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Nonlinear multiscale problems are ubiquitous in materials science and biology. Complicated interactions between nonlinearities and (nonseparable) multiple scales pose a major challenge for analysis and simulation. In this paper, we study the numerical homogenization for multiscale elliptic PDEs with monotone nonlinearity, in particular the Leray--Lions problem (a prototypical example is the p-Laplacian equation), where the nonlinearity cannot be parameterized with low dimensional parameters, and the linearization error is nonnegligible. We develop the iterated numerical homogenization scheme by combining numerical homogenization methods for linear equations and the so-called quasi-norm based iterative approach for monotone nonlinear equation. We propose a residual regularized nonlinear iterative method and, in addition, develop the sparse updating method for the efficient update of coarse spaces. A number of numerical results are presented to complement the analysis and validate the numerical method. [ABSTRACT FROM AUTHOR]

In this paper, we study the problem of computing the effective diffusivity for particles moving in chaotic flows. Instead of solving a convection-diffusion type cell problem in the Eulerian formulation (arising from homogenization theory for parabolic equations), we compute the motion of particles in the Lagrangian formulation, which is modeled by stochastic differential equations (SDEs). A robust numerical integrator based on a splitting method was proposed to solve the SDEs and rigorous error analysis for the numerical integrator was provided using the backward error analysis technique in our previous work. However, the upper bound on the error estimate is not sharp. To improve our result, we propose a new and uniform in time error analysis for the numerical integrator that allows us to get rid of the exponential growth factor in our previous error estimate. Our new error analysis is based on a probabilistic approach, which interprets the solution process generated by our numerical integrator as a Markov process. By exploring the ergodicity of the solution process, we prove the convergence analysis of our method in computing effective diffusivity over infinite time. We present numerical results to verify the accuracy and efficiency of the proposed method in computing effective diffusivity for several chaotic flows, especially the Arnold--Beltrami--Childress flow and Kolmogorov flow in three-dimensional space. [ABSTRACT FROM AUTHOR]

We consider the problem of determining the arrival statistics of unbiased planar random walkers to complex target configurations. In contrast to problems posed in finite domains, simple moments of the first passage time distribution, such as its mean and variance, are not defined. Therefore, it is necessary to obtain the full arrival statistics. We describe several methods to obtain these distributions and other associated quantities such as splitting probabilities. One approach combines a Laplace transform of the underlying parabolic equation with matched asymptotic analysis followed by numerical transform inversion. The second approach is similar but uses a boundary integral equation method to solve for the transformed variable. To validate the results of this theory, and to obtain the arrival time statistics in very general configurations of absorbers, we introduce an efficient kinetic Monte Carlo (KMC) method that describes trajectories as a combination of large but exactly solvable projection steps. The effectiveness of these methodologies is demonstrated on a variety of challenging examples highlighting the applicability of these methods to a variety of practical scenarios, such as source inference. A particularly useful finding arising from these results is that homogenization theories, in which complex configurations are replaced by equivalent simple ones, are remarkably effective at describing arrival time statistics. [ABSTRACT FROM AUTHOR]

We study optimal convergence rates in the periodic homogenization of linear elliptic equations of the form -A(x/ε) : D²uε = f subject to a homogeneous Dirichlet boundary condition. We show that the optimal rate for the convergence of uε to the solution of the corresponding homogenized problem in the W1,p-norm is O(ε). We further obtain gradient and Hessian bounds with correction terms taken into account in the Lp-norm and recover the known optimal convergence rate in the L∞-norm under weak assumptions on the data. We then provide an explicit c-bad diffusion matrix and use it to perform various numerical experiments, which demonstrate the optimality of the obtained rates. Finally, we discuss extensions of the results to the case of nonsmooth domains and their utility in regard to numerical homogenization. [ABSTRACT FROM AUTHOR]