Michael B. Giles

Multilevel Monte Carlo Path Simulation

We show that multigrid ideas can be used to reduce the computational complexity of estimating an expected value arising from a stochastic differential equation using Monte Carlo path simulations. In the simplest case of a Lipschitz payoff and a Euler discretisation, the computational cost to achieve an accuracy of O(e) is reduced from O(e-3) to O(e-2 (log e)2). The analysis is supported by numerical results showing significant computational savings.

Nonreflecting boundary conditions for Euler equation calculations

We present a unified theory for the construction of steady-state and unsteady nonreflecting boundary conditions for the Euler equations. These allow calculatios to be performed on truncated domains without the generation of spurious nonphysical reflections at the far-field boundaries.

An Introduction to the Adjoint Approach to Design

Optimal design methods involving the solution of an adjoint system of equations are an active area of research in computational fluid dynamics, particularly for aeronautical applications. This paper presents an introduction to the subject, emphasising the simplicity of the ideas when viewed in the context of linear algebra. Detailed discussions also include the extension to p.d.e.s, the construction of the adjoint p.d.e. and its boundary conditions, and the physical significance of the adjoint solution. The paper concludes with examples of the use of adjoint methods for optimising the design of business jets.

Adjoint methods for PDEs: a posteriori error analysis and postprocessing by duality

We give an overview of recent developments concerning the use of adjoint methods in two areas: the a posteriori error analysis of finite element methods for the numerical solution of partial differential equations where the quantity of interest is a functional of the solution, and superconvergent extraction of integral functionals by postprocessing.

Adjoint Recovery of Superconvergent Functionals from PDE Approximations

Motivated by applications in computational fluid dynamics, a method is presented for obtaining estimates of integral functionals, such as lift or drag, that have twice the order of accuracy of the computed flow solution on which they are based. This is achieved through error analysis that uses an adjoint PDE to relate the local errors in approximating the flow solution to the corresponding global errors in the functional of interest. Numerical evaluation of the local residual error together with an approximate solution to the adjoint equations may thus be combined to produce a correction for the computed functional value that yields the desired improvement in accuracy. Numerical results are presented for the Poisson equation in one and two dimensions and for the nonlinear quasi-one-dimensional Euler equations. The theory is equally applicable to nonlinear equations in complex multi-dimensional domains and holds great promise for use in a range of engineering disciplines in which a few integral quantities are a key output of numerical approximations.

On the utility of graphics cards to perform massively parallel simulation of advanced Monte Carlo methods.

We present a case study on the utility of graphics cards to perform massively parallel simulation of advanced Monte Carlo methods. Graphics cards, containing multiple Graphics Processing Units (GPUs), are self-contained parallel computational devices that can be housed in conventional desktop and laptop computers and can be thought of as prototypes of the next generation of many-core processors. For certain classes of population-based Monte Carlo algorithms they offer massively parallel simulation, with the added advantage over conventional distributed multicore processors that they are cheap, easily accessible, easy to maintain, easy to code, dedicated local devices with low power consumption. On a canonical set of stochastic simulation examples including population-based Markov chain Monte Carlo methods and Sequential Monte Carlo methods, we find speedups from 35- to 500-fold over conventional single-threaded computer code. Our findings suggest that GPUs have the potential to facilitate the growth of statistical modeling into complex data-rich domains through the availability of cheap and accessible many-core computation. We believe the speedup we observe should motivate wider use of parallelizable simulation methods and greater methodological attention to their design. This article has supplementary material online.

Multilevel Monte Carlo methods and applications to elliptic PDEs with random coefficients

We consider the numerical solution of elliptic partial differential equations with random coefficients. Such problems arise, for example, in uncertainty quantification for groundwater flow. We describe a novel variance reduction technique for the standard Monte Carlo method, called the multilevel Monte Carlo method, and demonstrate numerically its superiority. The asymptotic cost of solving the stochastic problem with the multilevel method is always significantly lower than that of the standard method and grows only proportionally to the cost of solving the deterministic problem in certain circumstances. Numerical calculations demonstrating the effectiveness of the method for one- and two-dimensional model problems arising in groundwater flow are presented.

Adjoint equations in CFD: duality, boundary conditions and solution behaviour

The first half of this paper derives the adjoint equations for inviscid and viscous compressible flow, with the emphasis being on the correct formulation of the adjoint boundary conditions and restrictions on the permissible choice of operators in the linearised functional. It is also shown that the boundary conditions for the adjoint problem can be simplified through the use of a linearised perturbation to generalised coordinates. The second half of the paper constructs the Greens functions for the quasi-lD and 2D Euler equations. These are used to show that the adjoint variables have a logarithmic singularity at the sonic line in the quasi-lD case, and a weak inverse square-root singularity at the upstream stagnation streamline in the 2D case, but are continuous at shocks in both cases.

Two-Dimensional Transonic Aerodynamic Design Method

This paper demonstrates the capabilities of a new transonic, two-dimensional design method, based on the simultaneous solution of multiple streamtubes, coupled through the position of, and pressure at, the streamline interfaces. This allows the specification of either the airfoil shape (direct, analysis mode) or the airfoil surface pressure distribution (inverse, design mode). The nonlinear system of equations is formulated in a conservative manner, which guarantees the correct treatment of shocks, and is solved by a rapid Newton solution method. Viscous effects can also be included through a coupled integral boundary-layer analysis. The first set of results shows the effect of different far-field treatments, demonstrating the improvement in accuracy obtained by including the second-order doublet terms in addition to the usual first-order vortex term. The results are also compared to those obtained by specifying straight far-field streamlines (corresponding to solid-wall wind-tunnel experiments) or constant far-field pressure (corresponding to freejet experiments) to show the sensitivity to the farfield distance. In the second set of results, the design method is used to design a transonic airfoil with C/ = 1.000 at A/oo = 0.70. It is shown that the off-design performance is improved by specifying a surface pressure distribution with a very weak shock.

Improved Multilevel Monte Carlo Convergence using the Milstein Scheme

In this paper we show that the Milstein scheme can be used to improve the convergence of the multilevel Monte Carlo method for scalar stochastic differential equations. Numerical results for Asian, lookback, barrier and digital options demonstrate that the computational cost to achieve a root-mean-square error of e is reduced to O(e -2). This is achieved through a careful construction of the multilevel estimator which computes the difference in expected payoff when using different numbers of timesteps.