The work within this area aims at setting up mathematical foundations
of using Markov chains for numerical analysis, a topic which is
relevant within the **project field**
Applied
Mathematical Finance, in particular for Monte
Carlo methods in finance.
However, this is also of general interest, thus fits
Numerical Methods.

Suppose that a Markov chain with invariant distribution is given by some transition kernel
*K*. The error of a sample from this Markov chain
used for integrating some square integrable function *f* is then
determined by

- Given a Markov chain
*K*, information on the rate of convergence is valuable only if*C*(*f*,*K*) is uniformly bounded over a possible class*F*of (square integrable) input functions. Therefore we aim at studying - Dependence of
*C*(*f*,*K*) on the kernel*K*is also crucial. While for independent sampling*C*(*f*,*K*)=*C*(*f*), long range dependencies have a possible huge impact on the actual error bound.- For
*uniformly ergodic Markov chains*the influence can be controlled successfully and leads to the notion of*auto-correlation time*. Here we refer to a recent paper. - A specific study concerning the efficiency of
Metropolis Markov chains was carried out recently, we refer to Simple Monte Carlo and the Metropolis algorithm. - In general, in particular for non-compact state spaces, the
designed Markov chain will not be uniformly ergodic. Therefore our
study focuses on appropriate error estimates for this more general
case.
For results in this direction we refer to Numerical integration using
*V*-uniformly ergodic Markov chains.

- For

This mathematical problem is accompanied with a numerical one. To this end we have to specify the class of admissible numerical methods. To this end it is assumed to be based on some design, say , which describes the way we obtain noisy observations. The resulting approximation based on such design may be obtained by any (measurable) mapping , hence the approximation is

It is the aim in this project to study efficiency issues for recovering
the unknown element *x* from indirect and noisy discrete observations,
as described above.

The studies within this sub-project are primarily carried out
within the **project field** Statistical
Data Analysis, specifically under the topic Numerics
of statistical ill-posed problems.

- Discretization issues;
- Optimality;
- Complexity.

A first discussion of these problems was given in the joint paper Optimal discretization of inverse problems in Hilbert scales. Regularization and self-regularization of projection methods. with Sergei V. Pereverzev.

Since then the collaboration continued, resulting a a series of papers in the same spirit, see

- Direct estimation of linear functionals from indirect noisy observations.
- Stable summation of noisy orthogonal series.

- Modulus of continuity of operator valued functions.
- Geometry of ill-posed problems in variable Hilbert scales
- Discretization strategy for ill-posed problems in variable Hilbert scales

- The discretized discrepancy principle under general source conditions,
- The Lepskii principle revisited

- Regularization of some linear ill-posed problems with discretized random noisy data,
- Regularization by projection in variable Hilbert scales.