Improved Estimation of the Stochastic Gradient with Quasi-Monte Carlo Methods

Sarma, P. V. L. N.; Chen, W.H.

doi:10.3997/2214-4609.20141779

Cited by 6 publications

(5 citation statements)

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“…In particular, there is no need to assume that they are generated from a Gaussian distribution. In fact, they could be generated by a Sobol sequence as recently suggested in . Then to obtain smoothing, one could multiply ∇ u ,sto J ( m k , u ℓ ) defined in Eq.…”

Section: Comments On Stochastic Simplex Approximate Gradientmentioning

confidence: 99%

A Stochastic Simplex Approximate Gradient (StoSAG) for optimization under uncertainty

Fonseca

Chen

Jansen

et al. 2016

Numerical Meth Engineering

146

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SUMMARYWe consider a technique to estimate an approximate gradient using an ensemble of randomly chosen control vectors, known as Ensemble Optimization (EnOpt) in the oil and gas reservoir simulation community. In particular, we address how to obtain accurate approximate gradients when the underlying numerical models contain uncertain parameters because of geological uncertainties. In that case, 'robust optimization' is performed by optimizing the expected value of the objective function over an ensemble of geological models. In earlier publications, based on the pioneering work of Chen et al. (2009), it has been suggested that a straightforward one-to-one combination of random control vectors and random geological models is capable of generating sufficiently accurate approximate gradients. However, this form of EnOpt does not always yield satisfactory results. In a recent article, formulate a modified EnOpt algorithm, referred to here as a Stochastic Simplex Approximate Gradient (StoSAG; in earlier publications referred to as 'modified robust EnOpt') and show, via computational experiments, that StoSAG generally yields significantly better gradient approximations than the standard EnOpt algorithm. Here, we provide theoretical arguments to show why StoSAG is superior to EnOpt.

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Section: Comments On Stochastic Simplex Approximate Gradientmentioning

confidence: 99%

A Stochastic Simplex Approximate Gradient (StoSAG) for optimization under uncertainty

Fonseca

Chen

Jansen

et al. 2016

Numerical Meth Engineering

146

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show abstract

“…The effect of the covariance matrix has been investigated recently in Fonseca et al (2013) and a theoretical foundation for the use of a varying covariance matrix has been provided in Stordal et al (2014). Sarma and Chen (2014) have investigated the applicability of different sampling techniques to improve the quality of a gradient estimate. However none of those studies have performed a detailed investigation into the effect of ensemble size on the estimated ensemble gradient quality.…”

Section: Introductionmentioning

confidence: 99%

Quantification of the Impact of Ensemble Size on the Quality of an Ensemble Gradient Using Principles of Hypothesis Testing

Fonseca

Kahrobaei

Gastel

et al. 2015

SPE Reservoir Simulation Symposium

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With an increase in the number of applications of ensemble optimization (EnOpt) for production optimization, the theoretical understanding of the gradient quality has received little attention. An important factor that influences the quality of the gradient estimate is the number of samples. In this study we use principles from statistical hypothesis testing to quantify the number of samples needed to estimate an ensemble gradient that is comparable in quality to an accurate adjoint gradient. We develop a methodology to estimate the necessary ensemble size to obtain an approximate gradient that is within a predefined angle compared to the adjoint gradient, with a predefined statistical confidence. The method is first applied to the Rosenbrock function (a standard optimization test problem), for a single realization, and subsequently for a case with uncertainty, represented by multiple realizations (robust optimization). The maximum allowed error applied in both experiments is a 10° angle between the directions of the EnOpt gradient and the exact gradient. For the single-realization case we need, depending on the perturbation size, 900, 5 and 3 samples to estimate a "good" gradient with 95% confidence at 50 points in the optimization space for 50 different random sequences. For the robust case, the conventional EnOpt approach is to couple one model realization with one control sample, which leads to a computationally efficient technique to estimate a mean gradient. However, our results show that in order to be 95% confident the original one-to-one model realization to control sample ratio formulation is not sufficient. To achieve the required confidence requires a ratio of 1:1100, i.e. each model realization is paired with 1100 control samples using the original formulation. However, using a modified formulation we need a ratio of 1:10 to stay within the maximum allowed error for 95% of the points in space, though a 1:1 ratio is sufficient for 85% of the points. We also tested our methodology on a reservoir case for deterministic and robust cases, where we observe similar trends in the results. Our results provide insight into the necessary number of samples required for EnOpt, in particular for robust optimization, to achieve a gradient comparable to an adjoint gradient.

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“…A method called CMA-EnOpt to adaptively adjust the perturbation size through covariance matrix adaptation (CMA) was found to improve the performance of ensemble gradients [13]. The impact of alternative distributions was considered by Sarma and Chen [31] who investigated the impact of a quasi-random sampling method (Sobol sampling, [25]) that avoids clustering of samples on SNR gradient estimates. They found Sobol sampling to lead to a faster rate of convergence relative to Gaussian sampling when applied to a deterministic reservoir optimization problem.…”

Section: Introductionmentioning

confidence: 99%

Improved sampling strategies for ensemble-based optimization

et al. 2020

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We are concerned with the efficiency of stochastic gradient estimation methods for large-scale nonlinear optimization in the presence of uncertainty. These methods aim to estimate an approximate gradient from a limited number of random input vector samples and corresponding objective function values. Ensemble methods usually employ Gaussian sampling to generate the input samples. It is known from the optimal design theory that the quality of sample-based approximations is affected by the distribution of the samples. We therefore evaluate six different sampling strategies to optimization of a high-dimensional analytical benchmark optimization problem, and, in a second example, to optimization of oil reservoir management strategies with and without geological uncertainty. The effectiveness of the sampling strategies is analyzed based on the quality of the estimated gradient, the final objective function value, the rate of the convergence, and the robustness of the gradient estimate. Based on the results, an improved version of the stochastic simplex approximate gradient method is proposed based on UE(s 2) sampling designs for supersaturated cases that outperforms all alternative approaches. We additionally introduce two new strategies that outperform the UE(s 2) designs previously suggested in the literature.

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Improved Estimation of the Stochastic Gradient with Quasi-Monte Carlo Methods

Cited by 6 publications

References 0 publications

A Stochastic Simplex Approximate Gradient (StoSAG) for optimization under uncertainty

A Stochastic Simplex Approximate Gradient (StoSAG) for optimization under uncertainty

Quantification of the Impact of Ensemble Size on the Quality of an Ensemble Gradient Using Principles of Hypothesis Testing

Improved sampling strategies for ensemble-based optimization

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