Related papers: AAA rational approximation on a continuum
To verify safety and robustness of neural networks, researchers have successfully applied abstract interpretation, primarily using the interval abstract domain. In this paper, we study the theoretical power and limits of the interval domain…
This paper explores artificial intelligence (AI) methods for the approximate compiling of unitaries, focusing on the use of fixed two-qubit gates and arbitrary single-qubit rotations typical in superconducting hardware. Our approach…
The classical universal approximation (UA) theorem for neural networks establishes mild conditions under which a feedforward neural network can approximate a continuous function $f$ with arbitrary accuracy. A recent result shows that neural…
This paper concerns a high-dimensional stochastic programming problem of minimizing a function of expected cost with a matrix argument. To this problem, one of the most widely applied solution paradigms is the sample average approximation…
Automatic differentiation is involved for long in applied mathematics as an alternative to finite difference to improve the accuracy of numerical computation of derivatives. Each time a numerical minimization is involved, automatic…
Unitary best approximation to the exponential function on an interval on the imaginary axis has been introduced recently. In the present work two algorithms are considered to compute this best approximant: an algorithm based on rational…
We study adaptive approximation algorithms for general multivariate linear problems where the sets of input functions are non-convex cones. While it is known that adaptive algorithms perform essentially no better than non-adaptive…
In this paper a first order analytical system of difference equations is considered. For an asymptotically stable fixed point x0 of the system a gradual approximation of the domain of attraction DA is presented in the case when the matrix…
The rapid expansion of scientific data has widened the gap between analytical capability and research intent. Existing AI-based analysis tools, ranging from AutoML frameworks to agentic research assistants, either favor automation over…
The exponential function maps the imaginary axis to the unit circle and, for many applications, this unitarity property is also desirable from its approximations. We show that this property is conserved not only by the (k,k)-rational…
In this article we present and analyse new multilevel adaptations of stochastic approximation algorithms for the computation of a zero of a function $f\colon D \to \mathbb R^d$ defined on a convex domain $D\subset \mathbb R^d$, which is…
This paper presents a novel algorithm, based on use of rational approximants of a randomly scalarized boundary integral resolvent in conjunction with an adaptive search strategy and an exponentially convergent secant-method termination…
Quantum Approximate Optimization algorithm (QAOA) aims to search for approximate solutions to discrete optimization problems with near-term quantum computers. As there are no algorithmic guarantee possible for QAOA to outperform classical…
A regularization algorithm allowing random noise in derivatives and inexact function values is proposed for computing approximate local critical points of any order for smooth unconstrained optimization problems. For an objective function…
Approximate Bayesian computation (ABC) is a family of computational techniques in Bayesian statistics. These techniques allow to fi t a model to data without relying on the computation of the model likelihood. They instead require to…
In this work we show a rational approximation of the Dawson's integral that can be implemented for high-accuracy computation of the complex error function in a rapid algorithm. Specifically, this approach provides accuracy exceeding $\sim…
The quantum approximate optimization algorithm (QAOA) applies two Hamiltonians to a quantum system in alternation. The original goal of the algorithm was to drive the system close to the ground state of one of the Hamiltonians. This paper…
We introduce a method to improve the tractability of the well-known Sample Average Approximation (SAA) without compromising important theoretical properties, such as convergence in probability and the consistency of an independent and…
Most commonly used \emph{adaptive} algorithms for univariate real-valued function approximation and global minimization lack theoretical guarantees. Our new locally adaptive algorithms are guaranteed to provide answers that satisfy a…
The quantum approximate optimization algorithm (QAOA) is a leading candidate algorithm for solving optimization problems on quantum computers. However, the potential of QAOA to tackle classically intractable problems remains unclear. Here,…