A parallel evolution strategy for an earth imaging problem in geophysics

Abstract

In this paper we propose a new way to compute a rough approximation solution, to be later used as a warm starting point in a more refined optimization process, for a challenging global optimization problem related to earth imaging in geophysics. The warm start consists of a velocity model that approximately solves a full-waveform inverse problem at low frequency. Our motivation arises from the availability of massively parallel computing platforms and the natural parallelization of evolution strategies as global optimization methods for continuous variables. Our first contribution consists of developing a new and efficient parametrization of the velocity models to significantly reduce the dimension of the original optimization space. Our second contribution is to adapt a class of evolution strategies to the specificity of the physical problem at hands where the objective function evaluation is known to be the most expensive computational part. A third contribution is the development of a parallel evolution strategy solver, taking advantage of a recently proposed modification of these class of evolutionary methods that ensures convergence and promotes better performance under moderate budgets. The numerical results presented demonstrate the effectiveness of the algorithm on a realistic 3D full-waveform inverse problem in geophysics. The developed numerical approach allows us to successfully solve an acoustic full-waveform inversion problem at low frequencies on a reasonable number of cores of a distributed memory computer.

Appendix

Theorem 6.1

Given $n \in {\mathbb {N}}, N \in {\mathbb {N}}$ with $n < N$ and $\delta =[N/n],$ the coefficient matrix $C \in {\mathbb {R}}^{n \times n}$ defined as

$$ C_{ij} \; = \; \frac{2N}{(j-1)\pi \delta } \cos \bigg (\frac{\pi }{N}(j-1) (i-\frac{1}{2})\delta \bigg ) \sin \bigg (\frac{\delta \pi }{2N}(j-1)\bigg ) $$

is nonsingular. (Note that for $j=1$ we have used the convention $sin(0)/0=1$ , i.e., $C_{i1} = 1$.)

Proof

A direct calculation shows that

$$ \det (C) \; = \; \prod ^{n}_{j=1} \frac{2N}{(j-1)\pi \delta } \sin \left( \frac{\delta \pi }{2N}(j-1) \right) \det (M), $$

where $M \in {\mathbb {R}}^{n \times n}$ is defined as $M_{ij}= \cos (\frac{\pi }{N}(j-1) (i-\frac{1}{2})\delta )$. The nonsingularity of M can be deduced from the properties of DCT-III (Britanak et al. 2006). $\square$