Ronald D. Haynes

A Computationally Stable Approach to Gaussian Process Interpolation of Deterministic Computer Simulation Data

For many expensive deterministic computer simulators, the outputs do not have replication error and the desired metamodel (or statistical emulator) is an interpolator of the observed data. Realizations of Gaussian spatial processes (GP) are commonly used to model such simulator outputs. Fitting a GP model to n data points requires the computation of the inverse and determinant of n×n correlation matrices, R, that are sometimes computationally unstable due to near-singularity of R. This happens if any pair of design points are very close together in the input space. The popular approach to overcome near-singularity is to introduce a small nugget (or jitter) parameter in the model that is estimated along with other model parameters. The inclusion of a nugget in the model often causes unnecessary over-smoothing of the data. In this article, we propose a lower bound on the nugget that minimizes the over-smoothing and an iterative regularization approach to construct a predictor that further improves the interpolation accuracy. We also show that the proposed predictor converges to the GP interpolator.

Simultaneous and sequential approaches to joint optimization of well placement and control

Determining optimal well placement and control is essential to maximizing production from an oil field. Most academic literature to date has treated optimal placement and control as two separate problems; well placement problems, in particular, are often solved assuming some fixed flow rate or bottom-hole pressure at injection and production wells. Optimal placement of wells, however, does depend on the control strategy being employed. Determining a truly optimal configuration of wells thus requires that the control parameters be allowed to vary as well. This presents a challenging optimization problem, since well location and control parameters have different properties from one another. In this paper, we address the placement and control optimization problem jointly using approaches that combine a global search strategy (particle swarm optimization, or PSO) with a local generalized pattern search (GPS) strategy. Using PSO promotes a full, semi-random exploration of the search space, while GPS allows us to locally optimize parameters in a systematic way. We focus primarily on two approaches combining these two algorithms. The first is to hybridize them into a single algorithm that acts on all variables simultaneously, while the second is to apply them sequentially to decoupled well placement and well control problems. We find that although the best method for a given problem is context-specific, decoupling the problem may provide benefits over a fully simultaneous approach.

Joint optimization of well placement and control for nonconventional well types

Abstract Optimal well placement and optimal well control are two important areas of study in oilfield development. Although the two problems differ in several respects, both are important considerations in optimizing total oilfield production, and so recent work in the field has considered the problem of addressing both problems jointly. Two general approaches to addressing the joint problem are a simultaneous approach, where all parameters are optimized at the same time, or a sequential approach, where a distinction between placement and control parameters is maintained by separating the optimization problem into two (or more) stages, some of which consider only a subset of the total number of variables. This latter approach divides the problem into smaller ones which are easier to solve, but may not explore search space as fully as a simultaneous approach. In this paper we combine a stochastic global algorithm (particle swarm optimization) and a local search (mesh adaptive direct search) to compare several simultaneous and sequential approaches to the joint placement and control problem. In particular, we study how increasing the complexity of well models (requiring more variables to describe the well׳s location and path) affects the respective performances of the two approaches. The results of several experiments with synthetic reservoir models suggest that the sequential approaches are better able to deal with increasingly complex well parameterizations than the simultaneous approaches.

A Parallel Space-Time Algorithm

With the continued evolution of computing architectures towards many-core computing, algorithms that can effectively and efficiently use many cores are crucial. In this paper, we propose, as a proof of principle, a parallel space-time algorithm that layers time parallelization together with a parallel elliptic solver to solve time dependent partial differential equations (PDEs). The parallel elliptic solver utilizes domain decomposition to divide a spatial grid into subdomains, and applies a parallel Schwarz iteration to find consistent solutions. The high-order parallel time integrator employed belongs to the family of revisionist integral deferred correction methods (RIDC) introduced by Christlieb, Macdonald, and Ong [SIAM J. Sci. Comput., 32 (2010), pp. 818--835], which allows for the small scale parallelization of solutions to initial value problems. The two established algorithms are combined in this proposed space-time algorithm to add parallel scalability. As a proof of concept, we utilize a framew...

Domain Decomposition Approaches for Mesh Generation via the Equidistribution Principle

Moving mesh methods based on the equidistribution principle are powerful techniques for the space-time adaptive solution of evolution problems. Solving the resulting coupled system of equations, namely the original PDE and the mesh PDE, however, is challenging in parallel. We propose in this paper several Schwarz domain decomposition algorithms for this task. We then study in detail the convergence properties of these algorithms applied to the nonlinear mesh PDE in one spatial dimension. We prove convergence for classical transmission conditions, and optimal and optimized variants for the generation of steady equidistributing grids. A classical, parallel, Schwarz algorithm is presented and analyzed for the generation of time dependent (moving) equidistributing grids. We conclude our study with numerical experiments.

A Schwarz Waveform Moving Mesh Method

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A multilevel coordinate search algorithm for well placement, control and joint optimization

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A Closer Look At Differential Evolution For The Optimal Well Placement Problem

-refinement (moving mesh) method is considered for solving time dependent partial differential equations (PDEs). The resulting coupled system, consisting of the physical PDE and a moving mesh PDE, is solved by a Schwarz waveform relaxation method. In particular, the computational space-time domain is decomposed into overlapping subdomains and the solution obtained by iteratively solving the system of PDEs on each subdomain. Dirichlet boundary conditions are used to pass solution information between neighboring regions. The efficacy of this approach is demonstrated for some model problems. For problems where the solutions evolve on disparate time scales in different regions of the spatial domain, this approach demonstrates the significant savings in computational time and effort which are possible.

MPI–OpenMP Algorithms for the Parallel Space–Time Solution of Time Dependent PDEs

Determining optimal well placements and controls are two important tasks in oil field development. These problems are computationally expensive, nonconvex, and contain multiple optima. The practical solution of these problems require efficient and robust algorithms. In this paper, the multilevel coordinate search (MCS) algorithm is applied for well placement and control optimization problems. MCS is a derivative-free algorithm that combines global and local search. Both synthetic and real oil fields are considered. The performance of MCS is compared to generalized pattern search (GPS), particle swarm optimization (PSO), and covariance matrix adaptive evolution strategy (CMA-ES) algorithms. Results show that the MCS algorithm is strongly competitive, and outperforms for the joint optimization problem and with a limited computational budget. The effect of parameter settings for MCS are compared for the test examples. For the joint optimization problem we compare the performance of the simultaneous and sequential procedures and show the utility of the latter.

Parallel stochastic methods for PDE based grid generation

Energy demand has increased considerably with the growth of world population, increasing the interest in the hydrocarbon reservoir management problem. Companies are concerned with maximizing oil recovery while minimizing capital investment and operational costs. A first step in solving this problem is to consider optimal well placement. In this work, we investigate the Differential Evolution (DE) optimization method, using distinct configurations with respect to population size, mutation factor, crossover probability, and mutation strategy, to solve the well placement problem. By assuming a bare control procedure, one optimizes the parameters representing positions of injection and production wells. The Tenth SPE Comparative Solution Project and MATLAB Reservoir Simulation Toolbox (MRST) are the benchmark dataset and simulator used, respectively. The goal is to evaluate the performance of DE in solving this important real-world problem. We show that DE can find high-quality solutions, when compared with a reference from the literature, and a preliminary analysis on the results of multiple experiments gives useful information on how DE configuration impacts its performance.