Marcin Łoś

Dynamics with Matrices Possessing Kronecker Product Structure.

Abstract In this paper we present an application of Alternating Direction Implicit (ADI) algorithm for solution of non-stationary PDE-s using isogeometric finite element method. We show that ADI algorithm has a linear computational cost at every time step. We illustrate this approach by solving two example non-stationary three-dimensional problems using explicit Euler and Newmark time-stepping scheme: heat equation and linear elasticity equations for a cube. The stability of the simulation is controlled by monitoring the energy of the solution.

Local Misfit Approximation in Memetic Solving of Ill-Posed Inverse Problems

The approximation of the objective function is a well known method of speeding up optimization process, especially if the objective evaluation is costly. This is the case of inverse parametric problems formulated as global optimization ones, in which we recover partial differential equation parameters by minimizing the misfit between its measured and simulated solutions. Typically, the approximation used to build the surrogate objective is rough but globally applicable in the whole admissible domain. The authors try to carry out a different task of detailed misfit approximation in the regions of low sensitivity (plateaus). The proposed complex method consists of independent \(C^0\) Lagrange approximation of the misfit and its gradient, based on the nodes obtained during the dedicated memetic process, and the subsequent projection of the obtained components (single or both) on the space of B-splines. The resulting approximation is globally \(C^1\), which allows us to use fast gradient-based local optimization methods. Another goal attained in this way is the estimation of the shape of plateau as an appropriate level set of the approximated objective. The proposed strategy can be applied for solving ill-conditioned real world inverse problems, e.g., appearing in the oil deposit investigation. We show the results of preliminary tests of the method on two benchmarks featuring convex and non-convex U-shaped plateaus.

An Algorithm for Tensor Product Approximation of Three-Dimensional Material Data for Implicit Dynamics Simulations

In the paper, a heuristic algorithm for tensor product approximation with B-spline basis functions of three-dimensional material data is presented. The algorithm has an application as a preconditioner for implicit dynamics simulations of a non-linear flow in heterogeneous media using alternating directions method. As the simulation use-case, a non-stationary problem of liquid fossil fuels exploration with hydraulic fracturing is considered. Presented algorithm allows to approximate the permeability coefficient function as a tensor product what in turn allows for implicit simulations of the Laplacian term in the partial differential equation. In the consequence the number of time steps of the non-stationary problem can be reduced, while the numerical accuracy is preserved.

Approximating landscape insensitivity regions in solving ill-conditioned inverse problems

Solving ill-posed continuous, global optimization problems is challenging. No well-established methods are available to handle the objective intensity that appears when studying the inversion of non-invasive tumor tissue diagnosis or geophysical applications. The paper presents a complex metaheuristic method that identifies regions of objective function’s insensitivity (plateaus). It is composed of a multi-deme hierarchic memetic strategy coupled with random sample clustering, cluster integration, and a special kind of local evolution processes using the multiwinner selection that allows to breed the demes to cover each plateau separately. The final phase consists in a smooth local objective approximation which determines the shape of the plateaus by analyzing the objective level sets. We test the method on benchmarks with multiple non-convex plateaus and in an actual geophysical application of magnetotelluric data inversion.

Memetic approach for irremediable ill-conditioned parametric inverse problems*

Abstract The paper introduces a new taxonomy of ill-posed parametric inverse problems, formulated as global optimization ones. It systematizes irremediable problems, which appear quite often in the real life but cannot be solved using the regularization method. The paper also shows a new way of solving irremediable inverse problems by a complex memetic approach including: genetic computation with adaptive accuracy, random sample clustering and a sophisticated local approximation of misfit plateau regions. Finally, we use a benchmark function featuring cross-shaped plateau to discuss some factors that influence the quality of plateau shape approximation.

Coupled isogeometric Finite Element Method and Hierarchical Genetic Strategy with balanced accuracy for solving optimization inverse problem

Abstract The liquid fossil fuel reservoir exploitation problem (LFFEP) has not only economical signification but also strong natural environment impact. When the hydraulic fracturing technique is considered from the mathematical point of view it can be formulated as an optimization inverse problem, where we try to find optimal locations of pumps and sinks to maximize the amount of the oil extracted and to minimize the contamination of the groundwater. In the paper, we present combined solver consisting of the Hierarchical Genetic Strategy (HGS) with variable accuracy for solving optimization problem and isogeometric finite element method (IGA-FEM) with different mesh size for modeling a non-stationary flow of the non-linear fluid in heterogeneous media. The algorithm was tested and compared with the strategy using Simple Genetic Algorithm (SGA) as optimization algorithm and the same IGA-FEM solver for solving a direct problem. Additionally, a parallel algorithm for non-stationary simulations with isogeometric L2 projections is discussed and preliminarily assessed for reducing the computational cost of the solvers consisting of genetic algorithm and IGA-FEM algorithm. The theoretical asymptotic analysis which shows the correctness of algorithm and allows to estimate computational costs of the strategy is also presented.

Algorithm for simultaneous adaptation and time step iterations for the electromagnetic waves propagation and heating of the human head induced by cell phone

Abstract In this paper, we propose a parallel algorithm for simultaneous adaptation and time step iterations for the solution of difficult coupled time-dependent problems. In particular, we focus on the problem of propagation of electromagnetic waves over the human head induced by cell phone antenna, coupled with the Pennes bio-heat equations modeling the heating of the human head. Our algorithm allows for utilization of multiple cores for faster solution of the time-dependent difficult problems. Each core is assigned to a single time step. We utilize hp-adaptive algorithm for the iterative solution of both Maxwell and Pennes equations, in every time step. We progress with parallel computations in subsequent time steps, by using the solutions from previous time steps with a given accuracy. At the same time, we increase the accuracy of intermediate time step solutions by performing hp-adaptive computations in parallel, in every time step.1

Parallel space–time hp adaptive discretization scheme for parabolic problems

Abstract We introduce an integration scheme for parabolic problems. Our parallelizable method uses adaptive h p -finite elements in space, and finite differences in time. The strategy can also be combined with a classical finite element method parallelization technique based on domain decomposition. We verified the performance of our method against two different benchmarks, in both two-dimensional (model problem on an L-shaped domain) and three-dimensional (Pennes bioheat equation) settings. Results show a significant speedup in computational time when compared with the sequential version of the algorithm. Moreover, we develop a mathematical framework to analyze similar schemes which include h p spatial adaptivity. Our framework describes error propagation rigorously, and as such allows to analyze convergence properties of these mixed methods.

Applications of Alternating Direction Solver for simulations of time-dependent problems

This paper deals with application of Alternating Direction solver (ADS) to nonstationary linear elasticity problem solved with isogeometric FEM. Employing tensor product B-spline basis in isogeometric analysis under some restrictions leads to system of linear equations with matrix possessing tensor product structure. Alternating Direction Implicit algorithm is a direct method that exploits this structure to solve the system in O (N ), where N is a number of degrees of freedom (basis functions). This is asymptotically faster than state-of-theart general purpose multi-frontal direct solvers. In this paper we also present the complexity analysis of ADS incorporating dependence on order of B-spline basis.