Advances in Numerical Methods and Algorithms

The field of numerical methods and algorithms is experiencing significant developments, with a focus on improving efficiency, accuracy, and scalability. Researchers are exploring new techniques for solving complex problems, such as tensor equations, singular kernel convolutions, and eigenvalue problems. The use of iterative methods, truncated decompositions, and fast convolution solvers is becoming increasingly popular. Additionally, there is a growing interest in developing algorithms that can take advantage of emerging machine number formats, such as posit and takum arithmetic. Noteworthy papers include: Fast Singular-Kernel Convolution on General Non-Smooth Domains via Truncated Fourier Filtering, which presents a fast and high-order numerical methodology for evaluating convolutions with singular kernels. Breaking the Barrier of Self-Concordant Barriers: Faster Interior Point Methods for M-Matrices, which shows that an interior point method with adaptive step sizes can solve certain optimization problems in fewer iterations than previously thought possible. Mixed Precision Orthogonalization-Free Projection Methods for Eigenvalue and Singular Value Problems, which introduces an approach that eliminates orthogonalization requirements in traditional Rayleigh-Ritz projection methods, allowing for significant computational advantages.

Sources

Efficient iterative techniques for solving tensor problems with the T-product

Fast Singular-Kernel Convolution on General Non-Smooth Domains via Truncated Fourier Filtering

On Symmetric Lanczos Quadrature for Trace Estimation

Efficient approximations of matrix multiplication using truncated decompositions

Fast convolution solver based on far-field smooth approximation

Fast LDL factorization for dense and sparse symmetric matrices over an arbitrary field

Breaking the Barrier of Self-Concordant Barriers: Faster Interior Point Methods for M-Matrices

Numerical Performance of the Implicitly Restarted Arnoldi Method in OFP8, Bfloat16, Posit, and Takum Arithmetics

Spectral Methods via FFTs in Emerging Machine Number Formats: OFP8, Bfloat16, Posit, and Takum Arithmetics

Arbitrary precision computation of hydrodynamic stability eigenvalues

Mixed Precision Orthogonalization-Free Projection Methods for Eigenvalue and Singular Value Problems

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