Maciej Smołka

An Existence Theorem for Wave-Type Hyperbolic Hemivariational Inequalities

In this paper we prove the existence of solutions for a hyperbolic hemivariationalinequality of the form u″ + Bu + ∂j (u) ∋ f where B is a linear elliptic operator and ∂j is the Clarke subdifferential of a locally Lipschitz function j. Our result is based on the parabolic regularization method.

A hybrid algorithm for solving inverse problems in elasticity

Abstract 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.

An Agent-Oriented Hierarchic Strategy for Solving Inverse Problems

Abstract The paper discusses the complex, agent-oriented hierarchic memetic strategy (HMS) dedicated to solving inverse parametric problems. The strategy goes beyond the idea of two-phase global optimization algorithms. The global search performed by a tree of dependent demes is dynamically alternated with local, steepest descent searches. The strategy offers exceptionally low computational costs, mainly because the direct solver accuracy (performed by the hp-adaptive finite element method) is dynamically adjusted for each inverse search step. The computational cost is further decreased by the strategy employed for solution inter-processing and fitness deterioration. The HMS efficiency is compared with the results of a standard evolutionary technique, as well as with the multi-start strategy on benchmarks that exhibit typical inverse problems’ difficulties. Finally, an HMS application to a real-life engineering problem leading to the identification of oil deposits by inverting magnetotelluric measurements is presented. The HMS applicability to the inversion of magnetotelluric data is also mathematically verified.

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

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 island model as a Markov dynamic system

Parallel multi-deme genetic algorithms are especially advantageous because they allow reducing the time of computations and can perform a much broader search than single-population ones. However, their formal analysis does not seem to have been studied exhaustively enough. In this paper we propose a mathematical framework describing a wide class of island-like strategies as a stationary Markov chain. Our approach uses extensively the modeling principles introduced by Vose, Rudolph and their collaborators. An original and crucial feature of the framework we propose is the mechanism of inter-deme agent operation synchronization. It is important from both a practical and a theoretical point of view. We show that under a mild assumption the resulting Markov chain is ergodic and the sequence of the related sampling measures converges to some invariant measure. The asymptotic guarantee of success is also obtained as a simple issue of ergodicity. Moreover, if the cardinality of each island population grows to infinity, then the sequence of the limit invariant measures contains a weakly convergent subsequence. The formal description of the island model obtained for the case of solving a single-objective problem can also be extended to the multi-objective case

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

HighlightsHybrid strategy hp-HGS for solving inverse problems.Global step solved by Hierarchical Genetic Search (HGS) coupled with self-adaptive hp-Finite Element Method (hp-FEM).Local search performed with Broyden-Fletcher-Goldfar-Shanno (BFGS).hp-HGS algorithm adapts the length of the genetic code and the accuracy of the direct solver.Application to 3D AC resistivity logging measurement simulations in deviated wells. In this paper, we propose the use of a hybrid algorithm for the inversion of 3D Alternate Current (AC) resistivity logging measurements. The forward problem is solved using a goal-oriented self-adaptive hp-Finite Element Method (hp-FEM) that provides exponential convergence of the numerical error with respect to the mesh size. The inverse problem is solved using a Hierarchical Genetic Search (HGS) coupled with a Broyden-Fletcher-Goldfar-Shanno (BFGS) method. Individuals from the genetic populations represent the resistivity of the formation layers. The fitness function is estimated based on hp-FEM results. The hybrid method controls the accuracy of evaluation of particular individuals, as well as the accuracy of the genetic coding. After finding those regions where the fitness function has small values, the local search method by means of BFGS algorithm is executed. The paper is concluded with numerical results for the hybrid algorithm.

Existence of solutions for wave-type hemivariational inequalities with noncoercive viscosity damping

Abstract In this paper we prove the existence of solutions for a hyperbolic hemivariational inequality of the form u″+Au′+Bu+∂j(u)∋f, where B is a linear elliptic operator and A is linear and nonnegative (not necessarily coercive).

Multiwinner Voting in Genetic Algorithms

Genetic algorithms are a group of powerful tools for solving ill-posed global optimization problems in continuous domains. When insensitivity in the fitness function is an obstacle, the most desired feature of a genetic algorithm is its ability to explore plateaus of the fitness function surrounding its minimizers. The authors suggest a way of maintaining diversity of the population in the plateau regions based on a new approach for selection according to the theory of multiwinner elections among autonomous agents. The article delivers a detailed description of the new selection algorithm, computational experiments that put the choice of the proper multiwinner rule to use, and a preliminary experiment showing the proposed algorithms effectiveness in exploring a fitness functions plateau.

Evolutionary multiobjective optimization algorithm as a Markov system

In the paper we consider the ranking given by the Pareto dominance relation as a basis to create a selection operator for the Evolutionary Multiobjective Optimization Algorithm (EMOA). Assuming that sampling to the next epoch is performed according to the generalized Bernoulli schema with regard to a selected type of the rank selection, a heuristic operator for EMOA is introduced. Having defined the heuristic operator, the transition probability matrix of the uniform Markov chain modeling EMOA can be explicitly obtained as in the Voses theory of the Simple Genetic Algorithm (SGA). This chain is ergodic if the mixing operator following the EMOA selection operator in each epoch is strictly positive. Moreover, we show that the measure on the space of populations imposed by the EMOA infinite population concentrates on the set of fixed points of the heuristic operator after infinite number of epochs, assuming that the heuristic operator is focusing.

The dynamics of computing agent systems

The paper presents the Multi Agent System (MAS) designed for the large scale parallel computations. The special kind of diffusion-based scheduling enables to decompose and allocate the migrable computing agents basing only of the local information. The paper introduces the formal model of the MAS under consideration in order to depict the roles of agent behavior and the whole system dynamics. The optimal scheduling problem for MAS as well as the way of its verification was presented in terms of such model. The brief report of the test results is stressed in the section 6.