$\mathcal{H}$-Matrix approximation for the operator exponential with applications

Summary.

We develop a data-sparse and accurate approximation to parabolic solution operators in the case of a rather general elliptic part given by a strongly P-positive operator [4].

In the preceding papers [12]–[17], a class of matrices ($\mathcal{H}$-matrices) has been analysed which are data-sparse and allow an approximate matrix arithmetic with almost linear complexity. In particular, the matrix-vector/matrix-matrix product with such matrices as well as the computation of the inverse have linear-logarithmic cost. In the present paper, we apply the $\mathcal{H}$-matrix techniques to approximate the exponent of an elliptic operator.

Starting with the Dunford-Cauchy representation for the operator exponent, we then discretise the integral by the exponentially convergent quadrature rule involving a short sum of resolvents. The latter are approximated by the $\mathcal{H}$-matrices. Our algorithm inherits a two-level parallelism with respect to both the computation of resolvents and the treatment of different time values. In the case of smooth data (coefficients, boundaries), we prove the linear-logarithmic complexity of the method.