David Echeverria

A trust-region strategy for manifold-mapping optimization

Studying the space-mapping technique by Bandler et al. [J. Bandler, R. Biernacki, S. Chen, P. Grobelny, R.H. Hemmers, Space mapping technique for electromagnetic optimization, IEEE Trans. Microwave Theory Tech. 42 (1994) 2536-2544] for the solution of optimization problems, we observe the possible difference between the solution of the optimization problem and the computed space-mapping solution. We repair this discrepancy by exploiting the correspondence with defect-correction iteration and we construct the manifold-mapping algorithm, which is as efficient as the space-mapping algorithm but converges to the exact solution. To increase the robustness of the algorithm we introduce a trust-region strategy (a regularization technique) based on the generalized singular value decomposition of the linearized fine and coarse manifold representations. The effect of this strategy is shown by the solution of a variety of small non-linear least squares problems. Finally we show the use of the technique for a more challenging engineering problem.

Optimisation in electromagnetics with the space-mapping technique

Purpose – Optimisation in electromagnetics, based on finite element models, is often very time‐consuming. In this paper, we present the space‐mapping (SM) technique which aims at speeding up such procedures by exploiting auxiliary models that are less accurate but much cheaper to compute.Design/methodology/approach – The key element in this technique is the SM function. Its purpose is to relate the two models. The SM function, combined with the low accuracy model, makes a surrogate model that can be optimised more efficiently.Findings – By two examples we show that the SM technique is effective. Further we show how the choice of the low accuracy model can influence the acceleration process. On one hand, taking into account more essential features of the problem helps speeding up the whole procedure. On the other hand, extremely simple auxiliary models can already yield a significant acceleration.Research limitations/implications – Obtaining the low accuracy model is not always straightforward. Some resear...

Two new variants of the manifold‐mapping technique

htmlabstractManifold-mapping is an efficient surrogate-based optimization technique aimed at the acceleration of very time-consuming design problems. In this paper we present two new variants of the original algorithm that make it applicable to a broader range of optimization scenarios. The first variant is useful when the optimization constraints are expressed by means of functions that are very expensive to compute. The second variant endows the original scheme with a trust-region strategy and the result is a much more robust algorithm. By two practical design problems from electromagnetics we eventually show that the proposed variants perform efficiently.

Robust scheme for inversion of seismic and production data for reservoir facies modeling

Summary This research aims at making optimal updates of geological models by jointly inverting flow and seismic data while honoring the geologic spatial continuity. Numerical models for reservoir characterization are increasing in complexity, due in part to the greater need to model the complex spatial heterogeneity and fluid flow in the subsurface. These models, once properly calibrated, can make better forecasts. This calibration process requires in essence the solving of an inverse problem. The inversion problem is formulated as mi nimizing the mismatch function between observations and the output of the numerical models. The optimal search is carried out by adjusting model parameters, typically one or more for each grid-point of the reservoir. The optimization problem is large-scale in nature, with a nonlinear and nonconvex objective function, that often involves time-expensive simulations. Additionally, this problem is generally ill-conditioned, because the number of degrees of freedom usually is larger than the number of observations. We present a robust and fairly efficient methodology to deal with these difficulties in the framework of oil reservoir characterization. The illconditioned character of the optimal search can be attenuated in two ways. By Principal Component Analysis (PCA) the search space can be projected to a subspace of much smaller dimension, while keeping consistency with prior spatial geological features already known for the reservoir. The number of optimal solutions can be reduced further by increasing the diversity of the data observed. We integrate two different types of data: time-lapse seismic (spatially distributed and of lower temporal periodicity) and production data (localized around wells and of high temporal periodicity). Production data provides an integrated response of the reservoir to fluid flow, while time-lapse seismic data yields a spatially distributed characterization of the changes in elastic velocities due to saturation and pressure variations. The reduction in the number of optimization variables by PCA allows the use of numerical derivatives of the cost function. Within a distributed computing framework these approximate derivatives can be calculated efficiently. We also consider derivative-free algorithms. We illustrate the methodology on a sector of the Stanford VI synthetic reservoir created for testing algorithms.

Manifold Mapping for Multilevel Optimization

We first show the idea behind a space-mapping iteration technique for the effi- cient solution of optimization problems. Then we show how space-mapping optimization can be understood in the framework of defect correction. We observe a difference between the solution of the optimization problem and the computed space-mapping solutions. We repair this discrepancy by exploiting the correspondence with defect correction iteration and we construct the manifold-mapping algorithm, which is as efficient as the space-mapping algorithm but converges to the accurate solution.