Numerical methods for non-local operators

Non-local operators appear in the context of integral equations, as solution operators of elliptic partial differential equations, or as solutions of certain matrix equations appearing in control theory.

Hierarchical matrices offer an elegant tool for handling these operators. The matrix is split into submatrices of low numerical rank that can be approximated efficiently in factorized form.

For integral operators, the factorized approximations can be constructed, e.g., by interpolation or multipole expansion.

For general matrices, optimal low-rank approximations can be obtained by algebraic techniques using singular value decompositions and orthogonal transformations.

In this talk, I will give a brief introduction into the basic concepts of hierarchical matrices and present recent results in the field of matrix arithmetic operations and high-frequency problems.