Abstract In this paper, we apply a randomized approach to the numerical solution of two chemotaxis models: the two-component minimal non-local Keller–Segel model and the three-component reaction-diffusion model. Both models are described by nonlinear reaction-diffusion systems. To solve these problems, we use an iterative process over time. We transform the reaction-diffusion system into a matrix form using a finite-difference approximation of the equations of the system. The resulting systems of linear algebraic equations are solved by the randomized projection method with refinement. We test the time-iterative algorithm and the randomized projection method with refinement on the minimal non-local Keller–Segel model with a known exact solution and apply these algorithms to solve the three-component reaction-diffusion model. It is shown that the iterative refinement algorithm makes it possible to increase the accuracy of solutions to systems of linear algebraic equations and the entire system describing the chemotaxis model as a whole. Numerical solutions obtained using the randomized projection method with refinement are compared with exact solutions and values calculated using the LAPACKE dgesv subroutine, and good agreement between these values is obtained.
Abstract Refined Descriptive Sampling (RDS) is one of the best methods that can be used in Monte Carlo simulation especially when the precision of the estimates is important. The RDS method provides lower-variance and more accurate estimates of multidimensional integrals than the standard Monte Carlo (MC), Descriptive Sampling, and Latin Hypercube Sampling (LHS) methods. In this paper, we focused first, on the practical Performance of the RDS Method by its comparison with MC and LHS multidimensional integral estimators, and second, on the use of RDS in the limit of error estimation to the Normal distribution which led to the proposal of the central limit theorem under RDS.
Abstract Feynman developed the path integral approach to quantum mechanics as an alternative to wave or matrix mechanics that has found fruitful implementation in quantum mechanics, quantum electrodynamics, and statistical physics. Despite its many successes, the usual Feynman path integral formalism suffers from a lack of mathematical rigor. Questions of mathematical rigor still exist, but Mark Kac provided a rigorous path integral approach to quantum mechanics. The Kac formulation for quantum systems is given in terms of Wiener measure for a path integral and employs the Brownian motion distribution. In this paper, we review the Feynman–Kac approach and other stochastic processes related to Brownian motion. We also give an account of the successes and difficulties we faced during our journey through numerical path integral problems involved in physics.
Abstract This paper proposes the application of the Neumann–Monte Carlo method to estimate the statistical moments of the solution to the bending problem of a Timoshenko beam resting on a stochastic Pasternak foundation. Uncertainty is modeled in the foundation’s stiffness coefficients via parameterized stochastic processes. The methodology proposed in this work differs from the usual approach, as it is developed based on theoretical results regarding the existence and uniqueness of the realizations. The consistency of the numerical approximations is ensured through rigorous analysis of the well-posedness of the underlying stochastic boundary value problem. This theoretical result guides the choice of approximation spaces via the Galerkin method, leading to realizations consistent with this result. Consequently, statistically consistent estimators of the expected value and variance of the transverse and angular displacement stochastic processes are obtained. The Monte Carlo simulation method is used to evaluate the performance of the proposed methodology.
Abstract Control variates are a classical variance-reduction technique for Monte Carlo estimators, yet the behavior of the associated variance estimators is less understood. We analyze the second-order properties of a single control variate Monte Carlo estimator under a simple multifidelity cost model. In the balanced case we obtain an exact finite-sample formula for Var [ Var ^ ( Q ^ CV ) ] {\operatorname{Var}[\widehat{\operatorname{Var}}(\widehat{Q}_{\mathrm{CV}})]} in terms of fourth-order moments and show that it decays as 𝒪 ( N - 3 ) {\mathcal{O}(N^{-3})} in the number of high-fidelity samples N , and hence as 𝒪 ( C tot - 3 ) {\mathcal{O}(C_{\mathrm{tot}}^{-3})} in the total cost. We then derive a compact asymptotic approximation via the multivariate delta method and construct a nonparametric bootstrap analogue. A simulation study on a polynomial benchmark with a low-cost surrogate confirms the predicted scaling and demonstrates that both a fourth-moment plug-in estimator and a bootstrap estimator provide accurate, comparable approximations to this variance-of-variance.
