John Jakeman | Sandia National Laboratories (original) (raw)

Papers by John Jakeman

International Journal for Uncertainty Quantification

Environmental Modelling & Software

Journal of Machine Learning for Modeling and Computing

International Journal for Uncertainty Quantification

Environmental Modelling & Software

SIAM/ASA Journal on Uncertainty Quantification

Journal of Computational Physics

SIAM Journal on Scientific Computing

Siam Journal on Scientific Computing, Jan 8, 2015

Geoscience Australia, in an open collaboration with the Mathematical Sciences Institute, The Aust... more Geoscience Australia, in an open collaboration with the Mathematical Sciences Institute, The Australian National University, is developing a software ap-plication, ANUGA, to model the hydrodynamics of floods, storm surges and tsunamis. The free source software implements a finite volume central-upwind Godunov method to solve the non-linear depth-averaged shallow water wave equations. In light of the renewed interest in tsunami forecasting and mitigation, this paper explores the use of ANUGA to model the inundation of the Indian Ocean tsunami of December 2004. The Method of Splitting Tsunamis (MOST) was used to simulate the initial tsunami source and the tsunami's propagation at depths greater than 100m. The resulting output was used to provide boundary conditions to the ANUGA model in the shallow water. Data with respect to 4-minute bathymetry, 2-minute bathymetry, 3-arc second bathymetry and elevation were used in the open ocean, shallow water and on land, respectively. A parti...

International Journal for Uncertainty Quantification

Environmental Modelling & Software

Journal of Machine Learning for Modeling and Computing

International Journal for Uncertainty Quantification

Environmental Modelling & Software

SIAM/ASA Journal on Uncertainty Quantification

Journal of Computational Physics

SIAM Journal on Scientific Computing

Siam Journal on Scientific Computing, Jan 8, 2015

ArXiv, 2018

We describe and analyze a variance reduction approach for Monte Carlo (MC) sampling that accelera... more We describe and analyze a variance reduction approach for Monte Carlo (MC) sampling that accelerates the estimation of statistics of computationally expensive simulation models using an ensemble of models with lower cost. These lower cost models-which are typically lower fidelity with unknown statistics-are used to reduce the variance in statistical estimators relative to a MC estimator with equivalent cost. We derive the conditions under which our proposed approximate control variate framework recovers existing multi-model variance reduction schemes as special cases. We demonstrate that these existing strategies use recursive sampling strategies, and as a result, their maximum possible variance reduction is limited to that of a control variate algorithm that uses only a single low-fidelity model with known mean. This theoretical result holds regardless of the number of low-fidelity models and/or samples used to build the estimator. We then derive new sampling strategies within our framework that circumvent this limitation to make efficient use of all available information sources. In particular, we demonstrate that a significant gap can exist, of orders of magnitude in some cases, between the variance reduction achievable by using a single low-fidelity model and our non-recursive approach. We also present initial sample allocation approaches for exploiting this gap. They yield the greatest benefit when augmenting the high-fidelity model evaluations is impractical because, for instance, they arise from a legacy database. Several analytic examples and an example with a hyperbolic PDE describing elastic wave propagation in heterogeneous media are used to illustrate the main features of the methodology. 1. Introduction. Numerical evaluation of integrals is a foundational aspect of mathematics that has impact on diverse areas such as finance, uncertainty quantification, stochastic programming, and many others. Monte Carlo (MC) sampling is arguably the most robust means of estimating such integrals and can be easily applied to arbitrary integration domains and measures. The MC estimate of an integral is unbiased, and its rate of convergence is independent of the number of variables and the smoothness of the integrand. Nevertheless, obtaining a moderately accurate estimate of an integral with MC is computationally intractable for integrands that are expensive to evaluate, e.g., those arising from a high-fidelity simulation. This intractability arises because the variance of a MC estimator is proportional to the ratio of the variance of the integrand and inversely proportional to the number of samples used. As such, techniques that retain the benefits of MC estimation while reducing its variance are important for extending the applicability of these sampling-based approaches. Control variates (CV) are a class of such techniques that have a long history of reducing MC variance by introducing additional estimators that are correlated with the MC estimator [15, 17, 16, 13]. The use of CV methods has recently seen a resurgence for uncertainty quantification (UQ) problems where the integrands are computationally expensive to evaluate. In these cases, CV approaches can leverage multiple simulation models to accelerate the convergence of statistics for both forward [20, 11, 28, 4, 8] and inverse [1] UQ. These additional simulation models arise from either different sets of equations (i.e., the multifidelity case of differing model forms) and/or from varying temporal and spatial discretizations (i.e., the multilevel case of differing numerical resolutions for the same set of equations). The model ensemble could include reduced-order models [29], dimension-reduction or surrogate models [25] (e.g., active subspace approximations), and even data from physical experiments [14]. Multiple conceptual dimensions can exist within a modeling hierarchy , leading to multi-index constructions [12] in the case of independent resolution controls. Finally, both multi-physics and multi-scale simulations can contribute additional combinatorial richness to the associated modeling ensemble. Traditional CV methods [15] require explicit knowledge of the statistics (for instance the expected value) of their approximate information sources. However, these estimates are frequently unavailable a priori in the UQ simulation-based context. Consequently, CV methods must be modified to balance the computational cost of evaluating lower fidelity models and the reduction in error that they each provide. There exist several strategies that explicitely pursue the goal of estimating the unknown expected values [31, 5, 26, 21] within a control variate framework; however the analysis of these approaches is limited to the case of a single control variate only. As a result, they do not consider how ensembles of low-fidelity information sources could be