Abstract
Besides Tikhonov regularization, which is probably the most wellknown regularization method for linear as well as nonlinear inverse problems, iterative regularization algorithms have more recently been investigated and applied successfully to the solution of, especially nonlinear and largescale problems. The design and analysis of problem adapted iterative regularization methods is one of the major tasks of this project. Additionally, the efficient implementation, coupling with discretization techniques (forward solvers) and the design of preconditioners are fields of ongoing research.
Analysis of regularization methods
Many iterative methods have been developed and analysed for wellposed problems. Their application to inverse, especially illposed problems, is not straight forward and requires an analysis in the framework of regularization methods. There,

Design and analysis of (new) iterative regularization
algorithms:
in particular Newtontype regularization methods including parameter choice strategies and discretization matters. Recently, a NewtonKaczmarz iteration has been proposed for the regularization of largescale problems, especially time dependent problems and parameter estimation from boundary data, where the data is (part of) the DirichlettoNeumann map.
 M. Burger, B. Kaltenbacher, Regularizing NewtonKaczmarz Methods for Nonlinear IllPosed Problems, SFBReport 200417. psfile pdffile
 H. Egger, Accelerated NewtonLandweber Iterations for Regularizing Nonlinear Inverse Problems, SFBReport 20053. psfile pdffile
 H. W. Engl, A. Leitão, A Mann iterative regularization method for elliptic Cauchy Problems, Numer. Funct. Anal. and Optimiz., 22(78):861884, 2001. Preprint: psfile
 B. Kaltenbacher, A. Neubauer, A. G. Ramm, Convergence rates of the continuous regularized GaussNewton method, Journal of Inverse and Illposed Problems, 10:261280, 2002. Preprint: psfile

SQPtype methods:
Such methods are closely related to Augmented Lagrangian or Allatonce methods in optimal control, shape and topology optimization (see also Subproject F1309)). SQPtype methods have been investigated as regularization methods for inverse problems governed by PDE's.
The main advantage of the SQP approach is that the sparse structure coming from finite element discretizations is preserved in the formulation of the inverse problem. The fast solution of indefinite linear systems naturally appearing in the SQP framework, in particular the design of effective preconditioners, will be the next step.
 M. Burger, W. Mühlhuber, Iterative regularization of parameter identification problems by SQPmethods, Inverse Problems, 18:943970, 2002. Preprint: psfile
 M. Burger, W. Mühlhuber, Numerical Approximation of an SQPtype Method for Parameter Identification, SIAM J. Numer. Anal., 40(5):17751797, 2002. Preprint: psfile
Acceleration of iterative regularization methods
One of the major drawbacks of iterative regularization
methods for illposed problems is that  due to the
illposedness, which results in illconditioned finite
dimensional approximations  usually a high number of
iterations is needed in order to reconstruct (order optimal)
approximations of a solution. This behavior also appears for
Newtontype iterations, if the linearized systems are again
solved by iterative regularization methods. In order to
reduce the overall numerical effort of the solution process,
several strategies are pursued:

Preconditioning iterative regularization methods for
linear and nonlinear inverse problems:
The design of efficient preconditioners has recently become one of the main topics in numerical mathematics, especially in the FEM community. While the theory of preconditioning wellposed problems arising in PDE's is well developed, preconditioning of inverse, in particular illposed, problems is not so well understood. Taking into account the illposed nature of the problems under consideration, the number of iterations needed to achieve optimal convergence rates for the solution of linear and nonlinear inverse problems by iterative regularization methods can essentially be reduced to the square root by appropriate preconditioning (in Hilbert scales).
Reconstruction of an unknown source term.
Left: standard iterations, right: preconditioned.
 H. Egger, A. Neubauer, Preconditioning Landweber Iteration in Hilbert Scales, SFBReport 200425. psfile pdffile
 H. Egger, Semiiterative Regularization in Hilbert Scales, SFBReport 200426. psfile pdffile

Efficient implementation and/or preconditioning the
solution of linearized systems arising in Newtontype
methods:
A direct application of Newton's method to the solution of inverse problems is not possible, since the illposedness of the nonlinear problem usually implies the illposedness of the linearized systems, which have to be solved in every Newton step. In order to ensure stability, the linearized equations have to be solved by regularization methods instead. For largescale problems, iterative regularization methods turn out to be appropriate. In a first step, acceleration of the NewtonLandweber method by by using faster semiiterative regularization methods for the stable solution of the linearized Newton equations has been investigated. In a second step, the effect of preconditioning will be considered.
 H. Egger Accelerated NewtonLandweber iterations for the solution of nonlinear inverse problems, SFBReport 20053. psfile pdffile
 Fast iterative solution and preconditioning of
indefinite systems arising in SQP methods:
Saddlepoint problems naturally appear in many applications, e.g., in (Navier)Stokes equations, or in optimal control problems. Thus fast solution and especially, preconditioning of saddlepoint problems has attracted significant interest in the last years. The saddlepoint problems stemming from an SQP formulation of parameter identification problems governed by PDEs show an additional illposedness, i.e., the part of the system, which is usually considered to be uniformly elliptic, now depends on a regularization parameter, which may become arbitrarily small.
Problem adapted regularization strategies and theory
The general theory for the regularization of inverse problems is formulated for very general problems. For special classes of problems, the results can be improved significantly. A problem adapted theory of regularization methods, and even the design of problem adapted regularization algorithms is thus an important task. Topics of ongoing research are, e.g.,

Derivation of a problem adapted convergence theory for
selected applications and problem classes.
 H. Egger, H. W. Engl, M. V. Klibanov, Global Uniqueness and Hölder Stability for Recovering a Nonlinear Source Term in a Parabolic Equation, Inverse Problems, 21:271290, 2005. Preprint: psfile pdffile
 H. Egger, H. W. Engl, Tikhonov Regularization Applied to the Inverse Problem of Option Pricing: Convergence Analysis and Rates Inverse Problems, 21, 2005, to appear. Preprint: psfile pdffile
 M. Burger: Iterative Regularization of a Parameter Identification Problem occurring in PolymerCrystallization, SIAM J. Numer. Anal., 39:10291055, 2001. Preprint: psfile
 H. W. Engl, P. Kügler, The Influence of the Equation Type on Iterative Parameter Identification Problems which are Elliptic or Hyperbolic in the Parameter, Europ. J. Appl. Math., 14:129163, 2003. Preprint psfile
 H. W. Engl, Identification of Parameters in Polymer Crystallization, Semiconductor Models and Elasticity via Iterative Regularization Methods, SFBReport 200202. psfile pdffile
 SQPtype methods: As already mentioned, SQPmethods are
specially well suited for the application to inverse
problems governed by PDE's. Their application to inverse
problems for differential inequalities is one of the
research topics this project.

The design and analysis of new methods, e.g.
derivativefree methods, which can be formulated without
a derivative of the forward operator or onlinealgorithms
for time dependent inverse problems.
 P. Kügler, A Derivative Free Landweber Method for Parameter Identification in Elliptic PDEs, Inverse Problems, 19:14071426, 2003. Preprint: psfile pdffile
 P. Kügler, A Derivative Free Landweber Method for Parameter Identification in Elliptic Partial Differential Equations with Application to the Manufacture of Car Windshields, PhD Thesis, Johannes Kepler University, 2003.
 P. Kügler, An approach to online parameter estimation in nonlinear dynamical systems, SFBReport 200418. psfile pdffile
 Level set methods for inverse problems.
Efficient discretization
Sophisticated discretization strategies become an important factor when it comes to the implementation of regularization algorithms. Especially for large scale problems, the efficient coupling of discretization and iteration process can significantly reduce the overall numerical effort. Topics of recent research are

Multilevel and multigrid techniques for regularization
methods.
 M. Burger, W. Mühlhuber, Numerical Approximation of an SQPtype Method for Parameter Identification, SIAM J. Numer. Anal., 40(5):17751797, 2002. Preprint: psfile
 B. Kaltenbacher, On the regularizing properties of a full multigrid method for illposed problems, Inverse Problems, 17:767788, 2001. Preprint: psfile
 Kaltenbacher, B.: A Multigrid Method with A Priori and A Posteriori Level Choice for the Regularization of Nonlinear IllPosed Problems. March 2000. Eds.: Heinz W. Engl, Ulrich Langer psfile

Regularization by (adaptive) discretization.
 H. Ben Ameur, B. Kaltenbacher, Regularization of parameter estimation by adaptive discretization using refinement and coarsening indicators, Journal of Inverse and IllPosed Problems, 10:561584, 2003. Preprint: psfile
 B. Kaltenbacher, Regularization by Projection with A Posteriori Discretization Level Choice for Linear and Nonlinear IllPosed Problems. Inverse Problems, 16:15231539, 2000. Preprint: psfile
 B. Kaltenbacher, A ProjectionRegularized Newton Method for Nonlinear IllPosed Problems and its Application to Parameter Identification Problems with Finite Element Discretization. November 1998. Eds.: H.W. Engl, U. Langer psfile
 B. Kaltenbacher, Regularization by Projection with A Posteriori Discretization Level Choice for Linear and Nonlinear IllPosed Problems, Inverse Problems, 16:15231539, 2000. Preprint: psfile
 B. Kaltenbacher and J. Schöberl, A saddle point variational formulation for projectionregularized parameter identification, Numerische Mathematik, 91:675697, 2002. Preprint: psfile
 B. Kaltenbacher, Vcycle convergence of some multigrid methods for illposed problems, Mathematics of Computation, 72:17111730, 2003. Preprint: psfile
 Kaltenbacher, B.: On a Multigrid Method for IllPosed Problems, SFBReport 199920. psfile
SpezialForschungsBereich SFB F013  Special Research Program of the FWF  Austrian Science Fund