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

Highlights

• We increase the accuracy of the local search in MO-HMS by means of gradient-based optimization methods.

• We use benchmarks to study the influence of parameters on our multi-objective selection operator.

• We solve the twin-objective magnetotelluric inverse problem using the enhanced MO-HMS.

• Using local methods delivers more good-quality Pareto solutions.

Abstract

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 measurements.

Introduction

Parametric inverse problems (IPs) for partial differential equations (PDEs) consist of restoring the values of PDE parameters (inverse solutions) from the known observation of their solution called forward solution over certain subdomains. IPs are fundamental in several applications, such as in oil and gas exploration, structure health monitoring, and cancer tissue diagnosis (see e.g. [1]).

The most popular mathematical formulation of IPs is in terms of global optimization problems (GOPs), where the decision variables belong to the admissible set of parameter functions representations, and the objective functionals to be minimized express the misfit between measured and simulated PDE forward solutions.

Solving IPs meets many obstacles caused by their ill-conditioning. If the ill-conditioning involves the lack of the global misfit convexity only, but still there exists a unique global minimizer, the misfit regularization (see e.g. [2]) can be the effective way to obtain its numerical solution. If the problem possesses more than one solution (i.e. the misfit is multimodal and has many global minimizers) and/or the misfit is insensitive with respect to several decision variables in the neighborhood of the global minimizer, the complex stochastic searches (see e.g. [3]) allow to overcome the above difficulties when solving them.

The misfit multimodality and insensitivity are generally caused by the lack of information about the phenomenon to be analyzed. It may result in a mathematical formulation of the problem that allows multiple solutions (see e.g. [4]) or the uncertain misfit representation due to the irreducible measurement errors (see e.g. [5], [6], [7]). The other obstacles are caused by artifacts that might be produced by deterministic and stochastic global optimization strategies (see e.g. [8]).

The most straightforward way to improve the IPs conditioning is by increasing the amount of information about the studied phenomenon. If the phenomenon is composed of multiple physical processes, then it is possible to consider many misfits simultaneously, each one associated with a separate physics. This approach leads to the multiobjective global optimization problem defined and discussed in Section 1.2.

The idea of solving ill conditioned IPs by finding Pareto solutions for misfits imposed by multiple physics is rarely described in the literature. The authors of [9] apply the inverse quantitative structure–property relationship for designing new chemical compounds. Optimal design of a magnetic pole is considered in [10]. In both cases, different objective functions are associated with two independent methods of solving the considered forward problem.

Some existing methods for the inversion of multi-physics measurements in the area of oil exploration are based on requesting that geometrical structures identified by single-physics measurements are correlated (see e.g. [11]). Other methods employ experimental laws such as Archie's and Gassman's equations to relate different physical measurements among each other (see e.g. [12], [13]). Unfortunately, the aforementioned experimental laws contain various parameters that need to be properly adjusted, which is not always possible.

The more common idea of applying multi-objective optimization for solving IPs leads only to improving the method utilized for minimizing single ill conditioned misfit. A two-objective parameter identification by genetic algorithms can be found in [14]. The second additional criterion penalizes the small diversity in the populations of candidate solutions. Another approach used in [15] consists in combining two objective functions with an immunological algorithm. These objectives become fitnesses of individuals and T-cells, respectively.

The first approach intensively utilized in the sequel of this paper is related to the set of n physical processes uⁱ(ω) ∈ Vⁱ, i = 1, …, n, which depend on the same, unknown parameter $\omega \in \mathcal{D}$. Typically, ω is a discrete representation of a distribution of some physical quantity (e.g. heat conductivity) on the dense domain of the forward problem. Vⁱ are proper Sobolev spaces and Aⁱ(uⁱ(ω)) = 0 the relevant equations representing forward problems, where Aⁱ : Vⁱ ⟶ (Vⁱ)′ is a family of differential operators from Vⁱ to their conjugate spaces.

We are able to measure the state of all processes, which results in the vector $d;d^i\in {\mathcal{O}}^i$, where ${\mathcal{O}}^i$ are the sets of observations specific to each physic. Next, we introduce the misfit functionals f(d, u(ω)), such that ${\mathcal{O}}^i\times \mathcal{D}\ni (d^i,u^i(\omega ))\to f^i(d^i,u^i(\omega ))\in {ℝ}_{+}$. Each coordinate fⁱ represents the particular physics i = 1, …, n.

The first multi-objective problem that represents the IPs associated with multiple physical processes dependent on the same parameter function consists of finding ω such that it minimizes all misfit functionals in the Pareto sense (see e.g. [16])

$$\underset{\omega \in \mathcal{D}}{min}\left\{f(d,u(\omega )):A(u(\omega ))=0\right\}.$$

The above approach might be generalized to the case of many physical processes that depend on different parametric functions. Let us assume similarly that we have n physical processes uⁱ(ωⁱ) ∈ Vⁱ, i = 1, …, n, which depend on a different, unknown parameter ${\omega }^i\in {\mathcal{D}}^i$ where ${\mathcal{D}}^i$ is the admissible set. Their states can be measured and their measurements can be denoted as previously by $d;d^i\in {\mathcal{O}}^i$. Now, we introduce the separate, metric space of features $\mathcal{F}$ that represents the phenomenon to be recognized (e.g. oil deposit, tumor tissue) and the operators extracting features $C^i:{\mathcal{D}}^i\to \mathcal{F}$. The value Cⁱ(ωⁱ) represents the information upon the phenomenon under interest obtained from the ith physics. Their values may represent the characteristic function of the deposit region (e.g. the region of the oil occurrence).

In the sequel, we introduce a new incidence criterion

$$f^{n+1}:{\mathcal{F}}^n\to {ℝ}_{+}$$

that penalizes the incoherency between the parameters assigned to the particular physics. In other words, the quantity fⁿ⁺¹(C¹(ω¹), …, Cⁿ(ωⁿ)) takes a small value if the parameters ω¹, …, ωⁿ represent a “similar” phenomenon to be searched. As the misfit operators fⁱ, i = 1, …, n, the incidence criterion is strongly related to the particular IPs. In the simplest case n = 2, we may set $f^3(C^1({\omega }^1),C^2({\omega }^2))={\rho }_{\mathcal{F}}(C^1({\omega }^1),C^2({\omega }^2))$, where ${\rho }_{\mathcal{F}}$ stands for the metric function in the space of features.

The generalized multi-objective problem that represents the IPs associated with many physical processes that depend on the different parametric functions is formulated in a way similar to the problem (1):

$$\underset{{\omega }^1\in {\mathcal{D}}^1,\dots ,{\omega }^n\in {\mathcal{D}}^n}{min}\{f^1(d^1,u^1({\omega }^1)),\dots ,f^n(d^n,u^n({\omega }^n)),f^{n+1}(C^1({\omega }^1),\dots ,C^n({\omega }^n)):A^i(u^i({\omega }^i))=0,i=1,\dots ,n\}.$$

We foresee that, due to the last objective fⁿ⁺¹, we will be able to seamlessly discriminate against those solutions obtained for the physical models that are too distant. More specifically, an artifact appearing in the particular ωⁱ would be considered only if the other processes would yield enough evidence that the particular candidate solution is indeed promising. Thus, we expect tremendous reduction of generated artifacts and, in consequence, we expect severe reduction of multimodality of the problem (as many apparent extrema arise from interactions between artifacts and true extrema). Generally, we expect that solving many models jointly would yield information about inverse solutions that would be much broader and useful than that obtained if we considered each model separately. We hope that because of this additional information, the computational cost of solving (1) or (3) would not be larger (or, in fact, would even be possibly smaller than) the cost of minimizing each misfit separately.

Our proposed approach provides a rigorous mathematical framework that complements and generalizes existing methods for the joint inversion of multi-physical measurements. In particular, functions Cⁱ defined above can be selected: (a) to impose some geometrical correlation, (b) to reproduce the behavior of certain experimental laws, or (c) to impose any other interrelationship of interest between different physical phenomena.

The generalized IP formulation (3) might be considered for solving oil deposit investigations on the base of the common inverting electrical resistivity and sound speed using independent measurements obtained by the electromagnetic and ultrasonic antennas.

The approach presented in Section 1.2 was briefly introduced in [17]. The current paper extends the first computational experiments contained in [18] and published in [19].

We apply the complex, multi-deme Hierarchic Memetic Strategy (HMS) [20] well suited for solving IPs for finding Pareto compromise solutions of the problem (1) (see Section 2) for the case of twin physics (n = 2). In Section 2.2, the special kind of the selection operator, a particular type of rank selection (cf. MOGA [21]) that prefers the coherent Pareto solutions is discussed.

In the sequel of the paper we perform the benchmark analysis (see Section 3) showing the exploratory power of the proposed strategy and the effective elimination of incoherent compromise solutions.

The last part of the paper (see Section 4) contains the solution of a real-world engineering problem of inverting magnetotelluric (MT) measurements (see [22]) in order to find oil deposits located under the Earth's surface. Two misfit functions are related to distinct frequencies of the electric and magnetic waves, for which the maximum sensitivity with respect to the impedance to be searched is expected. In this example, we hybridize for the first time HMS with a special kind of the local, convex optimization [23], [16] in order to increase the search accuracy.