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AdaptiveCrossApproximation

This package provides implementations of adaptive cross approximation and several of its variants [1, 3, 6]. It supports different pivoting strategies and convergence criteria [2, 4, 5]. The package also contains the incomplete adaptive cross approximation for efficient pivot selection in the construction of $\mathcal{H}^2$-matrices [7].

The package also assembles $\mathcal{H}$-matrices for point kernels and BEAST operators and provides their arithmetic: sums, products, recompression, and an LU factorization for direct solves and preconditioning [8, 9]. For transpose-symmetric operators, single-space assembly shares off-diagonal near blocks and far-field factors. See Symmetric Assembly on One Space.

Installation

Installing AdaptiveCrossApproximation is done by entering the package manager (enter ] at the julia REPL) and issuing:

pkg> add https://github.com/JoshuaTetzner/AdaptiveCrossApproximation.jl.git

References

  • [1] Bebendorf, M., and S. Rjasanow. “Adaptive Low-Rank Approximation of Collocation Matrices.” Computing 70, no. 1 (2003): 1–24. https://doi.org/10.1007/s00607-002-1469-6.
  • [2] De Marchi, Stefano. “On Leja Sequences: Some Results and Applications.” Applied Mathematics and Computation 152, no. 3 (2004): 621–47. https://doi.org/10.1016/S0096-3003(03)00580-0.
  • [3] Zhao, K., M. N. Vouvakis, and J. F. Lee. “The Adaptive Cross Approximation Algorithm for Accelerated Method of Moments Computations of EMC Problems.” IEEE Transactions on Electromagnetic Compatibility 47, no. 4 (2005): 763–73. https://doi.org/10.1109/TEMC.2005.857898.
  • [4] Heldring, Alexander, Eduard Ubeda, and Juan M. Rius. “Improving the Accuracy of the Adaptive Cross Approximation with a Convergence Criterion Based on Random Sampling.” IEEE Transactions on Antennas and Propagation 69, no. 1 (2021): 347–55. https://doi.org/10.1109/TAP.2020.3010857.
  • [5] Bauer, M., M. Bebendorf, and B. Feist. “Kernel-Independent Adaptive Construction of $\mathcal{H}^2$-Matrix Approximations.” Numerische Mathematik 150, no. 1 (2022): 1–32. https://doi.org/10.1007/s00211-021-01255-y.
  • [6] Tetzner, Joshua M., and Simon B. Adrian. “On the Adaptive Cross Approximation for the Magnetic Field Integral Equation.” IEEE Transactions on Antennas and Propagation 72, no. 12 (2024): 9366–77. https://doi.org/10.1109/TAP.2024.3483296.
  • [7] Tetzner, Joshua M., and Simon B. Adrian. “The Incomplete Adaptive Cross Approximation for the Fast Construction of $\mathcal{H}^2$-Matrices and Its Application to the Electric Field Integral Equation for Electrically Small Problems.” IEEE Transactions on Antennas and Propagation 74, no. 8 (2026): 7835–47. https://doi.org/10.1109/TAP.2026.3687274.
  • [8] Hackbusch, W. “A Sparse Matrix Arithmetic Based on $\mathcal{H}$-Matrices. Part I: Introduction to $\mathcal{H}$-Matrices.” Computing 62, no. 2 (1999): 89–108. https://doi.org/10.1007/s006070050015.
  • [9] Grasedyck, L., and W. Hackbusch. “Construction and Arithmetics of $\mathcal{H}$-Matrices.” Computing 70, no. 4 (2003): 295–334. https://doi.org/10.1007/s00607-003-0019-1.
  • [10] Bebendorf, M. Hierarchical Matrices: A Means to Efficiently Solve Elliptic Boundary Value Problems. Lecture Notes in Computational Science and Engineering 63. Berlin: Springer, 2008. https://doi.org/10.1007/978-3-540-77147-0.
  • [11] Halko, N., P. G. Martinsson, and J. A. Tropp. “Finding Structure with Randomness: Probabilistic Algorithms for Constructing Approximate Matrix Decompositions.” SIAM Review 53, no. 2 (2011): 217–88. https://doi.org/10.1137/090771806.
  • [12] Kriemann, R. “$\mathcal{H}$-LU Factorization on Many-Core Systems.” Computing and Visualization in Science 16, no. 3 (2013): 105–17. https://doi.org/10.1007/s00791-014-0226-7.