Michael B. Giles

Michael B. Giles, Takashi Goda

In this paper we develop a very efficient approach to the Monte Carlo estimation of the expected value of partial perfect information (EVPPI) that measures the average benefit of knowing the value of a subset of uncertain parameters involved in a decision model. The calculation of EVPPI is inherently a nested expectation problem, with an outer expectation with respect to one random variable $X$ and an inner conditional expectation with respect to the other random variable $Y$. We tackle this problem by using a Multilevel Monte Carlo (MLMC) method (Giles 2008) in which the number of inner samples for $Y$ increases geometrically with level, so that the accuracy of estimating the inner conditional expectation improves and the cost also increases with level. We construct an antithetic MLMC estimator and provide sufficient assumptions on a decision model under which the antithetic property of the estimator is well exploited, and consequently a root-mean-square accuracy of $\varepsilon$ can be achieved at a cost of $O(\varepsilon^{-2})$. Numerical results confirm the considerable computational savings compared to the standard, nested Monte Carlo method for some simple testcases and a more realistic medical application.

Wei Fang, Michael B. Giles

This paper proposes an adaptive timestep construction for an Euler-Maruyama approximation of the ergodic SDEs with a drift which is not globally Lipschitz over an infinite time interval. If the timestep is bounded appropriately, we show not only the stability of the numerical solution and the standard strong convergence order, but also that the bound for moments and strong error of the numerical solution are uniform in T, which allow us to introduce the adaptive multilevel Monte Carlo. Numerical experiments support our analysis.

Wei Fang, Michael B. Giles

This paper proposes a new multilevel Monte Carlo (MLMC) method for the ergodic SDEs which do not satisfy the contractivity condition. By introducing the change of measure technique, we simulate the path with contractivity and add the Radon-Nykodim derivative to the estimator. We can show the strong error of the path is uniformly bounded with respect to $T.$ Moreover, the variance of the new level estimators increase linearly in $T,$ which is a great reduction compared with the exponential increase in standard MLMC. Then the total computational cost is reduced to $O(\varepsilon^{-2}|\log \varepsilon|^{2})$ from $O(\varepsilon^{-3}|\log \varepsilon|)$ of the standard Monte Carlo method. Numerical experiments support our analysis.

Grigoris Katsiolides, Eike H. Müller, Robert Scheichl et al.

A common way to simulate the transport and spread of pollutants in the atmosphere is via stochastic Lagrangian dispersion models. Mathematically, these models describe turbulent transport processes with stochastic differential equations (SDEs). The computational bottleneck is the Monte Carlo algorithm, which simulates the motion of a large number of model particles in a turbulent velocity field; for each particle, a trajectory is calculated with a numerical timestepping method. Choosing an efficient numerical method is particularly important in operational emergency-response applications, such as tracking radioactive clouds from nuclear accidents or predicting the impact of volcanic ash clouds on international aviation, where accurate and timely predictions are essential. In this paper, we investigate the application of the Multilevel Monte Carlo (MLMC) method to simulate the propagation of particles in a representative one-dimensional dispersion scenario in the atmospheric boundary layer. MLMC can be shown to result in asymptotically superior computational complexity and reduced computational cost when compared to the Standard Monte Carlo (StMC) method, which is currently used in atmospheric dispersion modelling. To reduce the absolute cost of the method also in the non-asymptotic regime, it is equally important to choose the best possible numerical timestepping method on each level. To investigate this, we also compare the standard symplectic Euler method, which is used in many operational models, with two improved timestepping algorithms based on SDE splitting methods.

Michael B. Giles, Frances Y. Kuo, Ian H. Sloan

Building on previous research which generalized multilevel Monte Carlo methods using either sparse grids or Quasi-Monte Carlo methods, this paper considers the combination of all these ideas applied to elliptic PDEs with finite-dimensional uncertainty in the coefficients. It shows the potential for the computational cost to achieve an $O(\varepsilon)$ r.m.s. accuracy to be $O(\varepsilon^{-r})$ with $r<2$, independently of the spatial dimension of the PDE.

Michael B. Giles, Christoph Reisinger

In this article, we propose a Milstein finite difference scheme for a stochastic partial differential equation (SPDE) describing a large particle system. We show, by means of Fourier analysis, that the discretisation on an unbounded domain is convergent of first order in the timestep and second order in the spatial grid size, and that the discretisation is stable with respect to boundary data. Numerical experiments clearly indicate that the same convergence order also holds for boundary-value problems. Multilevel path simulation, previously used for SDEs, is shown to give substantial complexity gains compared to a standard discretisation of the SPDE or direct simulation of the particle system. We derive complexity bounds and illustrate the results by an application to basket credit derivatives.

Michael B. Giles, Mario Hefter, Lukas Mayer et al.

We study the approximation of expectations $\E(f(X))$ for Gaussian random elements $X$ with values in a separable Hilbert space $H$ and Lipschitz continuous functionals $f \colon H \to \R$. We consider restricted Monte Carlo algorithms, which may only use random bits instead of random numbers. We determine the asymptotics (in some cases sharp up to multiplicative constants, in the other cases sharp up to logarithmic factors) of the corresponding $n$-th minimal error in terms of the decay of the eigenvalues of the covariance operator of $X$. It turns out that, within the margins from above, restricted Monte Carlo algorithms are not inferior to arbitrary Monte Carlo algorithms, and suitable random bit multilevel algorithms are optimal. The analysis of this problem leads to a variant of the quantization problem, namely, the optimal approximation of probability measures on $H$ by uniform distributions supported by a given, finite number of points. We determine the asymptotics (up to multiplicative constants) of the error of the best approximation for the one-dimensional standard normal distribution, for Gaussian measures as above, and for scalar autonomous SDEs.

Michael B. Giles, Mario Hefter, Lukas Mayer et al.

We study the approximation of expectations $\E(f(X))$ for solutions $X$ of SDEs and functionals $f \colon C([0,1],\R^r) \to \R$ by means of restricted Monte Carlo algorithms that may only use random bits instead of random numbers. We consider the worst case setting for functionals $f$ from the Lipschitz class w.r.t.\ the supremum norm. We construct a random bit multilevel Euler algorithm and establish upper bounds for its error and cost. Furthermore, we derive matching lower bounds, up to a logarithmic factor, that are valid for all random bit Monte Carlo algorithms, and we show that, for the given quadrature problem, random bit Monte Carlo algorithms are at least almost as powerful as general randomized algorithms.

Michael B. Giles, Francisco Bernal

This paper proposes and analyses a new multilevel Monte Carlo method for the estimation of mean exit times for multi-dimensional Brownian diffusions, and associated functionals which correspond to solutions to high-dimensional parabolic PDEs through the Feynman-Kac formula. In particular, it is proved that the complexity to achieve an $\varepsilon$ root-mean-square error is $O(\varepsilon^{-2}\, |\!\log \varepsilon|^3)$.

Irina-Beatrice Haas, Michael B. Giles, Yuji Nakatsukasa

Sketch-and-solve (SAS) is a very successful method to efficiently estimate the solution of heavily overdetermined large linear least squares problems. It uses random sketching to reduce the size of the problem, hence reducing the computational cost. Several authors have shown that averaging several solutions from SAS further improves the accuracy, which is measured by the residual associated to the approximate solution. Going further, we combine solutions from sketch-and-solve in a multilevel manner, such that the approximate solution is a combination of SAS samples obtained from small sketches and more accurate correction terms obtained from larger sketches. We first consider the variance of the estimator, which depends on the variance of the coarse samples and the correction terms. We show that the variance of the correction terms on each level follows a trend and decreases faster than the variance of the simple SAS estimator. However, we then show that the overall computational cost of our multilevel framework is slightly higher than that of the simple average estimator, so a naive application of multilevel methods appears unattractive for least squares problems.

Michael B. Giles, Mario Hefter, Lukas Mayer et al.

We study the approximation of expectations $\operatorname{E}(f(X))$ for solutions $X$ of stochastic differential equations and functionals $f$ on the path space by means of Monte Carlo algorithms that only use random bits instead of random numbers. We construct an adaptive random bit multilevel algorithm, which is based on the Euler scheme, the Lévy-Ciesielski representation of the Brownian motion, and asymptotically optimal random bit approximations of the standard normal distribution. We numerically compare this algorithm with the adaptive classical multilevel Euler algorithm for a geometric Brownian motion, an Ornstein-Uhlenbeck process, and a Cox-Ingersoll-Ross process.

Matteo Croci, Michael B. Giles, Marie E. Rognes et al.

When solving stochastic partial differential equations (SPDEs) driven by additive spatial white noise, the efficient sampling of white noise realizations can be challenging. Here, we present a new sampling technique that can be used to efficiently compute white noise samples in a finite element method and multilevel Monte Carlo (MLMC) setting. The key idea is to exploit the finite element matrix assembly procedure and factorize each local mass matrix independently, hence avoiding the factorization of a large matrix. Moreover, in a MLMC framework, the white noise samples must be coupled between subsequent levels. We show how our technique can be used to enforce this coupling even in the case of non-nested mesh hierarchies. We demonstrate the efficacy of our method with numerical experiments. We observe optimal convergence rates for the finite element solution of the elliptic SPDEs of interest in 2D and 3D and we show convergence of the sampled field covariances. In a MLMC setting, a good coupling is enforced and the telescoping sum is respected.

Michael B. Giles, Mateusz B. Majka, Lukasz Szpruch et al.

We develop a framework that allows the use of the multi-level Monte Carlo (MLMC) methodology (Giles2015) to calculate expectations with respect to the invariant measure of an ergodic SDE. In that context, we study the (over-damped) Langevin equations with a strongly concave potential. We show that, when appropriate contracting couplings for the numerical integrators are available, one can obtain a uniform in time estimate of the MLMC variance in contrast to the majority of the results in the MLMC literature. As a consequence, a root mean square error of $\mathcal{O}(\varepsilon)$ is achieved with $\mathcal{O}(\varepsilon^{-2})$ complexity on par with Markov Chain Monte Carlo (MCMC) methods, which however can be computationally intensive when applied to large data sets. Finally, we present a multi-level version of the recently introduced Stochastic Gradient Langevin Dynamics (SGLD) method (Welling and Teh, 2011) built for large datasets applications. We show that this is the first stochastic gradient MCMC method with complexity $\mathcal{O}(\varepsilon^{-2}|\log {\varepsilon}|^{3})$, in contrast to the complexity $\mathcal{O}(\varepsilon^{-3})$ of currently available methods. Numerical experiments confirm our theoretical findings.