A Unified 2D/3D Large-Scale Software Environment for Nonlinear Inverse Problems

Abstract

Large-scale parameter estimation problems are among some of the most computationally demanding problems in numerical analysis. An academic researcher’s domain-specific knowledge often precludes that of software design, which results in inversion frameworks that are technically correct but not scalable to realistically sized problems. On the other hand, the computational demands for realistic problems result in industrial codebases that are geared solely for high performance, rather than comprehensibility or flexibility. We propose a new software design for inverse problems constrained by partial differential equations that bridges the gap between these two seemingly disparate worlds. A hierarchical and modular design reduces the cognitive burden on the user while exploiting high-performance primitives at the lower levels. Our code has the added benefit of actually reflecting the underlying mathematics of the problem, which lowers the cognitive load on the user using it and reduces the initial startup period before a researcher can be fully productive. We also introduce a new preconditioner for the 3D Helmholtz equation that is suitable for fault-tolerant distributed systems. Numerical experiments on a variety of 2D and 3D test problems demonstrate the effectiveness of this approach on scaling algorithms from small- to large-scale problems with minimal code changes.

Feature-Oriented Joint Time-Lapse Seismic and Electromagnetic History Matching Using Ensemble Methods

“…where u is the pressure field; η is the Lamé coefficient related to seismic velocities through the relation V p = η/ρ; q is the monopole source; ω is the angular frequency and i 2 = −1; r denotes spatial positions; r s denotes source positions; ∇ is the spatial differential operator with respect to r; and • is the vector inner product. The above equation is solved using the frequency-domain finite-difference approach with a perfectly matched layer absorbing boundary condition [45,49]. Seismic measurements in crosswell setting are defined as a vector of pressure field responses, generated by the sources at positions , where N rs and N rr are the numbers of the source locations and the receiver locations, respectively.…”

Efficient Joint 4D Seismic and Electromagnetic History Matching Using an Ensemble-Based Approach

“…where is the pressure field; is the monopole source; is the angular frequency and 2 = −1; is the Lamé coefficient related to seismic velocity through the relation = √ / ; denotes spatial positions; denotes source positions, and ∇ is the spatial differential operator with respect to . The above equation is solved using the frequency-domain finite-difference approach with a perfectly matched layer absorbing boundary condition (Operto et al, 2007;Silva and Herrmann, 2019). Seismic measurements in crosswell setting are defined as a vector of pressure field responses, generated by the sources at positions and observed at receiver locations along boreholes, denoted by…”

Enabling numerically exact local solver for waveform inversion—a low-rank approach

“…For this example, we use the full Marmousi velocity model, which is 2 km deep and 12 km wide, sampled at 10 m. The synthetic data contains 600 sources and 600 receivers sampled at 20 m. We use frequencies from 3 to 10 Hz, where the source-signature is a Ricker wavelet with a central frequency of 20 Hz. We use a frequency domain finite difference code [10] to simulate the synthetic data.…”

“…This is written as the following optimization problem: minimize X rank(X) subject to A(X)−b 2 ≤ ǫ, where rank is defined as the maximum number of linearly independent rows or column of a matrix, b is a set of blended measurements and A represents the sampling-blending operator. Since rankminimization problems are NP hard and therefore computationally intractable, [46] showed that solutions to rank-minimization problems can be found by solving the following nuclear-norm minimization problem: minimize X X * subject to A(X) − b 2 ≤ ǫ, (10) where . * = σ 1 and σ is the vector of singular values for each monochromatic data matricization.…”