[Paper Review] Randomized Approach to Nonlinear Inversion Combining Simultaneous Random and Optimized Sources and Detectors

Selin Aslan, Eric de Sturler+1 more|arXiv (Cornell University)|Jun 17, 2017

Sparse and Compressive Sensing Techniques28 references3 citations

TL;DR

This paper proposes a hybrid randomized and optimized approach to accelerate nonlinear inverse problems in diffuse optical tomography (DOT) by reducing the number of large-scale PDE solves. By combining random simultaneous sources/detectors with optimized ones that maximize Jacobian sensitivity, the method achieves solution quality comparable to full-data inversion while reducing PDE solves by a factor of 12, significantly improving convergence and efficiency.

ABSTRACT

In partial differential equations-based (PDE-based) inverse problems with many measurements, many large-scale discretized PDEs must be solved for each evaluation of the misfit or objective function. In the nonlinear case, evaluating the Jacobian requires solving an additional set of systems. This leads to a tremendous computational cost, and this is by far the dominant cost for these problems. Several authors have proposed randomization and stochastic programming techniques to drastically reduce the number of system solves by estimating the objective function using only a few appropriately chosen random linear combinations of the sources. While some have reported good solution quality at a greatly reduced cost, for our problem of interest, diffuse optical tomography, the approach often does not lead to sufficiently accurate solutions. We propose two improvements. First, to efficiently exploit Newton-type methods, we modify the stochastic estimates to include random linear combinations of detectors, drastically reducing the number of adjoint solves. Second, after solving to a modest tolerance, we compute a few simultaneous sources and detectors that maximize the Frobenius norm of the sampled Jacobian to improve the rate of convergence and obtain more accurate solutions. We complement these optimized simultaneous sources and detectors by random simultaneous sources and detectors constrained to a complementary subspace. Our approach leads to solutions of the same quality as obtained using all sources and detectors but at a greatly reduced computational cost, as the number of large-scale linear systems to be solved is significantly reduced.

Motivation & Objective

• Address the high computational cost of solving many large-scale PDEs in nonlinear inverse problems, particularly in DOT with thousands of sources and detectors.
• Overcome the poor convergence and accuracy of standard stochastic approximation (SAA) with random simultaneous sources and detectors in DOT applications.
• Improve the efficiency and accuracy of Newton-type methods by reducing adjoint solves through randomized detector combinations.
• Enhance solution quality and convergence rate by introducing optimized simultaneous sources and detectors that capture dominant Jacobian components.
• Develop a robust, low-cost framework that combines random and optimized sources/detectors to maintain accuracy while minimizing computational effort.

Proposed method

• Use the Sample Average Approximation (SAA) method to estimate the objective function and gradient using random linear combinations of sources and detectors.
• Extend SAA to the adjoint solve phase by randomizing detector combinations, drastically reducing the number of adjoint systems required per Jacobian evaluation.
• After initial optimization with random sources/detectors, compute a small number of optimized simultaneous sources and detectors that maximize the Frobenius norm of the sampled Jacobian.
• Constrain optimized sources and detectors to a complementary subspace to avoid redundancy and improve sensitivity capture.
• Combine random and optimized sources/detectors in a single framework to balance exploration and exploitation of the inverse problem’s sensitivity structure.
• Use the TREGS trust-region algorithm with Gauss-Newton regularization to solve the nonlinear least-squares problem efficiently.

Experimental results

Research questions

• RQ1 Can randomization of both sources and detectors reduce the number of PDE solves in DOT inverse problems without sacrificing solution quality?
• RQ2 Why does standard SAA with random sources and detectors fail to converge to the noise level in DOT, and how can this stagnation be overcome?
• RQ3 Can optimized simultaneous sources and detectors that maximize Jacobian sensitivity improve convergence and accuracy in nonlinear inverse problems?
• RQ4 What is the optimal balance between random and optimized sources/detectors to minimize computational cost while maintaining high reconstruction fidelity?
• RQ5 How does combining random and optimized sources/detectors compare to dynamic sample size adjustment in terms of convergence speed and accuracy?

Key findings

• The proposed method reduces the total number of large-scale PDE solves by approximately a factor of 12 compared to using all sources and detectors in a 32×32×32 mesh with 225 sources and detectors.
• Including 2 optimized simultaneous sources and detectors improved the SAA approach’s convergence, achieving a residual norm of δ² with only 726 PDE solves, compared to 9225 for the full-data case.
• Adding 4 optimized sources and detectors further reduced the number of PDE solves to 762 while maintaining high reconstruction quality.
• The method achieved reconstructions of the same quality as full-data inversion, with the true anomaly shape well approximated even at intermediate tolerances.
• The combination of random and optimized sources/detectors led to faster convergence and robust performance, especially in later optimization stages where standard SAA stagnates.
• The approach is scalable and expected to yield even greater computational gains in larger problems with many sources, detectors, and multiple frequencies.