On optimization techniques for solving nonlinear inverse problems
Bibliographic record
Abstract
This paper considers optimization techniques for the solution of nonlinear inverse problems where the forward problems, like those encountered in electromagnetics, are modelled by differential equations. Such problems are often solved by utilizing a Gauss-Newton method in which the forward model constraints are implicitly incorporated. Variants of Newton's method which use second-derivative information are rarely employed because their perceived disadvantage in computational cost per step offsets their potential benefits of faster convergence. In this paper we show that, by formulating the inversion as a constrained or unconstrained optimization problem, and by employing sparse matrix techniques, we can carry out variants of sequential quadratic programming and the full Newton iteration with only a modest additional cost. By working with the differential equation explicitly we are able to relate the constrained and the unconstrained formulations and discuss the advantages of each. To make the comparisons meaningful we adopt the same global optimization strategy for all inversions. As an illustration, we focus upon a 1D electromagnetic (EM) example simulating a magnetotelluric survey. This problem is sufficiently rich that it illuminates most of the computational complexities that are prevalent in multi-source inverse problems and we therefore describe its solution process in detail. The numerical results illustrate that variants of Newton's method which utilize second-derivative information can produce a solution in fewer iterations and, in some cases where the data contain significant noise, requiring fewer floating point operations than Gauss-Newton techniques. Although further research is required, we believe that the variants proposed here will have a significant impact on developing practical solutions to large-scale 3D EM inverse problems.
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How this classification was reachedexpand
Full frame machine prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. The Gemma side is a direct model label for every work in the frame, read from the title-only record. The Codex side is a classifier learned from the 10,348 direct Codex labels and calibrated to design-weighted sample rates; fields without enough sample support carry no Codex call. Candidate is the union of the two sides; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels.
Distilled classifier scores by category (both heads)
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.004 | 0.009 |
| Meta-epidemiology (narrow) | 0.003 | 0.001 |
| Meta-epidemiology (broad) | 0.002 | 0.002 |
| Bibliometrics | 0.002 | 0.003 |
| Science and technology studies | 0.001 | 0.002 |
| Scholarly communication | 0.002 | 0.002 |
| Open science | 0.001 | 0.003 |
| Research integrity | 0.002 | 0.004 |
| Insufficient payload (model declined to judge) | 0.005 | 0.003 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from itClassification
machine, unvalidatedMachine predicted; a candidate call from one source (direct Gemma or distilled Codex), not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".