A Control Variate Approach for Improving Efficiency of Ensemble Monte Carlo

arXiv:0809.3187v1 [cs.CE] 18 Sep 2008 A Control Variate Approach for Improving Efficiency of Ensemble Monte Carlo ⋆ Tarik Borogovac a,b,∗Francis J. Alexander b Pirooz Vakili c aElectrical and Computer Engineering Department, Boston University, 8 Saint Mary’s St. Boston, MA 02215, USA bCCS-3, Los Alamos National Laboratory, MS-B256, Los Alamos, NM 87545, USA cDivision of Systems Engineering & Mechanical Engineering Department, Boston University, 15 Saint Mary’s St. Brookline, MA 02446, USA Abstract In this paper we present a new approach to control variates for improving compu- tational efficiency of Ensemble Monte Carlo. We present the approach using sim- ulation of paths of a time-dependent nonlinear stochastic equation. The core idea is to extract information at one or more nominal model parameters and use this information to gain estimation efficiency at neighboring parameters. This idea is the basis of a general strategy, called DataBase Monte Carlo (DBMC), for improving efficiency of Monte Carlo. In this paper we describe how this strategy can be imple- mented using the variance reduction technique of Control Variates (CV). We show that, once an initial setup cost for extracting information is incurred, this approach can lead to significant gains in computational efficiency. The initial setup cost is justified in projects that require a large number of estimations or in those that are to be performed under real-time constraints. Key words: Monte Carlo, Variance Reduction, Control Variates PACS: S05.10.Ln, 02.70.Uu, 02.70.Tt ⋆Portions of this work, LA-UR-08-05399, were carried out at Los Alamos National Laboratory under the auspices of the US National Nuclear Security Administra- tion of the US Department of Energy. Tarik Borogovac and Pirooz Vakili were supported in part by the National Science Foundation grants CMMI-0620965 and DGE-0221680. ∗Corresponding author Email address: tarikb@bu.edu (Tarik Borogovac). Preprint submitted to Elsevier 17 November 2018 1 Introduction The purpose of this paper is to present a novel approach for efficient estimation via the Monte Carlo (MC) method. The approach is very broadly applicable but here, to present the main ideas, we narrow the focus to Ensemble Monte Carlo where estimation is based on stochastically independent trajectories of a system. To illustrate, we use simulation of time-dependent nonlinear processes for which Monte Carlo is a particularly general and powerful numerical method compared to available alternatives. Time-dependent nonlinear processes are very general models used, among others, in statistical mechanics [1], data assimilation in climate, weather and ocean modeling [2], financial modeling [3], and quantitative biology [4]. Hence developing efficient MC methods may significantly impact a wide range of applications. A known weakness of MC is its slow rate of convergence. Assume Y is a random quantity defined on paths of a process and let σY denote its standard deviation. The convergence rate of MC for estimating the expected value of Y is ≈σY /√n where n is the number of independent paths of the process. In general the canonical n−1/2 rate of convergence cannot be improved upon, hence, since the inception of the MC method, a number of variance reduction (VR) techniques have been devised to reduce σY (see, [5] for an early account and [3] and [6] for more recent discussions). Most VR techniques lead to estimators of the form w1Y1 + · · · + wnYn, i.e., a weighted average of the samples. These techniques prescribe (i) a recipe for selecting samples Y1, · · · , Yn and (ii) a set of weights w1, · · ·, wn. To arrive at these prescriptions, one must rely on the existence of specific problem features and the ability of the user of the method to discover and effectively exploit such features. This lack of generality has significantly limited the applicability of VR techniques. The point of departure of a new strategy, called DataBase Monte Carlo (DBMC), is to address this shortcoming and to devise generic VR techniques that can be generically applied [7]. All VR techniques bring additional information to bear on the estimation problem, however, as mentioned above, this informa- tion is problem specific and relies on exploiting special features of the prob- lem at hand. By contrast, as will be clarified in this paper, DBMC adds a generic computational exploration phase to the estimation problem that re- lies on gathering information at one (or more) nominal model parameter(s) to achieve estimation efficiency at neighboring parameters. The advantage of this approach is its generality and wide applicability: it is quite easy to im- 2 plement and it can wrap existing ensemble MC codes. On the other hand, the computational exploration phase of the DBMC approach may require ex- tensive simulations and can be computationally costly. Therefore, the initial setup cost needs justification. The setup cost may be justified in projects that involve estimations at many model parameters and

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