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.
The paper offers a new approach to handling difficult parametric inverse problems in elasticity and thermo-elasticity, formulated as global optimization ones. The proposed strategy is composed of two phases. In the first, global phase, the stochastic hp-HGS algorithm recognizes the basins of attraction of various objective minima. In the second phase, the local objective minimizers are closer approached by steepest descent processes executed singly in each basin of attraction. The proposed complex strategy is especially dedicated to ill-posed problems with multimodal objective functionals. The strategy offers comparatively low computational and memory costs resulting from a double-adaptive technique in both forward and inverse problem domains. We provide a result on the Lipschitz continuity of the objective functional composed of the elastic energy and the boundary displacement misfits with respect to the unknown constitutive parameters. It allows common scaling of the accuracy of solving forward and inverse problems, which is the core of the introduced double-adaptive technique. The capability of the proposed method of finding multiple solutions is illustrated by a computational example which consists in restoring all feasible Young modulus distributions minimizing an objective functional in a 3D domain of a photo polymer template obtained during step and flash imprint lithography.Keywords: inverse problem, hierarchic genetic strategy, hybrid optimization, automatic hp-adaptive finite element method. MotivationConsiderable progress has been achieved recently in the field of inverse problems and, as a result, this area is one of the fastest growing domains in applied mathematics and computer science. It is concerned with problems that consist in finding an unknown property of a medium, or an object, from the observation of a response of this medium, or object, to a probing signal. A general framework for inverse problems provides analytical means of estimating constants in mathematical models given appropriate measurements, building mathematical models, and giving insight into the design of experiments (see, for instance, the works of Banks and Kunisch (1989), Isakov (2006), Tarantola (2005) and the references therein). Typically, inverse problems lead to mathematical models which are not well-posed in the sense of Hadamard, i.e., their solution might not be unique and/or might be unstable under data perturbations, and therefore they pose severe numerical difficulties.The growth of the area of inverse problems has largely been driven by the needs of applications both in sciences and in industry. We refer to Engl et al. (2000), Samarski and Vabishchevich (2007) or Tikhonov et al. (1995) for a description of inverse problems of different types, including inverse problems in which the equation is not specified completely as some equation coefficients are unknown, boundary inverse problems in which boundary conditions are unknown, and evolutionary inverse problems in which initial conditions are unknown.In this p...
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