Nonlinear inverse problem by T-matrix completion. I. Theory

We propose a conceptually new method for solving nonlinear inverse scattering problems (ISPs) such as are commonly encountered in tomographic ultrasound imaging, seismology and other applications. The method is inspired by the theory of nonlocality of physical interactions and utilizes the relevant formalism. We formulate the ISP as a problem whose goal is to determine an unknown interaction potential $V$ from external scattering data. Although we seek a local (diagonally-dominated) $V$ as the solution to the posed problem, we allow $V$ to be nonlocal at the intermediate stages of iterations. This allows us to utilize the one-to-one correspondence between $V$ and the T-matrix of the problem, $T$. Here it is important to realize that not every $T$ corresponds to a diagonal $V$ and we, therefore, relax the usual condition of strict diagonality (locality) of $V$. An iterative algorithm is proposed in which we seek $T$ that is (i) compatible with the measured scattering data and (ii) corresponds to an interaction potential $V$ that is as diagonally-dominated as possible. We refer to this algorithm as to the data-compatible T-matrix completion (DCTMC). This paper is Part I in a two-part series and contains theory only. Numerical examples of image reconstruction in a strongly nonlinear regime are given in Part II. The method described in this paper is particularly well suited for very large data sets that become increasingly available with the use of modern measurement techniques and instrumentation.Comment: This is Part I of a paper series containing theory only. Part II contains simulations and is available as arXiv:1505.06777 [math-ph]. Accepted in this form to Phys. Rev.

Massive Database Generation for 2.5D Borehole Electromagnetic Measurements using Refined Isogeometric Analysis

Borehole resistivity measurements are routinely inverted in real-time during geosteering operations. The inversion process can be efficiently performed with the help of advanced artificial intelligence algorithms such as deep learning. These methods require a massive dataset that relates multiple Earth models with the corresponding borehole resistivity measurements. In here, we propose to use an advanced numerical method —refined isogeometric analysis (rIGA)— to perform rapid and accurate 2.5D simulations and generate databases when considering arbitrary 2D Earth models. Numerical results show that we can generate a meaningful synthetic database composed of 100,000 Earth models with the corresponding measurements in 56 hours using a workstation equipped with two CPUs.European POCTEFA 2014–2020 Project PIXIL (EFA362/19); The grant ‘‘Artificial Intelligence in BCAM number EXP. 2019/0043

Cognitive effort and active inference

This paper aims to integrate some key constructs in the cognitive neuroscience of cognitive control and executive function by formalising the notion of cognitive (or mental) effort in terms of active inference. To do so, we call upon a task used in neuropsychology to assess impulse inhibition—a Stroop task. In this task, participants must suppress the impulse to read a colour word and instead report the colour of the text of the word. The Stroop task is characteristically effortful, and we unpack a theory of mental effort in which, to perform this task accurately, participants must overcome prior beliefs about how they would normally act. However, our interest here is not in overt action, but in covert (mental) action. Mental actions change our beliefs but have no (direct) effect on the outside world—much like deploying covert attention. This account of effort as mental action lets us generate multimodal (choice, reaction time, and electrophysiological) data of the sort we might expect from a human participant engaging in this task. We analyse how parameters determining cognitive effort influence simulated responses and demonstrate that—when provided only with performance data—these parameters can be recovered, provided they are within a certain range

A Novel Iterative Algorithm for Solving Nonlinear Inverse Scattering Problems

We introduce a novel iterative method for solving nonlinear inverse scattering problems. Inspired by the theory of nonlocality, we formulate the inverse scattering problem in terms of reconstructing the nonlocal unknown scattering potential V from scattered field measurements made outside a sample. Utilizing the one-to-one correspondence between V and T, the T-matrix, we iteratively search for a diagonally dominated scattering potential V corresponding to a data compatible T-matrix T. This formulation only explicitly uses the data measurements when initializing the iterations, and the size of the data set is not a limiting factor. After introducing this method, named data-compatible T-matrix completion (DCTMC), we detail numerous improvements the speed up convergence. Numerical simulations are conducted that provide evidence that DCTMC is a viable method for solving strongly nonlinear ill-posed inverse problems with large data sets. These simulations model both scalar wave diffraction and diffuse optical tomography in three dimensions. Finally, numerical comparisons with the commonly used nonlinear iterative methods Gauss-Newton and Levenburg-Marquardt are provided