A new biased estimation method based on Neumann series for solving ill-posed problems

The ill-posed least squares problems often arise in many engineering applications such as machine learning, intelligent navigation algorithms, surveying and mapping adjustment model, and linear regression model. A new biased estimation (BE) method based on Neumann series is proposed in this article to solve the ill-posed problems more effectively. Using Neumann series expansion, the unbiased estimate can be expressed as the sum of infinite items. When all the high-order items are omitted, the proposed method degenerates into the ridge estimation or generalized ridge estimation method, whereas a series of new biased estimates can be acquired by including some high-order items. Using the comparative analysis, the optimal biased estimate can be found out with less computation. The developed theory establishes the essential relationship between BE and unbiased estimation and can unify the existing unbiased and biased estimate formulas. Moreover, the proposed algorithm suits for not only ill-conditioned equations but also rank-defect equations. Numerical results show that the proposed BE method has improved accuracy over the existing robust estimation methods to a certain extent.

On the convergence of the Neumann series for electrostatic fracture response

The feasibility of Neumann-series expansion of Maxwell’s equations in the electrostatic limit is investigated for potentially rapid and approximate subsurface imaging of geologic features proximal to metallic infrastructure in an oilfield environment. Although generally useful for efficient modeling of mild conductivity perturbations in uncluttered settings, we have raised the question of its suitability for situations such as oilfields, in which metallic artifacts are pervasive and, in some cases, in direct electrical contact with the conductivity perturbation on which the Neumann series is computed. Convergence of the Neumann series and its residual error are computed using the hierarchical finite-element framework for a canonical oilfield model consisting of an L-shaped, steel-cased well, energized by a steady-state electrode, and penetrating a small set of mildly conducting fractures near the heel of the well. For a given node spacing [Formula: see text] in the finite-element mesh, we find that the Neumann series is ultimately convergent if the conductivity is small enough — a result consistent with previous presumptions on the necessity of small conductivity perturbations. However, we also determine that the spectral radius of the Neumann series operator grows as approximately [Formula: see text], thus suggesting that in the limit of the continuous problem [Formula: see text], the Neumann series is intrinsically divergent for all conductivity perturbations, regardless of their smallness. The hierarchical finite-element methodology itself is critically analyzed and shown to possess the [Formula: see text] error convergence of traditional linear finite elements, thereby supporting the conclusion of an inescapably divergent Neumann series for this benchmark example. Application of the Neumann series to oilfield problems with metallic clutter should therefore be done with careful consideration to the coupling between infrastructure and geology. The methods used here are demonstrably useful in such circumstances.

Biobjective Optimization Algorithms Using Neumann Series Expansion for Engineering Design

In this paper, two novel algorithms are designed for solving biobjective optimization engineering problems. In order to obtain the optimal solutions of the biobjective optimization problems in a fast and accurate manner, the algorithms, which have combined Newton’s method with Neumann series expansion as well as the weighted sum method, are applied to deal with two objectives, and the Pareto optimal front is achieved through adjusting weighted factors. Theoretical analysis and numerical examples demonstrate the validity and effectiveness of the proposed algorithms. Moreover, an effective biobjective optimization strategy, which is based upon the two algorithms and the surrogate model method, is developed for engineering problems. The effectiveness of the optimization strategy is proved by its application to the optimal design of the dummy head structure in the car crash experiments.