Abstract: The objective of traditional goal-oriented strategies is to construct an optimal mesh that minimizes the problem size needed to achieve a user prescribed tolerance error for a given quantity of interest (QoI). Typical geophysical resistivity measurement acquisition systems can easily record electromagnetic (EM) fields. However, depending upon the application, EM fields are sometimes loosely related to the quantity that is to be inverted (conductivity or resistivity), and therefor they become inadequate for inversion. In the present work, we study the impact of the selection of the QoI in our inverse problem. We focus on two different acquisition systems: marine controlled source electromagnetic (CSEM), and magnetotellurics (MT). For both applications, numerical results illustrate the benefits of employing adequate QoI. Specifically, the use as QoI of the impedance matrix on MT measurements provides huge computational savings, since one can replace the existing absorbing boundary conditions (BCs) by a homogeneous Dirichlet BC to truncate the computational domain, something that is not possible when considering EM fields as QoI.

Abstract: We propose a multi-objective approach for solving challenging inverse parametric problems. The objectives are misfits for several physical descriptions of a phenomenon under consideration, whereas their domain is a common set of admissible parameters. The resulting Pareto set, or parameters close to it, constitute various alternatives of minimizing individual misfits. A special type of selection applied to the memetic solution of the multi-objective problem narrows the set of alternatives to the ones that are sufficiently coherent. The proposed strategy is exemplified by solving a real-world engineering problem consisting of the magnetotelluric measurement inversion that leads to identification of oil deposits located about 3 km under the Earth's surface, where two misfit functions are related to distinct frequencies of the electric and magnetic waves.

Abstract: In this paper we present the optimization of the energy consumption for the multi-frontal solver algorithm executed over two dimensional grids with point singularities. The multi-frontal solver algorithm is controlled by so-called elimination tree, defining the order of elimination of rows from particular frontal matrices, as well as order of memory transfers for Schur complement matrices. For a given mesh there are many possible elimination trees resulting in different number of floating point operations (FLOPs) of the solver or different amount of data transferred via memory transfers. In this paper we utilize the dynamic programming optimization procedure and we compare elimination trees optimized with respect to FLOPs with elimination trees optimized with respect to energy consumption.

Abstract: In this paper we analyze the optimality of the volume and neighbors algorithm constructing elimination trees for three dimensional h adaptive finite element method codes. The algorithm is a greedy algorithm that constructs the elimination trees based on the bottom up analysis of the computational mesh. We compare the results of the volume and neighbors greedy algorithm with the global dynamic programming optimization performed on a class of elimination trees. The comparison is based on the Directed Acyclic Graph (DAG) constructed for model grids. We construct DAGs for a two model grids, two dimensional grid with point singularity and two dimensional grid with edge singularity. We show that the quasi-optimal trees created by the volume and neighbors algorithm for considered grids are also captured by the dynamic programming procedure. It means that created elimination trees are optimal in the considered class of elimination trees. We show that different ordering of elements at the input of the volume and neighbors algorithm results in different computational costs of the multi-frontal solver algorithm executed over the resulting elimination trees. Finally we present the ordering of elements that results in optimal (in the considered class) elimination trees. The theoretical results are verified with numerical experiments performed on a three dimensional grids with point, edge and face singularities.

Abstract: In this paper we present a new integration scheme that can be applied for solution of dicult non-stationary problems. The scheme results from linearization of the Cranck-Nicolson scheme that is unconditionally stable but needs to solve non-linear equation at each time step. We test our linearized time integration scheme on the challenging Cahn-Hilliard equations, modeling the separation of two phase fluids. The problem is solved using higher order isogeometric fintie element method with B-spline basis functions. We implement our linear scheme in PETIGA framework interfaced via PETSc toolkit. We utilize a GMRES iterative solver for solution of a linear system at every time step. We also define a simple time adaptivity scheme, which increases the time step size when number of GMRES iterations is less than 30. We compare our linear scheme with simple time adaptation algorithm with non-linear scheme with sophisticated time adaptivity, on the two dimensional Cahn-Hilliard equations. We control the stability of our simulations by monitoring the Ginzberg-Landau free energy functional. We conclude that our simple scheme with simple time adaptivitiy outperforms the non-linear one with advanced time adaptivity by means of the execution time, while providing similar history of the evolution of the free energy functional.