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No longer afraid of numbers! Do you know what the probabilistic numerical method is?
In today's world of mathematics and computational mathematics, probabilistic numerical methods, as an interdisciplinary research field, have gradually attracted people's attention. This field combines applied mathematics, statistics, and machine learning and revolves around computational uncertainty. In probabilistic numerical methods, common numerical analysis tasks such as numerical integration, linear algebra, optimization, simulation, and solution of differential equations are treated as statistical, probabilistic, or Bayesian inference problems.
Numerical methods are algorithms used to approximate the solution of mathematical problems, including solving systems of linear equations, computing integrals, solving differential equations, and minimizing functions of multiple variables.
Traditional numerical algorithms are based on deterministic methods, while probabilistic numerical algorithms regard this process as an estimation or learning problem and implement it within the framework of probabilistic inference. This means that a prior distribution can be used to describe the computational problem, and by comparing the calculated numbers (such as matrix-vector products, gradients in optimization, integral function values, etc.) Make assumptions about the relationship and return the posterior distribution as output.
In fact, many classic numerical algorithms can be reinterpreted under the probabilistic framework, such as the conjugate gradient method, Nordsieck method, Gaussian integration rule, and quasi-Newton method. The advantage of these techniques is that they not only provide structured error estimates but also use hierarchical Bayesian inference to set and control internal hyperparameters.
Probabilistic numerical methods allow combining data from multiple sources of information, effectively removing nested loops from calculations.
Numerical tasks
Integration
In terms of numerical integration, probabilistic numerical methods have developed many techniques, the most famous of which is the Bayesian integration method. In this process, the integrated value of a function is estimated by evaluating it at a given series of points. In this case, choosing a prior distribution and conditioning on the observed data leads to a posterior distribution, which is especially useful for functions that are computationally expensive.
Optimization
In terms of mathematical optimization, probabilistic numerical methods have also been studied in depth. Bayesian optimization is a general method based on Bayesian inference. These algorithms help find the minimum or maximum by maintaining probabilistic beliefs about the objective function to guide subsequent observation selection.
Local optimization
In stochastic optimization in the context of deep learning, probabilistic numerical techniques have studied many important issues such as learning rate adjustment, mini-batch selection, etc., and achieved automatic decision-making by explicitly modeling these uncertainties.
Linear algebra
In applications of linear algebra, probabilistic numerical algorithms focus on solving systems of linear equations of the form A x = b. Such methods are usually iterative in nature, gathering information through repeated matrix-vector multiplications.
Ordinary differential equations
For ordinary differential equations, a variety of probabilistic numerical methods have been developed, which can be divided into methods based on randomization and Gaussian process regression, which can effectively handle initial value and boundary value problems.
Partial differential equations
Similarly, as technology evolves, probabilistic numerical methods for partial differential equations have also improved, and these methods effectively take advantage of the properties of Gaussian process regression.
Historical background and related fields
The development of probabilistic numerical methods did not happen overnight, but was closely related to other areas of mathematics such as information complexity, game theory and statistical decision theory. From the late 19th century to the early 20th century, the intersection of probability and numerical analysis began to receive attention. The contributions of many mathematicians, from Henri Poincaré to Albert Suldin to Mike Larkin, paved the way for the development of this field.
When we face complex data, have you ever thought about applying probabilistic numerical methods to improve your calculation efficiency?
