Related papers: Approximability in the GPAC
We consider the General Purpose Analog Computer (GPAC), introduced by Claude Shannon in 1941 as a mathematical model of Differential Analysers, that is to say as a model of continuous-time analog (mechanical, and later one electronic)…
In 1941, Claude Shannon introduced the General Purpose Analog Computer(GPAC) as a mathematical model of Differential Analysers, that is to say as a model of continuous-time analog (mechanical, and later one electronic) machines of that…
The Church-Turing thesis states that any sufficiently powerful computational model which captures the notion of algorithm is computationally equivalent to the Turing machine. This equivalence usually holds both at a computability level and…
In the "post-Moore era", the growing challenges in traditional computing have driven renewed interest in analog computing, leading to various proposals for the development of general-purpose analog computing (GPAC) systems. In this work, we…
We generalize the PAC (probably approximately correct) learning model to the quantum world by generalizing the concepts from classical functions to quantum processes, defining the problem of \emph{PAC learning quantum process}, and study…
We introduce new Gaussian Process (GP) high-order approximations to linear operations that are frequently used in various numerical methods. Our method employs the kernel-based GP regression modeling, a non-parametric Bayesian approach to…
Analog computers can be revived as a feasible technology platform for low precision, energy efficient and fast computing. We justify this statement by measuring the performance of a modern analog computer and comparing it with that of…
In many real-world applications we are interested in approximating costly functions that are analytically unknown, e.g. complex computer codes. An emulator provides a fast approximation of such functions relying on a limited number of…
Designing scalable estimation algorithms is a core challenge in modern statistics. Here we introduce a framework to address this challenge based on parallel approximants, which yields estimators with provable properties that operate on the…
Simulation of materials at the atomistic level is an important tool in studying microscopic structure and processes. The atomic interactions necessary for the simulation are correctly described by Quantum Mechanics. However, the…
A myriad of applications ranging from engineering and scientific simulations, image and signal processing as well as high-sensitive data retrieval demand high processing power reaching up to teraflops for their efficient execution. While a…
Relative to digital computation, analog computation has been neglected in the philosophical literature. To the extent that attention has been paid to analog computation, it has been misunderstood. The received view -- that analog…
Operator learning offers a powerful paradigm for solving parametric partial differential equations (PDEs), but scaling probabilistic neural operators such as the recently proposed Gaussian Processes Operators (GPOs) to high-dimensional,…
The outcomes of this paper are twofold. Implicit complexity. We provide an implicit characterization of polynomial time computation in terms of ordinary differential equations: we characterize the class PTIME of languages computable in…
It is known that standard stochastic Galerkin methods encounter challenges when solving partial differential equations with high-dimensional random inputs, which are typically caused by the large number of stochastic basis functions…
This paper serves as a bridge between quantum computing and analogical modeling (a general theory for predicting categories of behavior in varying contexts). Since its formulation in the early 1980s, analogical modeling has been…
Analogue computers use continuous properties of physical system for modeling. In the paper is described possibility of modeling by analogue quantum computers for some model of data analysis. It is analogue associative memory and a formal…
Parallel computing is omnipresent in today's scientific computer landscape, starting at multicore processors in desktop computers up to massively parallel clusters. While domain decomposition methods have a long tradition in computational…
Despite a large corpus of recent work on scaling up Gaussian processes, a stubborn trade-off between computational speed, prediction and uncertainty quantification accuracy, and customizability persists. This is because the vast majority of…
The PAC-Bayesian approach is a powerful set of techniques to derive non- asymptotic risk bounds for random estimators. The corresponding optimal distribution of estimators, usually called the Gibbs posterior, is unfortunately intractable.…