AFEM is a contraction for the total error between two consecutive adaptive loops. This strict error reduction property then leads to the optimality of AFEM, that is to a total error decay in terms of number of degrees of freedom as predicted by the best approximation.

Bibliography

[1] P. Dinev, W. Dahmen, and R. DeVore Adaptive finite element methods with convergence rate, Numer. Math., 97(2), pp.~219-268 (2004).

[2] P. Morin, R.H. Nochetto, and K.G. Siebert, Data Oscillation and Convergence of adaptive FEM. SIAM J. Numer. Anal. 38, pp.~466--488 (2000).

[3] R. Stevenson, Optimality of a standard adaptive finite element method (to appear).

(October 24) Professor Leszek Demkowitz: hp-FINITE ELEMENTS FOR MAXWELL EQUATIONS: Discrete Compactness and hp-Convergence for Maxwell Eigenvalues - We will present the Finite Element discretization of time-harmonic Maxwell equations. We shall start with a variational formulation, and discuss the so-called stabilized formulation that reveals that we are dealing with a mixed problem. The satisfaction of two Brezzi's inf-sup conditions leads then to the discrete exact sequence property, and the analysis of the corresponding Maxwell eigenvalue problem. We shall introduce then the Kikuchi's idea of discrete compactness property, and discuss how it implies the convergence of the Maxwell eigenvalues. Finally, we will present a proof of the discrete compactness for a two-dimensional hp-method based on Nedelec's rectangle of the first kind. The result is a joint work with D. Boffi, M. Dauge and M. Costabel.

(October 31) Professor Luc Mieussens: Numerical simulations of rarefied gases in curved channels: thermal creep, circulating flow, and pumping effect - Thermal creep is a flow of a slightly rarefied gas caused by the temperature gradient along a wall. Although it is one of the classical phenomena in rarefied gas dynamics, its importance was revived in recent years in connection with micro gas flows.

The aim of this work is to study the possibility to use the thermal creep to make a pump without moving part for a gas under rarefied conditions or in micro-scales. We first use an original 2D numerical solving of a Boltzmann kinetic model equation. We also derive an asymptotic model (by the diffusion approximation theory) in case of thin channels. The results of these two methods are compared.

In collaboration with K. Aoki, S. Takata and H. Yoshida (Kyoto, Japan) and P. Degond (Toulouse, France)

(November 07) Dr. Alexei Lozinski: Modeling and simulations of gas bubbles in a fluid - In this talk we shall discuss a model allowing to describe motion and coalescence of gas bubbles in a liquid under the action of gravitation and surface tension. The shape of the bubbles is described by a pre-defined family of mappings specified by a small number of parameters, and the effects of the gas motions inside the bubbles are neglected. The motion of a bubble is obtained then in a Lagrangian form using the principle of virtual works. The set of equations is numerically solved with the help of the fictitious domain technique with Lagrange multipliers on the bubbles boundaries. Numerical results in 2D and 3D will be presented.

(November 14) Ms. Olena Shevchenko: Some Aspects of the Theory of Long-Step Interior Point Methods Beyond Symmetric Cone Programming - Interior-Point methods play an important role in modern optimization. These methods are popular because of their efficiency and polynomial-time complexity. The algorithms that allow long steps are the most efficient among various interior-point methods. They were first developed for linear programming and then generalized to semidefinite and symmetric cone programming. Later, Guler showed that it is possible to extend long-step methods to larger classes of problems like hyperbolic or homogeneous programming problems.

The interior-point methods are based on generating iterates that follow some path in the interior of the feasible region that consists of the minimizers of a family of self-concordant barrier functions. These are auxiliary functions that keep the iterations inside the feasible region. They satisfy certain (restrictive) relations between the first, the second, and the third directional derivatives. It is crucial to have efficiently computable self-concordant barriers preferably with the best possible complexity value. In this talk, we introduce a new recursive formula for an optimal dual barrier function for homogeneous cones and discuss some of its properties. If time permits, we will talk about the work done jointly with Osman Guler on hyperbolic programming problems, hyperbolic polynomials, the Lax conjecture, and its applications.

(November 21) Professor Igor Griva: Case Studies in Shape and Trajectory Optimization: Catenary Problem - This talk presents a case study in modern large-scale constrained optimization to illustrate how recent advances in algorithms and modeling languages have made it easy to solve difficult problems using optimization software. We consider the shape of a hanging chain, which, in equilibrium, minimizes the potential energy of the chain. We emphasize the importance of the modeling aspect, present several models of the problem and demonstrate differences in iteration numbers and solution time.

(November 30) Professor George Papanicolaou: Optimal Illumination in Array Imaging - I will discuss the mathematical problem of optimally illuminating an object for imaging by an array. In a certain regime of parameters and for a special class of objects this can be done by using spheroidal wave functions. In general situations one must use algorithms that optimally image the object by detecting its edges. Such algorithms are very different from the ones that maximize the energy of the signals received by the array so as to enhance detectability. I will analyze and compare the two types of algorithms. (Joint work with L. Borcea and C. Tsogka.)

(December 01) Professor George Papanicolaou: Imaging in Random Media - 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.

(December 05) Dr. Christian Kreuser: Convergence of Adaptive Finite Element Methods for $p$-Laplace Equation - We consider the homogeneous Dirichlet Problem for the $p$-Laplacian, $p\in(1,\infty)$. We propose an adaptive algorithm with continuous piecewise affine finite elements and prove an error reduction rate of approximate solutions to the exact one. We improve the a posteriori estimations for quasi-norms and generalize the error reduction property of the linear case to an energy reduction property in the nonlinear case. For adaptive refinement we use a marking strategy incorporating only error estimators. Thus we obtain a strict reduction of energy differences plus oscillation. Since energy differences are proportional to the error measured in quasi-norms we get linear convergence.

(December 12) Dr. Jing Zou: The Super Fast Spectral Method for Multiscale PDEs - We develop a new sublinear super fast spectral method. It takes time $O(log N)$, as compared with the O(N log N) cost of the traditional spectral method, where N is the number of gird points. We apply this new method to solve the following 1-dimensional multiscale problem: u_t - (a(x, x/eps) u_x)_x =0, u(0,x)=f(x), where the coefficient a is bounded, periodic, continuous and uniformly positive, a and f are each well represented by sparse Fourier representations. Both theoretical analysis and numerical results show the encouraging advantage of the new method in speed over the traditional spectral methods, when the number of grid points N is very large. Finally, we also apply this method to solve elliptic and hyperbolic multiscale PDEs.