Theory of Numerical
Monte Carlo and
Theory of Numerical Analysis
theory (IBC) aims at finding the minimally necessary effort which is
for the numerical solution of problems of analysis. On a theoretical
lower bounds are established. Upper bounds are derived by either
the efficiency and optimality of known algorithms or by developing new
algorithms which are optimal in this setting. In a number of situations
this approach led to completely new types of algorithms. Within this
we study basic numerical problems like integration, approximation,
integration, solution of integral and differential equations.
and Quasi-Monte Carlo Algorithms
The main feature of
(or Monte Carlo) algorithms is the use of randomness in the calculation
process. Many highly complex applications, where traditional methods
can only be treated by stochastic techniques. We investigate the
of such algorithms both in theory and in numerical experiments. On the
basis of IBC results, new Monte Carlo algorithms for integration and
equations are developed and tested for practical applications.
A related field of
is that of quasi-Monte Carlo methods. Here random numbers are replaced
by low discrepancy sequences, leading to deterministic algorithms.
is a crucial quantity describing the efficiency (uniform distribution)
of such sequences. We study various types of discrepancy from the
point of view. We are also concerned with applications of
A new challenging
task is the
investigation of the potential power of quantum computers. Worldwide
efforts are concentrated on discrete and algebraic problems. Our
on quantum computing is concerned with the possible speed-up a quantum
computer could bring for numerical problems of analysis. For this
purpose we recently developed an IBC approach to quantum computation.
this frame we seek to obtain matching lower and upper quantum
bounds, which also includes the construction of optimal quantum
First results concern numerical integration, which show that in the
setting considerable speed-ups are possible as compared to the
deterministic and randomized setting.