Russel E. Caflisch

Monte Carlo and quasi-Monte Carlo methods

Monte Carlo is one of the most versatile and widely used numerical methods. Its convergence rate, O ( N −1/2 ), is independent of dimension, which shows Monte Carlo to be very robust but also slow. This article presents an introduction to Monte Carlo methods for integration problems, including convergence theory, sampling methods and variance reduction techniques. Accelerated convergence for Monte Carlo quadrature is attained using quasi-random (also called low-discrepancy) sequences, which are a deterministic alternative to random or pseudo-random sequences. The points in a quasi-random sequence are correlated to provide greater uniformity. The resulting quadrature method, called quasi-Monte Carlo, has a convergence rate of approximately O ((log N ) k N −1 ). For quasi-Monte Carlo, both theoretical error estimates and practical limitations are presented. Although the emphasis in this article is on integration, Monte Carlo simulation of rarefied gas dynamics is also discussed. In the limit of small mean free path (that is, the fluid dynamic limit), Monte Carlo loses its effectiveness because the collisional distance is much less than the fluid dynamic length scale. Computational examples are presented throughout the text to illustrate the theory. A number of open problems are described.

Quasi-random sequences and their discrepancies

Quasi-random (also called low discrepancy) sequences are a deterministic alternative to random sequences for use in Monte Carlo methods, such as integration and particle simulations of transport processes. The error in uniformity for such a sequence of N points in the s-dimensional unit cube is measured by its discrepancy, which is of size

Effective equations for wave propagation in bubbly liquids

(\log N)^s N^{ - 1}

Variance in the sedimentation speed of a suspension

for large N, as opposed to discrepancy of size

Uniformly Accurate Schemes for Hyperbolic Systems with Relaxation

(\log \log N)^{1/2} N^{ - 1/2}

Smoothness and dimension reduction in Quasi-Monte Carlo methods

for a random sequence (i.e., for almost any randomly chosen sequence). Several types of discrepancies, one of which is new, are defined and analyzed. A critical discussion of the theoretical bounds on these discrepancies is presented. Computations of discrepancies are presented for a wide choice of dimension s, number of points N, and different quasi-random sequences. In particular for moderate or large s, there is an intermediate regime in which the discrepancy of a quasi-random sequence is almost exactly the same as that of a randomly chosen sequence. A simplified p...

A Level Set Approach for the Numerical Simulation of Dendritic Growth

We derive a system of effective equations for wave propagation in a bubbly liquid. Starting from a microscopic description, we obtain the effective equations by using Foldys approximation in a nonlinear setting. We discuss in detail the range of validity of the effective equations as well as some of their properties.

The Boltzmann equation with a soft potential. I. Linear, spatially-homogeneous

The variance in the sedimentation speed for a homogeneous suspension of solid spheres in a Stokes fluid is calculated for a particular choice of the distribution function of the spheres. In the infinite particle number limit, the variance is found to be infinite.

Singular solutions and ill-posedness for the evolution of vortex sheets

We develop high-resolution shock-capturing numerical schemes for hyperbolic systems with relaxation. In such systems the relaxation time may vary from order-1 to much less than unity. When the relaxation time is small, the relaxation term becomes very strong and highly stiff, and underresolved numerical schemes may produce spurious results. Usually one cannot decouple the problem into separate regimes and handle different regimes with different methods. Thus it is important to have a scheme that works uniformly with respect to the relaxation time. Using the Broadwell model of the nonlinear Boltzmann equation we develop a second-order scheme that works effectively, with a fixed spatial and temporal discretization, for all ranges of the mean free path. Formal uniform consistency proof for a first-order scheme and numerical convergence proof for the second-order scheme are also presented. We also make numerical comparisons of the new scheme with some other schemes. This study is motivated by the reentry problem in hypersonic computations.

Shock profile solutions of the Boltzmann equation

Monte Carlo integration using quasirandom sequences has theoretical error bounds of size O (N^-^1 log^dN) in dimension d, as opposed to the error of size O (N^-^1^2) for random or pseudorandom sequences. In practice, however, this improved performance for quasirandom sequences is often not observed. The degradation of performance is due to discontinuity or lack of smoothness in the integrand and to large dimension of the domain of integration, both of which often occur in Monte Carlo methods. In this paper, modified Monte Carlo methods are developed, using smoothing and dimension reduction, so that the convergence rate of nearly O (N^-^1) is regained. The standard rejection method, as used in importance sampling, involves discontinuities, corresponding to the decision to accept or reject. A smoothed rejection method, as well as a method of weighted uniform sampling, is formulated below and found to have error size of almost O (N^-^1) in quasi-Monte Carlo. Quasi-Monte Carlo evaluation of Feynman-Kac path integrals involves high dimension, one dimension for each discrete time interval. Through an alternative discretization, the effective dimension of the integration domain is drastically reduced, so that the error size close to O(N^-^1) is again regained.