New types of adaptivity for the cross approximation of non-local operators

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In many applications from physics, engineering and economy, non-local operators are used to model the respective phenomenon. Such operators are characterised by the property that each output datum depends on each input datum when the operator is applied to a data set. Examples are integral operators arising from the boundary integral method, the Gauss transform, the Lippmann-Schwinger equation in quantum perturbation theory, non-integer powers and the inverse of differential operators. Also integral operators for modelling Levy processes in risk management belong to this class. Their discretization leads to fully populated matrices which due to the underlying geometry or the desired accuracy of the solution are large scale in general. Already storing such matrices can be a problem. However, the numerical solution of linear systems in which they appear as a coefficient matrix can currently not be done in acceptable time.In this project a new approach for the efficient numerical treatment of non-local operators will be developed and investigated. Both, the fast multipole method and hierarchical matrices can be employed to treat large scale discretizations of such operators with logarithmic-linear complexity. Depending on the respective method, the operator is approximated locally or blockwisewith the prescribed accuracy. The resulting approximation is universally applicable to any right hand side of linear systems in which it appears as a coefficient matrix. If many systems with the same operator are to be solved, then this kind of approximation is particularly efficient. However, often (probably in most cases) only a single system is to be solved for one operator, because it may, for instance, change in the course of a simulation. In such a situation, the universality of the approximation cannot be taken advantage of. On the contrary, the universality is paid for by generating and storing dispensable information. Since there are currently few alternatives, this kind of approximation is still used in practise. The aim of this project is to improve this situation by developing a new technique which tailors the approximation to the right hand side. Both, fast multipole methods and hierarchical matrices will be able to benefit from this new approach. Hence, succesful and widely recognised methods will be extended to significant problems to which they have not been efficiently applicable yet.