**Are you an EPFL student looking for a semester project?**

Work with us on data science and visualisation projects, and deploy your project as an app on top of GraphSearch.

Concept# Probabilistic numerics

Summary

Probabilistic numerics is an active field of study at the intersection of applied mathematics, statistics, and machine learning centering on the concept of uncertainty in computation. In probabilistic numerics, tasks in numerical analysis such as finding numerical solutions for integration, linear algebra, optimization and simulation and differential equations are seen as problems of statistical, probabilistic, or Bayesian inference.
A numerical method is an algorithm that approximates the solution to a mathematical problem (examples below include the solution to a linear system of equations, the value of an integral, the solution of a differential equation, the minimum of a multivariate function). In a probabilistic numerical algorithm, this process of approximation is thought of as a problem of estimation, inference or learning and realised in the framework of probabilistic inference (often, but not always, Bayesian inference).
Formally, this means casting the setup of the computational problem in terms of a prior distribution, formulating the relationship between numbers computed by the computer (e.g. matrix-vector multiplications in linear algebra, gradients in optimization, values of the integrand or the vector field defining a differential equation) and the quantity in question (the solution of the linear problem, the minimum, the integral, the solution curve) in a likelihood function, and returning a posterior distribution as the output. In most cases, numerical algorithms also take internal adaptive decisions about which numbers to compute, which form an active learning problem.
Many of the most popular classic numerical algorithms can be re-interpreted in the probabilistic framework. This includes the method of conjugate gradients, Nordsieck methods, Gaussian quadrature rules, and quasi-Newton methods. In all these cases, the classic method is based on a regularized least-squares estimate that can be associated with the posterior mean arising from a Gaussian prior and likelihood.

Official source

This page is automatically generated and may contain information that is not correct, complete, up-to-date, or relevant to your search query. The same applies to every other page on this website. Please make sure to verify the information with EPFL's official sources.

Related concepts (1)

Related courses (6)

Numerical integration

In analysis, numerical integration comprises a broad family of algorithms for calculating the numerical value of a definite integral, and by extension, the term is also sometimes used to describe the numerical solution of differential equations. This article focuses on calculation of definite integrals. The term numerical quadrature (often abbreviated to quadrature) is more or less a synonym for numerical integration, especially as applied to one-dimensional integrals.

Related people (43)

Related units (8)

MATH-251(c): Numerical analysis

Le cours présente des méthodes numériques pour la résolution de problèmes mathématiques comme des systèmes d'équations linéaires ou non linéaires, approximation de fonctions, intégration et dérivation

ChE-340: The engineering of chemical reactions

This course applies concepts from chemical kinetics and mass and energy balances to address chemical reaction engineering problems, with a focus on industrial applications. Students develop the abilit

CH-351: Molecular dynamics and Monte-Carlo simulation

Introduction to molecular dynamics and Monte-Carlo simulation methods.

, , , , , , , , ,

Related publications (474)

Related lectures (85)

Numerical Differentiation: Methods and ErrorsMATH-251(d): Numerical analysis

Explores numerical differentiation methods and round-off errors in computer computations.

Numerical Differentiation and IntegrationMATH-251(d): Numerical analysis

Explores numerical differentiation and integration methods, emphasizing the accuracy of finite differences in computing derivatives and integrals.

Solving the Quintic: Dominant BalanceME-201: Continuum mechanics

Explores dominant balance analysis in solving the quintic polynomial, with a focus on non-dimensionalization and series expansions.

The method of moments (MOM), as introduced by R. F. Harrington more than 50 years ago, is reviewed in the context of the classic potential integral equation (PIE) formulations applied to both electrostatic (part 1) and electrodynamic, or full-wave, problem ...

The method of moments (MOM), as introduced by Roger F. Harrington more than 50 years ago, is reviewed in the context of the classic potential integral equation (IE) formulations applied to both electrostatic (part 1) and electrodynamic or full-wave problem ...

Geometric properties of lattice quantum gravity in two dimensions are studied numerically via Monte Carlo on Euclidean Dynamical Triangulations. A new computational method is proposed to simulate gravity coupled with fermions, which allows the study of int ...