We present a fast algorithm that constructs a data-sparse approximation of matrices arising in the context of integral equation methods for elliptic partial differential equations.The new algorithm uses Green's representation formula in combination with quadrature to obtain a first approximation of the kernel function, and then applies nested cross approximation to obtain a more efficient representation.The resulting H 2 -matrix representation requires O(nk) units of storage for an n × n matrix, where k depends on the prescribed accuracy.MSC: 65N38, 65N80, 65D30, 45B05.

Magnetostatic field calculations in micromagnetic simulations can be numerically expensive, particularly in the case of large-scale finite element simulations. The established finite element / boundary element method (FEM/BEM) by Fredkin & Koehler [IEEE Trans. Mag. 26, 1518(1990] involves a densely populated matrix with unacceptable numerical costs for problems involving a large number of degrees of freedom N . By using hierarchical matrices of H 2 type, we show that the memory requirements of this FEM/BEM method can be reduced dramatically, effectively converting the quadratic complexity O(N 2 ) of the problem to a linear one O(N ). We obtain matrix size reductions of nearly 99% in test cases with more than 10 6 degrees of freedom, and we test the computed magnetostatic energy values by means of comparison with analytic values. The efficiency of the H 2 -matrix compression opens the way to large-scale magnetostatic field calculations in micromagnetic modeling, all while preserving the accuracy of the established FEM/BEM formalism.

A new method to construct task graphs for Hmatrix arithmetic is introduced, which uses the information associated with all tasks of the standard recursive H-matrix algorithms, e.g., the block index set of the matrix blocks involved in the computation. Task re nement, i.e., the replacement of tasks by sub-computations, is then used to proceed in the H-matrix hierarchy until the matrix blocks containing the actual matrix data are reached. is process is a natural extension of the classical, recursive way in which H-matrix arithmetic is de ned and thereby simpli es the e cient usage of many-core systems. Examples for standard and accumulator based H-arithmetic are shown for model problems with di erent block structures.

We address the parallelization of the LU factorization of hierarchical matrices (H-matrices) arising from boundary element methods. Our approach exploits task-parallelism via the OmpSs programming model and runtime, which discovers the data-flow parallelism intrinsic to the operation at execution time, via the analysis of data dependencies based on the memory addresses of the tasks' operands. This is especially challenging for H-matrices, as the structures containing the data vary in dimension during the execution. We tackle this issue by decoupling the data structure from that used to detect dependencies. Furthermore, we leverage the support for weak operands and early release of dependencies, recently introduced in OmpSs-2, to accelerate the execution of parallel codes with nested task-parallelism and fine-grain tasks.

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