Advances in Numerical Methods and Algorithms

The field of numerical methods and algorithms is witnessing significant developments, with a focus on improving the efficiency and accuracy of various numerical techniques. Researchers are exploring new approaches to solve complex problems, such as elliptic inverse problems, eigenvalue computations, and transport equations. Notably, the integration of physics-based priors with data-driven learning is leading to more robust and accurate methods. Additionally, the development of novel iterative methods, preconditioning techniques, and adaptive mesh refinement strategies is enhancing the performance of existing algorithms. These advancements have the potential to impact various fields, including materials science, biology, and engineering. Noteworthy papers include: The paper on FlowTIE, which introduces a neural-network-based framework for phase reconstruction from 4D-STEM data, demonstrating improved accuracy and robustness. The paper on A Stable Iterative Direct Sampling Method, which develops a novel method for solving elliptic inverse problems with partial Cauchy data, showing remarkable robustness and flexibility.

Sources

Momentum accelerated power iterations and the restarted Lanczos method

FlowTIE: Flow-based Transport of Intensity Equation for Phase Gradient Estimation from 4D-STEM Data

A Stable Iterative Direct Sampling Method for Elliptic Inverse Problems with Partial Cauchy Data

Pointwise A Posteriori Error Estimators for Multiple and Clustered Eigenvalue Computations

Fast integral methods for the Neumann Green's function: applications to capture and signaling problems in two dimensions

A Particle method for stationary transport equations

Dual Weighted Residual-driven adaptive mesh refinement to enhance biomechanical simulations

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