Euler scheme for approximation of solution of nonlinear ODEs under inexact information

We investigate error of the Euler scheme in the case when the right-hand side function of the underlying ODE satisfies nonstandard assumptions such as local one-sided Lipschitz condition and local H\"older continuity. Moreover, we assume two cases in regards to information availability: exact and noisy with respect to the right-hand side function. Optimality analysis of the Euler scheme is also provided. Finally, we present the results of some numerical experiments.Comment: 18 pages, 9 figure

Bayesian Probabilistic Numerical Methods in Time-Dependent State Estimation for Industrial Hydrocyclone Equipment

The use of high-power industrial equipment, such as large-scale mixing equipment or a hydrocyclone for separation of particles in liquid suspension, demands careful monitoring to ensure correct operation. The fundamental task of state-estimation for the liquid suspension can be posed as a time-evolving inverse problem and solved with Bayesian statistical methods. In this article, we extend Bayesian methods to incorporate statistical models for the error that is incurred in the numerical solution of the physical governing equations. This enables full uncertainty quantification within a principled computation-precision trade-off, in contrast to the over-confident inferences that are obtained when all sources of numerical error are ignored. The method is cast within a sequential Monte Carlo framework and an optimized implementation is provided in Python

The Complexity of Definite Elliptic Problems With Noisy Data

. We study the complexity of 2mth order definite elliptic problems Lu = f (with homogeneous Dirichlet boundary conditions) over a d-dimensional domain\Omega\Gamma error being measured in the H m(\Omega\Gamma9 norm. The problem elements f belong to the unit ball of W r;p(\Omega\Gamma3 where p 2 [2; 1] and r ? d=p. Information consists of (possibly-adaptive) noisy evaluations of f or the coefficients of L. The absolute error in each noisy evaluation is at most ffi. We find that the nth minimal radius for this problem is proportional to n \Gammar=d + ffi, and that a noisy finite element method with quadrature (FEMQ), which uses only function values, and not derivatives, is a minimal error algorithm. This noisy FEMQ can be efficiently implemented using multigrid techniques. Using these results, we find tight bounds on the "-complexity (minimal cost of calculating an "-approximation) for this problem, said bounds depending on the cost c(ffi) of calculating a ffi-noisy information v..