Advancements in Numerical Methods for Complex Problems

The field of numerical methods is rapidly advancing, with a focus on developing efficient and accurate techniques for solving complex problems. Recent developments have centered around improving the performance of existing methods, such as model order reduction and Krylov subspace techniques, as well as exploring new approaches like layer potential methods and hybridizable discontinuous Galerkin methods. These advancements have the potential to significantly impact various areas of research, including seismic wave propagation, electronic structure calculations, and time-harmonic electromagnetic problems. Notably, the development of fast and reliable algorithms for solving linear response calculations and the real Monge-Ampere equation has shown promising results. Furthermore, the introduction of enrichment functions in immersed finite element methods has improved the accuracy and stability of interface problems. The development of multigrid methods for ghost finite element approximations has also shown great potential for solving elliptic problems on arbitrary domains. Overall, these advancements demonstrate the ongoing efforts to improve the efficiency and accuracy of numerical methods for complex problems. Noteworthy papers include: The paper on MOR-T L, which presents a novel model order reduction method for parametrized problems with application to seismic wave propagation, demonstrating significant efficiency gains and enhanced performance. The paper on Efficient Krylov methods for linear response in plane-wave electronic structure calculations, which proposes a novel algorithm based on inexact GMRES methods, achieving superlinear convergence and reducing computational time by about 40%. The paper on Fast Bellman algorithm for real Monge-Ampere equation, which introduces a new numerical algorithm that runs considerably faster than existing methods, improving running time by a factor of 3-10 for smooth examples and 20-100 or more for mildly degenerate examples.

Sources

MOR-T L : A Novel Model Order Reduction Method for Parametrized Problems with Application to Seismic Wave Propagation

Comparison of FMM and $\mathcal{H}$-matrix based 3D-ACA for a time domain boundary element method

A generalization of the Gauss-Seidel iteration method for generalized absolute value equations

Efficient Krylov methods for linear response in plane-wave electronic structure calculations

Layer Potential Methods for Doubly-Periodic Harmonic Functions

An Enriched Immersed Finite Element Method for 3D Interface Problems

A hybridizable discontinuous Galerkin method with transmission variables for time-harmonic electromagnetic problems

Fast Bellman algorithm for real Monge-Amp\`ere equation

Multigrid methods for the ghost finite element approximation of elliptic problems

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