An algorithm for solving the inverse problem in total internal reflection microscopy

Abstract

The inverse problem encountered in total internal reflection microscopy is particularly challenging due to the complex relationship between the data and the state vector. To address the computational difficulties and compute all possible solutions to this inverse problem, we propose an algorithm for solving problems of the form $F\left(x\right)=y$, where $F$ is a nonlinear and continuous forward model, $x$ is the state vector, and $y$ is the data vector. The forward model is computationally expensive, lacks derivatives, and is treated as a black-box function, with no prior information available about the statistical properties of $x$, aside from a box constraint. Additionally, the data vector $y$ is affected by uncertainties, making the retrieval of accurate solutions more challenging. To address these issues, the algorithm incorporates a neural network-based surrogate model to reduce computational costs and manage the absence of derivatives. A whitening transformation is applied to handle isotropic noise, and the residual is minimized within simple bounds. The method is further enhanced by a multistart solver that generates starting points through a hitting set problem, improving the exploration of the solution space and increasing the likelihood of finding all solutions. The solver iterates over the absolute function tolerance to identify both global and local minima, minimizes a least-squares function while adhering to simple bounds on variables during the local search phase, and includes a double box stopping rule. Additionally, the algorithm employs various sampling techniques and optimization methods, with the covariance matrix estimated using either a standard analytical approach or a Monte Carlo analysis.