Research topics
This is an indicative list of the current research topics in Nines. As is usual in research, this list is incomplete and continuously changing. For more information about any of these topics, please consult our list of publications or ask one of the group members.Multivariate and high-dimensional integrals
Quasi Monte-Carlo and lattice rules
Lattice rules scale very well to higher dimensions. They can exhibit a lot of structure, which enables fast algorithms for their construction. In spite of this structure well-designed lattice rules appear to use almost random function evaluations, hence the name Quasi Monte-Carlo. They typically converge much faster than traditional Monte-Carlo simulations.
Software for the approximation of multi-dimensional integrals
Nines has a history of producing robust, efficient (and popular) packages for general-purpose adaptive numerical integration. Following in the footsteps of Quadpack, current work focuses on Cubpack. See software.
Chebyshev approximations
Extensive results have appeared in the literature of numerical integration on Chebyshev point sets and subsets thereof that scale favourably with increasing dimension. These point sets are useful as well for function approximation purposes.
An encyclopaedia of cubature formulas
Ronald Cools has compiled an extensive list of cubature rules that have been described in literature: the Encyclopaedia of Cubature Formulas.
Highly oscillatory integrals
Complex-plane methods
Highly oscillatory integrals are usually considered computationally demanding. Yet, in a growing number of cases, the exact opposite is true: it is cheaper to evaluate highly oscillatory integrals then their non-oscillatory counterparts. Efficient evaluation methods are nowadays based on results in asymptotic analysis, both old and new. A family of numerical methods of current interest in Nines is based on the classical method of steepest descent in the complex plane.
Integral equations
Boundary element methods
Integral equations and boundary integral equations are often closely linked to numerical integration. With the unknown function appearing under the integral formula, the problem eventually leads to solving a system of equations which is usually dense. Its solution can be sped up using Fast Multipole or H-Matrix methods. Still, constructing the equations typically requires the evaluation of thousands or even millions of multivariate integrals. Many of these integrals are weakly singular or strongly singular.
High-frequency methods
Modern numerical methods for highly oscillatory integrals have led to new methods for oscillatory integral equations. As is the case in numerical integration, methods that are robust or even improving with increasing frequency are based on asymptotic analysis.
