Dimensionally adaptive hp-finite element simulation and inversion of 2D magnetotelluric measurements

Magnetotelluric (MT) problems often contain different subdomains where the conductivity of the media depends upon one, two, or three spatial variables. Traditionally, when a MT problem incorporates a three-dimensional (3D) subdomain, the numerical method employed for simulation and inversion was 3D over then entire domain. In here, we propose to take advantage of the possibly lower dimensionality of certain subdomains during the inversion process. By doing so, we obtain significant computational savings (up to 75% in some scenarios) and increased accuracy on the results. We numerically illustrate this method by employing two dimensional (2D) computations based on a multi-goal oriented . hp-adaptive Finite Element Method (FEM) that exhibits superior convergence properties. Additionally, we provide a formulation for implementing an efficient adjoint based method for the computation of the derivatives of the impedance, and we show the importance of the (a) proper selection of the inversion variable, and (b) the advantages of using both the Transverse Electric (TE) and Transverse Magnetic (TM) measurements for the proper inversion of MT data

Automatically Adapted Perfectly Matched Layers for Problems with High Contrast Materials Properties

AbstractFor the simulation of wave propagation problems, it is necessary to truncate the computational domain. Perfectly Matched Layers are often employed for that purpose, especially in high contrast layered materials where absorbing boundary conditions are difficult to design. In here, we define a Perfectly Matched Layer that automatically adjusts its parameters without any user interaction. The user only has to indicate the desired decay in the surrounding layer. With this Perfectly Matched Layer, we show that even in the most complex scenarios where the material contrast properties are as high as sixteen orders of magnitude, we do not introduce numerical reflections when truncating the domain, thus, obtaining accurate solutions

A Secondary Field Based hp-Finite Element Method for the Simulation of Magnetotelluric Measurements

In some geophysical problems, it is sometimes possible to divide the subsurface resistivity distribution as a one dimensional (1D) contribution plus some two dimensional (2D) inhomogeneities. Assuming this scenario, we split the electromagnetic fields into their primary and secondary components, the former corresponding to the 1D contribution, and the latter to the 2D inhomogeneities. While the primary field is solved via an analytical solution, for the secondary field we employ a multi-goal oriented self-adaptive hp-Finite Element Method (FEM). To truncate the computational domain, we design a Perfectly Matched Layer (PML) that automatically adapts to high-contrast materials that appear in the subsurface and in the air–ground interface. Numerical results illustrate the robustness of the proposed PML and the gains of the secondary field approach, where we obtain results with comparable accuracy than with a full field based formulation but with a much lower computational cost

Using Covariance Matrix Adaptation Evolutionary Strategy to boost the search accuracy in hierarchic memetic computations

Many global optimization problems arising naturally in science and engineering exhibit some form of intrinsic ill-posedness, such as multimodality and insensitivity. Severe ill-posedness precludes the use of standard regularization techniques and necessitates more specialized approaches, usually comprised of two separate stages - global phase, that determines the problem's modality and provides rough approximations of the solutions, and a local phase, which re fines these approximations. In this work, we attempt to improve one of the most efficient currently known approaches - Hierarchic Memetic Strategy (HMS) - by incorporating the Covariance Matrix Adaptation Evolutionary Strategy (CMA-ES) into its local phase. CMA-ES is a stochastic optimization algorithm that in some sense mimics the behavior of population-based evolutionary algorithms without explicitly evolving the population. This way, it avoids, to an extent, the associated cost of multiple evaluations of the objective function. We compare the performance of the HMS on relatively simple multimodal benchmark problems and on an engineering problem. To do so, we consider two con gurations: the CMA-ES and the standard SEA (Simple Evolutionary Algorithm). The results demonstrate that the HMS with CMA-ES in the local phase requires less objective function evaluations to provide the same accuracy, making this approach more efficient than the standard SEA

Quantities of interest for surface based resistivity geophysical measurements

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 therefore 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 significant 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

An Agent-Oriented Hierarchic Strategy for Solving Inverse Problems

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 painless multi-level automatic goal-oriented hp-adaptive coarsening strategy for elliptic and non-elliptic problems

This work extends an automatic energy-norm $hp$-adaptive strategy based on performing quasi-optimal unrefinements to the case of non-elliptic problems and goal-oriented adaptivity. The proposed approach employs a multi-level hierarchical data structure and alternates global $h$- and $p$-refinements with a coarsening step. Thus, at each unrefinement step, we eliminate the basis functions with the lowest contributions to the solution. When solving elliptic problems using energy-norm adaptivity, the removed basis functions are those with the lowest contributions to the energy of the solution. For non-elliptic problems or goal-oriented adaptivity, we propose an upper bound of the error representation expressed in terms of an inner product of the specific equation, leading to error indicators that deliver quasi-optimal $hp$-unrefinements. This unrefinement strategy removes unneeded unknowns possibly introduced during the pre-asymptotical regime. In addition, the grids over which we perform the unrefinements are arbitrary, and thus, we can limit their size and associated computational costs. We numerically analyze our algorithm for energy-norm and goal-oriented adaptivity. In particular, we solve two-dimensional ($2$D) Poisson, Helmholtz, convection-dominated equations, and a three-dimensional ($3$D) Helmholtz-like problem. In all cases, we observe \revb{exponential} convergence rates. Our algorithm is robust and straightforward to implement; therefore, it can be easily adapted for industrial applications.BERC.2022-202

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

We propose a new memetic strategy that can solve the multi-physics, complex inverse problems, formulated as the multi-objective optimization ones, in which objectives are misfits between the measured and simulated states of various governing processes. The multi-deme structure of the strategy allows for both, intensive, relatively cheap exploration with a moderate accuracy and more accurate search many regions of Pareto set in parallel. The special type of selection operator prefers the coherent alternative solutions, eliminating artifacts appearing in the particular processes. The additional accuracy increment is obtained by the parallel convex searches applied to the local scalarizations of the misfit vector. The strategy is dedicated for solving ill-conditioned problems, for which inverting the single physical process can lead to the ambiguous results. The skill of the selection in artifact elimination is shown on the benchmark problem, while the whole strategy was applied for identification of oil deposits, where the misfits are related to various frequencies of the magnetic and electric waves of the magnetotelluric measurement

Multi-objective hierarchic memetic solver for inverse parametric problems

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