Misfit Landforms Imposed by Ill-Conditioned Inverse Parametric Problems

Abstract

In this paper, we put forward a new topological taxonomy that allows us to distinguish and separate multiple solutions to ill-conditioned parametric inverse problems appearing in engineering, geophysics, medicine, etc. This taxonomy distinguishes the areas of insensitivity to parameters called the landforms of the misfit landscape, be it around minima (lowlands), maxima (uplands), or stationary points (shelves). We have proven their important separability and completeness conditions. In particular, lowlands, uplands, and shelves are pairwise disjoint, and there are no other subsets of the positive measure in the admissible domain on which the misfit function takes a constant value. The topological taxonomy is related to the second, "local" one, which characterizes the types of ill-conditioning of the particular solutions. We hope that the proposed results will be helpful for a better and more precise formulation of ill-conditioned inverse problems and for selecting and profiling complex optimization strategies used in solving these problems.

Understanding measure-driven algorithms solving irreversibly ill-conditioned problems

“…It can be proven (see Theorem 5 in [30]), that if y is a local minimizer of f in D, then B y is a connected component of f jD ð Þ À1 ðf ðyÞÞ that contains y. Moreover B y 0 ¼ B y 8y 0 2 B y and B y ¼ B y .…”

“…Moreover B y 0 ¼ B y 8y 0 2 B y and B y ¼ B y . Additionally if f is differentiable, then rf jx ¼ 0 8x 2 B y (see Observations 6 and 11 in [30]).…”

Using Covariance Matrix Adaptation Evolutionary Strategy to boost the search accuracy in hierarchic memetic computations

“…and is presented on Figure 2. It has minima in (1, 1), (6,1) and (7,8). The four-dimension benchmark used in Test 4 is defined by:…”

“…It has minima in (2, 2, 2, 2), (6, 1, 2, 3) and (7,8,2,3). The radii were scaled up by 2 compared to f 1 , in order to maintain a similar susceptible region.…”

“…Typical symptoms of such ill-conditioning are the objective multimodality or/and the regions of objective insensitivity, which might be called "lowlands" in the optimization landscape (see [7] for details). The traditional formulations and solving methods are difficult to apply for such problems.…”