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The sparse approximation of high-frequency Helmholtz-type integral operatorshas many important physical applications such as problems in wave propagationand wave scattering. The discrete system matrices are huge and denselypopulated; hence their sparse approximation is of outstanding importance. Inour paper we will generalize the directional $\mathcal{H}^{2}$-matrixtechniques from the \textquotedblleft pure\textquotedblright\ Helmholtzoperator $\mathcal{L}u=-\Delta u+\zeta^{2}u$ with $\zeta=-\operatorname*{i}k$,$k\in\mathbb{R}$, to general complex frequencies $\zeta\in\mathbb{C}$ with$\operatorname{Re}\zeta>0$. In this case, the fundamental solution decreasesexponentially for large arguments. We will develop a new admissibilitycondition which contains $\operatorname{Re}\zeta$ in an explicit way andintroduce the approximation of the integral kernel function on admissibleblocks in terms of frequency-dependent \textit{directional expansionfunctions}. We develop an error analysis which is explicit with respect to theexpansion order and with respect to $\operatorname{Re}\zeta$ and$\operatorname{Im}\zeta$. This allows to choose the \textit{variable }expansionorder in a quasi-optimal way depending on $\operatorname{Re}\zeta$ butindependent of, possibly large, $\operatorname{Im}\zeta$. The complexityanalysis is explicit with respect to $\operatorname{Re}\zeta$ and$\operatorname{Im}\zeta$ and shows how higher values of $\operatorname{Re}%\zeta$ reduce the complexity. In certain cases, it even turns out that thediscrete matrix can be replaced by its nearfield part.  Numerical experiments illustrate the sharpness of the derived estimates andthe efficiency of our sparse approximation.

Variable Order, Directional H2-Matrices for Helmholtz Problems with Complex Frequency