Sparse Approximate Multifrontal Factorization with Butterfly Compression for High Frequency Wave Equations

We present a fast and approximate multifrontal solver for large-scale sparse linear systems arising from finite-difference, finite-volume or finite-element discretization of high-frequency wave equations. The proposed solver leverages the butterfly algorithm and its hierarchical matrix extension for compressing and factorizing large frontal matrices via graph-distance guided entry evaluation or randomized matrix-vector multiplication-based schemes. Complexity analysis and numerical experiments demonstrate $\mathcal{O}(N\log^2 N)$ computation and $\mathcal{O}(N)$ memory complexity when applied to an $N\times N$ sparse system arising from 3D high-frequency Helmholtz and Maxwell problems

A Parallel Hierarchical Blocked Adaptive Cross Approximation Algorithm

This paper presents a hierarchical low-rank decomposition algorithm assuming any matrix element can be computed in $O(1)$ time. The proposed algorithm computes rank-revealing decompositions of sub-matrices with a blocked adaptive cross approximation (BACA) algorithm, followed by a hierarchical merge operation via truncated singular value decompositions (H-BACA). The proposed algorithm significantly improves the convergence of the baseline ACA algorithm and achieves reduced computational complexity compared to the full decompositions such as rank-revealing QR decompositions. Numerical results demonstrate the efficiency, accuracy and parallel efficiency of the proposed algorithm

Open Problems in (Hyper)Graph Decomposition

Large networks are useful in a wide range of applications. Sometimes problem instances are composed of billions of entities. Decomposing and analyzing these structures helps us gain new insights about our surroundings. Even if the final application concerns a different problem (such as traversal, finding paths, trees, and flows), decomposing large graphs is often an important subproblem for complexity reduction or parallelization. This report is a summary of discussions that happened at Dagstuhl seminar 23331 on "Recent Trends in Graph Decomposition" and presents currently open problems and future directions in the area of (hyper)graph decomposition

A Supernodal Approach to Incomplete LU Factorization with Partial Pivoting

We present a new supernode-based incomplete LU factorization method to construct a preconditioner for solving sparse linear systems with iterative methods. The new algorithm is primarily based on the ILUTP approach by Saad, and we incorporate a number of techniques to improve the robustness and performance of the traditional ILUTP method. These include the new dropping strategies that accommodate the use of supernodal structures in the factored matrix. We present numerical experiments to demonstrate that our new method is competitive with the other ILU approaches and is well suited for today's high performance architectures

Detecting resonance of radio-frequency cavities using fast direct integral equation solvers and augmented Bayesian optimization

This paper presents a computationally efficient framework for identifying resonance modes of 3D radio-frequency (RF) cavities with damping waveguide ports. The proposed framework relies on surface integral equation (IE) formulations to convert the task of resonance detection to the task of finding resonance frequencies at which the lowest few eigenvalues of the system matrix is close to zero. For the linear eigenvalue problem \rev{with a fixed frequency}, we propose leveraging fast direct solvers to efficiently invert the system matrix; for the frequency search problem, we develop a hybrid optimization algorithm that combines Bayesian optimization with down-hill simplex optimization. The proposed IE-based resonance detection framework (IERD) has been applied to detection of high-order resonance modes (HOMs) of realistic accelerator RF cavities to demonstrate its efficiency and accuracy

Open Problems in (Hyper)Graph Decomposition

Large networks are useful in a wide range of applications. Sometimes problem instances are composed of billions of entities. Decomposing and analyzing these structures helps us gain new insights about our surroundings. Even if the final application concerns a different problem (such as traversal, finding paths, trees, and flows), decomposing large graphs is often an important subproblem for complexity reduction or parallelization. This report is a summary of discussions that happened at Dagstuhl seminar 23331 on "Recent Trends in Graph Decomposition" and presents currently open problems and future directions in the area of (hyper)graph decomposition