Related papers: Computational Complexity and Numerical Stability o…
$ $[This paper is a (self contained) chapter in a new book, Mathematics and Computation, whose draft is available on my homepage at https://www.math.ias.edu/avi/book ]. We survey some concrete interaction areas between computational…
A method is suggested for treating those complicated physical problems for which exact solutions are not known but a few approximation terms of a calculational algorithm can be derived. The method permits one to answer the following rather…
Genetic Programming (GP) has found various applications. Understanding this type of algorithm from a theoretical point of view is a challenging task. The first results on the computational complexity of GP have been obtained for problems…
Linear algebraic primitives are at the core of many modern algorithms in engineering, science, and machine learning. Hence, accelerating these primitives with novel computing hardware would have tremendous economic impact. Quantum computing…
We analyse the computational complexity of three problems in judgment aggregation: (1) computing a collective judgment from a profile of individual judgments (the winner determination problem); (2) deciding whether a given agent can…
We discuss computability and computational complexity of conformal mappings and their boundary extensions. As applications, we review the state of the art regarding computability and complexity of Julia sets, their invariant measures and…
Stability plays a central role in arithmetic. In this article, we explain some basic ideas and present certain constructions for such studies. There are two aspects: namely, general Class Field Theories for Riemann surfaces using…
Support for arithmetic in multiple precisions and number formats is becoming increasingly common in emerging high-performance architectures. From a computational scientist's perspective, our goal is to determine how and where we can safely…
We discuss the computational complexity of solving linear programming problems by means of an analog computer. The latter is modeled by a dynamical system which converges to the optimal vertex solution. We analyze various probability…
In this manuscript, we derive the principle of conservation of computational complexity. We measure computational complexity as the number of binary computations (decisions) required to solve a problem. Every problem then defines a unique…
Symmetries and reductions of some algebraic equations are considered. Transformations that preserve the form of several algebraic equations, as well as transformations that reduce the degree of these equations, are described. Illustrative…
We survey research that studies the connection between the computational complexity of optimization problems on the one hand, and the duality gap between the primal and dual optimization problems on the other. To our knowledge, this is the…
Compositionality is a key property for dealing with complexity, which has been studied from many points of view in diverse fields. Particularly, the composition of individual computations (or programs) has been widely studied almost since…
These notes contain, among others, a proof that the average running time of an easy solution to the satisfiability problem for propositional calculus is, under some reasonable assumptions, linear (with constant 2) in the size of the input.…
A strong link between information geometry and algebraic statistics is made by investigating statistical manifolds which are algebraic varieties. In particular it it shown how first and second order efficient estimators can be constructed,…
Using three different approaches, we analyze the complexity of various birational maps constructed from simple operations (inversions) on square matrices of arbitrary size. The first approach consists in the study of the images of lines,…
We first study birational mappings generated by the composition of the matrix inversion and of a permutation of the entries of $ 3 \times 3 $ matrices. We introduce a semi-numerical analysis which enables to compute the Arnold complexities…
We deliver a call to arms for probabilistic numerical methods: algorithms for numerical tasks, including linear algebra, integration, optimization and solving differential equations, that return uncertainties in their calculations. Such…
The assignment of personnel to teams is a fundamental and ubiquitous managerial function, typically involving several objectives and a variety of idiosyncratic practical constraints. Despite the prevalence of this task in practice, the…
Scientists have demonstrated that quantum computing has presented novel approaches to address computational challenges, each varying in complexity. Adapting problem-solving strategies is crucial to harness the full potential of quantum…