A hybrid method for inversion of 3D DC resistivity logging measurements

Abstract

This paper focuses on the application of hp hierarchic genetic strategy (hp–HGS) for solution of a challenging problem, the inversion of 3D direct current (DC) resistivity logging measurements. The problem under consideration has been formulated as the global optimization one, for which the objective function (misfit between computed and reference data) exhibits multiple minima. In this paper, we consider the extension of the hp–HGS strategy, namely we couple the hp–HGS algorithm with a gradient based optimization method for a local search. Forward simulations are performed with a self-adaptive hp finite element method, hp–FEM. The computational cost of misfit evaluation by hp–FEM depends strongly on the assumed accuracy. This accuracy is adapted to the tree of populations generated by the hp–HGS algorithm, which makes the global phase significantly cheaper. Moreover, tree structure of demes as well as branch reduction and conditional sprouting mechanism reduces the number of expensive local searches up to the number of minima to be recognized. The common (direct and inverse) accuracy control, crucial for the hp–HGS efficiency, has been motivated by precise mathematical considerations. Numerical results demonstrate the suitability of the proposed method for the inversion of 3D DC resistivity logging measurements.

An agent-oriented hierarchic strategy for solving inverse problems

“…Then, we can adapt the solver accuracy to the assumed accuracy of HMS tree demes; see Algorithm 5. Finally, we know the dependency between the solver accuracy and the computational cost of the direct problem solution (for details, see Barabasz et al, 2014;Gajda-Zagórska et al, 2015) compute new value of e rel 7: end while 8: return objective computed by means of final mesh policy (e.g., a diffusion-based one (Grochowski et al, 2006)). …”

A multi-objective memetic inverse solver reinforced by local optimization methods

“…the misfit is multimodal and has many global minimizers) and/or the misfit is insensitive with respect to several decision variables in the neighborhood of the global minimizer, the complex stochastic searches (see e.g. [3]) allow to overcome the above difficulties when solving them.…”

“…For the MT problem it was proven in [22,Remark 1]. Appropriate versions for other inverse problems can be found in papers [29,3,30]. Furthermore, in a few important cases, we can determine the dependency between the solver accuracy and the computational cost of the forward problem solution (cf.…”

“…Furthermore, in a few important cases, we can determine the dependency between the solver accuracy and the computational cost of the forward problem solution (cf. [29,3]), which is the main unit term of the overall HMS computational cost. Hence, by modulating the deme accuracy, we can optimize this overall cost.…”

On the Computational Cost and Complexity of Stochastic Inverse Solvers

“…Among metaheuristic search strategies, we highlight evolutionary algorithms (EAs; we point the reader to the works of Burczyński and Beluch [10], Burczyński et al [11], and Meruane and Heylen [36] for some examples of using EAs to solve inverse problems). Our particular interest is devoted to EAs that can handle misfit multimodality, such as niching and sequential niching strategies (see, e.g., the work of Mahfoud [35]), Hierarchic Genetic Strategy HGS [5,7,24,29,30,62], and adaptive, stochastic multi-start method (see, e.g., the work of Telega et al [76]). …”