Related papers: A new algebraic and arithmetic framework for inter…

Abstraction is essential for reducing the complexity of systems across diverse fields, yet designing effective abstraction methodology for probabilistic models is inherently challenging due to stochastic behaviors and uncertainties. Current…

Most numerical solvers and libraries nowadays are implemented to use mathematical models created with language-specific built-in data types (e.g. real in Fortran or double in C) and their respective elementary algebra implementations.…

We reexamine Smale's alpha theory as a way to certify a numerical solution to an analytic system. For a given point and a system, Smale's alpha theory determines whether Newton's method applied to this point shows the quadratic convergence…

An original approach to solving rather difficult probabilistic problems arising in studying the readout of random discrete fields and having no exact analytical solutions at the moment is proposed. Several algorithms for direct, iterative,…

This work presents a novel matrix-based method for constructing an approximation Hessian using only function evaluations. The method requires less computational power than interpolation-based methods and is easy to implement in matrix-based…

Using recent results from the theory of integer points close to smooth curves, we give an asymptotic formula for the distribution of values of a class of integer-valued prime-independent multiplicative functions.

This paper deals with exploiting symmetry for solving linear and integer programming problems. Basic properties of linear representations of finite groups can be used to reduce symmetric linear programming to solving linear programs of…

We give an explicit geometric way to build mixed-integer programming (MIP) formulations for unions of polyhedra. The construction is simply described in terms of spanning hyperplanes in an r-dimensional linear space. The resulting MIP…

Statistics and Optimization are foundational to modern Machine Learning. Here, we propose an alternative foundation based on Abstract Algebra, with mathematics that facilitates the analysis of learning. In this approach, the goal of the…

The ideas behind the concept of algebraic ("integration-by-parts") algorithms for multiloop calculations are reviewed. For any topology and mass pattern, a finite iterative algebraic procedure is proved to exist which transforms the…

Many applications in computational physics that use numerical integrators based on splitting and composition can benefit from the development of optimized algorithms and from choosing the best ordering of terms. The cost in programming and…

We propose a sparse algebra for samplet compressed kernel matrices, to enable efficient scattered data analysis. We show the compression of kernel matrices by means of samplets produces optimally sparse matrices in a certain S-format. It…

Inspired by computer assisted proofs in analysis, we present an interval approach to real-number computations.

We present a novel class of methods to compute functions of matrices or their action on vectors that are suitable for parallel programming. Solving appropriate simple linear systems of equations in parallel (or computing the inverse of…

There is a growing need for the ability to analyse interval-valued data. However, existing descriptive frameworks to achieve this ignore the process by which interval-valued data are typically constructed; namely by the aggregation of…

This article addresses a fundamental problem faced by the ab initio community: the lack of an effective formalism for the rapid exploration and exchange of new methods. To rectify this, we introduce a novel, basis-set independent,…

Dense and sparse tensors allow the representation of most bulk data structures in computational science applications. We show that sparse tensor algebra can also be used to express many of the transformations on these datasets, especially…

Integer programming is concerned with solving linear systems of equations over the non-negative integers. The basic question is to find a solution which minimizes a given linear objective function for a fixed right hand side. Here we also…

The multiplication of matrices is an important arithmetic operation in computational mathematics. In the context of hierarchical matrices, this operation can be realized by the multiplication of structured block-wise low-rank matrices,…

We discuss pattern languages for closed pattern mining and learning of interval data and distributional data. We first introduce pattern languages relying on pairs of intersection-based constraints or pairs of inclusion based constraints,…