More usually inverse problems involving PDEs are dealing with inappropriate or noisy data in trying to find out more about an entity which undoubtedly exists. An attempt is made here to categorise the various routes to identifying robust industrial design procedures, given that it may prove impossible to achieve the required design or anything near it.

We specifically examine a type of PDE-constrained optimization problem, the parameter estimation problem. We present three model parameter estimation problems, each containing a different type of partial differential equation as the constraint.

We also shortly describe discretization and solution techniques for each problem. We further supply a simple to modify matlab code with the talk.

In this talk I will give a brief overview of the design, theory and application of derivative based optimization algorithms for the solution of PDE constrained optimization problems. In particular, I will discuss the interaction between the optimization problem, discretization of the PDEs, and optimization algorithms. I will highlight some recent mathematical developments and research needs.

For such problems, an adaptive method based on wavelets is proposed. It builds on a recent paper by Schwab and Stevenson where a single linear parabolic evolution problem is formulated in a weak space-time form and where an adaptive wavelet method is designed for which optimal convergence rates can be shown.

We present a non-monotone filter method that offers greater freedom at accepting points. The methods are implemented in FASTr, our filter active-set trust-region framework. We present recent numerical experience comparing various globalization strategies, as well as different step-computation techniques.

The Gauss-Newton (GN) method is a well known iterative technique for solving nonlinear least squares problems subject to dynamical system constraints. Such problems arise commonly in optimal state estimation where the systems may be stochastic. Variational data assimilation techniques for state estimation in very large environmental systems currently use approximate GN methods. The GN method solves a sequence of linear least squares problems subject to linearized system constraints. For very large systems, low resolution linear approximations to the model dynamics are used to improve the efficiency of the algorithm. We propose a new approach for deriving low order system approximations based on model reduction techniques from control theory. We show how this technique can be combined with the GN method to retain the response of the dynamical system more accurately and improve the performance of the approximate GN method.

SFEMs and mixed SFEMs have desirable convergence properties but have suffered from bad press due to a lack of robust solvers and preconditioners for the large coupled stochastic linear systems of equations that arise.

In this talk, we report on fast, robust linear algebra techniques and preconditioning schemes based on geometric and algebraic multigrid methods, for solving the stochastic Galerkin formulation of the Darcy flow problem (the steady-state diffusion problem in mixed form) which leads to massive systems of equations of saddle-point type. We consider stochastic formulations which necessitate a large number of deterministic solves, and more challenging models where one large saddle-point system has to be solved, in which the spatial and stochastic degrees of freedom are coupled.

1. The influence of the spectral properties of the blocks of A and P on the spectrum of P^-1A and the role of the proximity of P to A;
2. The practical advantages and shortcomings of replacing the "ideal" P with a computationally more feasible preconditioning matrix P_I;
3. The actual usefulness of a spectral analysis in the prediction of the qualitative convergence behavior of iterative solvers based on Krylov subspaces.

We give an introduction to some basic concepts in numerical bifurcation theory and give examples of how these techniques have had spectacular success in explaining complicated flow behaviour in the Taylor problem in fluid dynamics. Next, we look in detail at the efficient numerical implementation of these bifurcation techniques for large sparse discretised PDEs. Finally, we consider some new developments in the preconditioning of linear systems arising from large sparse eigenvalue problems.

Derivation of the RCWA method must rigorously address two subtle points, in particular truncation of the Fourier expansions and physical interpretation of the solution of the discretized problem. The latter must take into account that in passive materials the signal in the direction of its propagation can only be damped, but not amplified. We briefly present the main points in derivation of the RCWA method and then examine a possible potential of matrix computations in building efficient RCWA-based solvers for practical problems.

This work performed under the auspices of the U.S. Department of Energy by Lawrence Livermore National Laboratory under Contract DE-AC52-07NA27344.

It seems to be an accepted fact that in order to properly design numerical methods for systems of partial differential equations one has to rely on the properties of the underlying differential systems themselves. In this series of talks we will argue that the same is true for the construction of preconditioners, i.e., we will argue that the structure of the preconditioners for the discrete systems are in some sense dictated by the properties of the corresponding continuous system.

In our first talk we will mainly discuss the concept of preconditioners for the continuous systems, and how these operators should be constructed. We will consider various examples. In particular, we will discuss parameter dependent problems, and how we can construct preconditioners that give rise to condition numbers which are independent of these parameters. As a consequence, we obtain iterative methods which converge with a rate which is bounded uniformly with respect to the parameters.

The second talk will be devoted to preconditioners for discrete problems obtained by finite element discretizations of systems of partial differential equations. Examples we will discuss include the stationary and time dependent Stokes problem, the Darcy--Stokes problem, and the Reissner--Mindlin plate model. We will also discuss examples of inverse problems and nonlinear problems. Furthermore, a disscussion of preconditioning in $H(div)$ and $H(curl)$ will also be given.

In the third talk will give a review of finite element exterior calculus and its connections to the Hellinger--Reissner mixed formulations of linear elasticity. We will discuss preconditioners for the general Hodge--Laplace problem. Furthermore, we will present various mixed finite elements for linear elasticity based on weak and strong symmetry requirements, and discuss how to construct preconditioners for these systems.

One way is to use an outer iteration, typically a block-preconditioned Richardson method, applied to the (discretized) mixed problem. Multigrid techniques (as an inner iteration or approximation) can be used for setting up the blocks of the block-preconditioner.

The other way is to use multigrid methods directly applied to the (discretized) mixed problem as an outer iteration based on appropriate smoothers (as a sort of inner iteration). This approach is also known as one-shot multigrid strategy. One of the most important ingredients of such a multigrid method is an appropriate smoother.

We consider typical examples for these two approaches applied to an elliptic optimal control problem and discuss the convergence analysis of these methods.