Dynamical Monte Carlo Methods (MCMC)
The usage of Markov chains in numerical simulation may serve different
purposes. While on the one-hand side it may be desirable to see a
certain Markovian system evolving in time, it is on the other-hand
side just used as a tool for approximate numerical integration.
Depending on the purpose, different error criteria are relevant.
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
Mathematical Finance, in particular for Monte
Carlo methods in finance.
However, this is also of general interest, thus fits
is on using Markov chains for numerical integration, when
direct simulation is not feasible. An appropriate error criterion is
introduced as follows.
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
Typically we expect this to behave like .
For application it is important to control the dependencies C(f,K).
- 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
- 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
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
For results in this direction we refer to Numerical integration using V-uniformly ergodic
Complexity of Ill-posed Problems
In this sub-project we study ill-posed problems, where we wish to recover some element
x from some Hilbert space from indirectly observed data near
y=Ax, where A is some injective compact linear operator acting
from X to X. In practice indirect
observations cannot be observed exactly but only in discretized and
noisy form, such that we have only a vector
where denotes the inner product in X, is some orthonormal system, usually called design, and
is the noise, which is assumed to be normalized.
For deterministic noise this means . In the stochastic setting we assume Gaussian white noise for simplicity,
i.e., the family consists of independent standard normal variables.
The operator A determines the way the
observations are indirect.
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.
Focus is on
- Discretization issues;
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
Recently, focus is on equations with variable source conditions, which
leads to equations in variable Hilbert scales. Results in this
direction can be found in
Several papers deal with the a posteriori choice of the
Finally we mention the studies on discretization of ill-posed