Special Session 139: New Developments in Computational Imaging, Learning, and Inverse Problems

Inverse radiative transfer via goal-oriented hp-adaptive mesh refinement

Shukai Du Syracuse University USA Co-Author(s):    Shukai Du, Samuel N. Stechmann

Abstract: The inverse problem in radiative transfer is critical for many applications, such as optical tomography and remote sensing. However, solving it numerically presents significant challenges, including high memory requirements and computational expense due to the problem`s high dimensionality and the iterative nature of the solution process. To address these challenges, we propose a goal-oriented hp-adaptive mesh refinement method for solving the inverse radiative transfer problem. A novel aspect of this approach is that it simultaneously combines two optimization processes -- one for inversion and one for mesh adaptivity. By leveraging the connection between duality-based mesh adaptivity and adjoint-based inversion techniques, we introduce a goal-oriented error estimator that is computationally inexpensive and can efficiently guide mesh refinement to solve the inverse problem numerically. For discretizing both the forward and adjoint problems, we employ discontinuous Galerkin spectral element methods. Using the goal-oriented error estimator, we then design an hp-adaptive algorithm to refine the meshes. Numerical experiments demonstrate that this method accelerates convergence and reduces memory usage, highlighting the efficiency of the goal-oriented mesh adaptive approach.

On computational passive imaging in the frequency domain

Thorsten Hohage University of Goettingen Germany Co-Author(s):    Bjoern Mueller

Abstract: We consider passive imaging with randomly excited waves in order to reconstruct coefficients of a differential operator or the shape of a domain. Primary data are measurements of waves excited by independent realizations of the source. From this one can compute correlations as approximations of the covariance operator of the random solution to the differential equation restricted to the measurement domain, which serve is input data for the inverse problem. Challenges occur in the huge size of correlation data, often too large to be stored or computed, and very large pointwise noise levels. We present a computational technique which addresses both of these challenges by using only the primary data while exploiting the full information content of the correlation data and respecting the distribution of the correlation data by taking into account the forth order moments of the primary data. The efficiency of this technique is demonstrated on real and synthetic data from helioseismology.

Parameter Reconstruction in Kinetic Equations: an Inverse Problem for Chemotaxis

Christian Klingenberg Wuerzburg University Germany Co-Author(s):    Kathrin Hellmuth, Qin Li, Min Tang

Abstract: On the mesoscopic level, motion of individual particles can be modeled by a kinetic transport equation for the population density $f(t,x,v)$ as a function of time $t$, space $x$ and velocity $v \in V$. A relaxation term on the right hand side accounts for scattering due to self-induced velocity changes and typically involves a parameter $K(x,v,v`)$ encoding the probability of changing from velocity $v`$ to $v$ at location $x$: \begin{equation} \nonumber \partial_t f(t,x,v) + v \cdot \nabla f(t,x,v) = \int K(x,v,v`) f(t,x,v`) - K(x,v`,v)f(t,x,v) dv` . \end{equation} This hyperbolic model is widely used to model bacterial motion, called chemotaxis. We study the inverse parameter reconstruction problem whose aim is to recover the scattering parameter $K$ and that has to be solved when fitting the model to a real situation. We restrict ourselves to macroscopic, i.e. velocity averaged data $\rho = \int f dv$ as a basis of our reconstruction. This introduces additional difficulties, which can be overcome by the use of short time interior domain data. In this way, we can establish theoretical existence and uniqueness of the reconstruction, study its macroscopic limiting behavior and numerically conduct the inversion under suitable data generating experimental designs. This work based on a collaboration with Kathrin Hellmuth (W\urzburg, Germany), Qin Li (Madison, Wisc., USA) and Min Tang (Shanghai, China).

LEARNING IN-BETWEEN IMAGERY DYNAMICS VIA PHYSICAL LATENT SPACES

Yoonsang Lee Dartmouth College USA Co-Author(s):    Jihun Han, Anne Gelb

Abstract: We present a framework designed to learn the underlying dynamics between two images observed at consecutive time steps. The complex nature of image data and the lack of temporal information pose significant challenges in capturing the unique evolving patterns. Our proposed method focuses on estimating the intermediary stages of image evolution, allowing for interpretability through latent dynamics while preserving spatial correlations with the image. By incorporating a latent variable that follows a physical model expressed in partial differential equations, our approach ensures the interpretability of the learned model and provides insight into corresponding image dynamics. We demonstrate the robustness and effectiveness of our learning framework through a series of numerical tests using geoscientific imagery data.

Bi-level iterative regularization for inverse problems in nonlinear PDEs

Tram Nguyen Max Planck Institute for Solar System Research Germany Co-Author(s):

Abstract: We investigate the ill-posed inverse problem of recovering unknown spatially dependent parameters in nonlinear evolution PDEs. We propose a bi-level Landweber scheme, where the upper-level parameter reconstruction embeds a lower-level state approximation. This can be seen as combining the classical reduced setting and the newer all-at-once setting, allowing us to, respectively, utilize well-posedness of the parameter-to-state map, and to bypass having to solve nonlinear PDEs exactly. Using this, we derive stopping rules for lower- and upper-level iterations and convergence of the bi-level method. We discuss application to parameter identification for the Landau-Lifshitz-Gilbert equation in magnetic particle imaging, as well as to several reaction-diffusion applications, in which the nonlinear reaction law needs to be determined.

Sediment Measurement: an Inverse Problem Formulation

Lingyun Qiu Tsinghua University Peoples Rep of China Co-Author(s):    Jiwei Li, Zhongjing Wang, Hui Yu, Siqin Zheng

Abstract: In this work, we present a novel approach for sediment concentration measurement in water flow, modeled as a multiscale inverse medium problem. To address the multiscale nature of the sediment distribution, we treat it as an inhomogeneous random field and use the homogenization theory in deriving the effective medium model. The inverse problem is formulated as the reconstruction of the effective medium model, specifically, the sediment concentration, from partial boundary measurements. Additionally, we develop numerical algorithms to improve the efficiency and accuracy of solving this inverse problem. Our numerical experiments demonstrate the effectiveness of the proposed model and methods in producing accurate sediment concentration estimates, offering new insights into sediment measurement in complex environments.

A New Sparse, Connected, and Rigid Graph Representations of Point Clouds and Beyond

Bao Wang University of Utah USA Co-Author(s):

Abstract: Graph neural networks (GNNs) -- learn graph representations by exploiting graph`s sparsity, connectivity, and symmetries -- have become indispensable for learning geometric data like molecules. However, the most used graphs (e.g., radial cutoff graphs) in molecular modeling lack theoretical guarantees for achieving connectivity and sparsity simultaneously, which are essential for the performance and scalability of GNNs. Furthermore, existing widely used graph construction methods for molecules lack rigidity, limiting GNNs` ability to exploit graph nodes` spatial arrangement. We will present a new hyperparameter-free graph construction of molecules and beyond with sparsity, connectivity, and rigidity guarantees.

Numerical Analysis of Quantitative Photoacoustic Tomography in a Diffusive Regime

Zhi Zhou The Hong Kong Polytechnic University Hong Kong Co-Author(s):    Giovanni S. Alberti, Siyu Cen

Abstract: In this talk, we explore the numerical analysis of quantitative photoacoustic tomography, modeled as an inverse problem to reconstruct the diffusion and absorption coefficients in a second-order elliptic equation using multiple internal measurements. We establish conditional stability in the $L^2$ norm, under a provable nonzero condition, with randomly chosen boundary excitation data. Building on this stability, we propose and analyze a numerical reconstruction scheme based on an output least-squares formulation, using finite element discretization. We provide an \textit{a priori} error estimate for the numerical reconstruction, serving as a guideline for selecting algorithmic parameters. Several numerical examples will be presented to illustrate the theoretical results.