Framework For Inverse Problems Research Articles

Determination of Initial Data in the Time-Fractional Pseudo-Hyperbolic Equation

We examine a time-fractional pseudo-hyperbolic equation involving positive operators. We explore the determination of initial velocity and perturbation. It is demonstrated that these initial inverse problems are ill posed. Additionally, we prove that under certain conditions, the inverse problems exhibit well-posedness properties. Our focus is on developing a theoretical framework for these initial inverse problems associated with time-fractional pseudo-hyperbolic equations, laying the groundwork for future studies on numerical algorithms to solve these problems. This investigation is crucial for understanding the fundamental behavior of the equations under various initial conditions and perturbations. By establishing a rigorous theoretical framework, we pave the way for future research to focus on practical numerical methods and simulations. Our results provide a deeper insight into the mathematical structure of time-fractional pseudo-hyperbolic equations, ensuring that future computational approaches are built on a solid theoretical foundation.

A variational inference framework for inverse problems

A framework is presented for fitting inverse problem models via variational Bayes approximations. This methodology guarantees flexibility to statistical model specification for a broad range of applications, good accuracy and reduced model fitting times. The message passing and factor graph fragment approach to variational Bayes that is also described facilitates streamlined implementation of approximate inference algorithms and allows for supple inclusion of numerous response distributions and penalizations into the inverse problem model. Models for one- and two-dimensional response variables are examined and an infrastructure is laid down where efficient algorithm updates based on nullifying weak interactions between variables can also be derived for inverse problems in higher dimensions. An image processing application and a simulation exercise motivated by biomedical problems reveal the computational advantage offered by efficient implementation of variational Bayes over Markov chain Monte Carlo.

Single-stage approach for estimating optical parameters in spectral quantitative photoacoustictomography.

In quantitative photoacoustic tomography, the optical parameters of a target, most importantly the concentrations of chromophores such as deoxygenated and oxygenated hemoglobin, are estimated from photoacoustic data measured on the boundary of the target. In this work, a numerical approximation of a forward model for spectral quantitative photoacoustic tomography is constructed by utilizing the diffusion approximation for light propagation, the acoustic wave equation for ultrasound propagation, and spectral models of optical absorption and scattering to describe the wavelength dependence of the optical parameters. The related inverse problem is approached in the framework of Bayesian inverse problems. Concentrations of four chromophores (deoxygenated and oxygenated hemoglobin, water, and lipid), two scattering parameters (reference scattering and scattering power), and the Grüneisen parameter are estimated in a single-stage from photoacoustic data. The methodology is evaluated using numerical simulations in different full-view and limited-view imaging settings. The results show that, utilizing spectral data and models, the spectral optical parameters and the Grüneisen parameter can be simultaneously estimated. Furthermore, the approach can also be utilized in limited-view imaging situations.

A unified Bayesian inversion approach for a class of tumor growth models with different pressure laws

Abstract. In this paper, we use the Bayesian inversion approach to study the data assimilation problem for a family of tumor growth models described by porous-medium type equations. The models contain uncertain parameters and are indexed by a physical parameter m, which characterizes the constitutive relation between density and pressure. Based on these models, we employ the Bayesian inversion framework to infer parametric and nonparametric unknowns that affect tumor growth from noisy observations of tumor cell density. We establish the well-posedness and the stability theories for the Bayesian inversion problem and further prove the convergence of the posterior distribution in the so-called incompressible limit, m →∞. Since the posterior distribution across the index regime m ∈ [2,∞) can thus be treated in a unified manner, such theoretical results also guide the design of the numerical inference for the unknown. We propose a generic computational framework for such inverse problems, which consists of a typical sampling algorithm and an asymptotic preserving solver for the forward problem. With extensive numerical tests, we demonstrate that the proposed method achieves satisfactory accuracy in the Bayesian inference of the tumor growth models, which is uniform with respect to the constitutive relation.

Analysis of a Computational Framework for Bayesian Inverse Problems: Ensemble Kalman Updates and MAP Estimators under Mesh Refinement

Analysis of a Computational Framework for Bayesian Inverse Problems: Ensemble Kalman Updates and MAP Estimators under Mesh Refinement

Multi-Scale Energy (MuSE) Framework for Inverse Problems in Imaging

Multi-Scale Energy (MuSE) Framework for Inverse Problems in Imaging

A Box-Spline Framework for Inverse Problems with Continuous-Domain Sparsity Constraints

A Box-Spline Framework for Inverse Problems with Continuous-Domain Sparsity Constraints

A methodology of phenotyping ICU patients from EHR data: High-fidelity, personalized, and interpretable phenotypes estimation

Objective:Computing phenotypes that provide high-fidelity, time-dependent characterizations and yield personalized interpretations is challenging, especially given the complexity of physiological and healthcare systems and clinical data quality. This paper develops a methodological pipeline to estimate unmeasured physiological parameters and produce high-fidelity, personalized phenotypes anchored to physiological mechanics from electronic health record (EHR). Methods:A methodological phenotyping pipeline is developed that computes new phenotypes defined with unmeasurable computational biomarkers quantifying specific physiological properties in real time. Working within the inverse problem framework, this pipeline is applied to the glucose-insulin system for ICU patients using data assimilation to estimate an established mathematical physiological model with stochastic optimization. This produces physiological model parameter vectors of clinically unmeasured endocrine properties, here insulin secretion, clearance, and resistance, estimated for individual patient. These physiological parameter vectors are used as inputs to unsupervised machine learning methods to produce phenotypic labels and discrete physiological phenotypes. These phenotypes are inherently interpretable because they are based on parametric physiological descriptors. To establish potential clinical utility, the computed phenotypes are evaluated with external EHR data for consistency and reliability and with clinician face validation. Results:The phenotype computation was performed on a cohort of 109 ICU patients who received no or short-acting insulin therapy, rendering continuous and discrete physiological phenotypes as specific computational biomarkers of unmeasured insulin secretion, clearance, and resistance on time windows of three days. Six, six, and five discrete phenotypes were found in the first, middle, and last three-day periods of ICU stays, respectively. Computed phenotypic labels were predictive with an average accuracy of 89%. External validation of discrete phenotypes showed coherence and consistency in clinically observable differences based on laboratory measurements and ICD 9/10 codes and clinical concordance from face validity. A particularly clinically impactful parameter, insulin secretion, had a concordance accuracy of 83%±27%. Conclusion:The new physiological phenotypes computed with individual patient ICU data and defined by estimates of mechanistic model parameters have high physiological fidelity, are continuous, time-specific, personalized, interpretable, and predictive. This methodology is generalizable to other clinical and physiological settings and opens the door for discovering deeper physiological information to personalize medical care.

Physics-informed UNets for discovering hidden elasticity in heterogeneous materials

Soft biological tissues often have complex mechanical properties due to variation in structural components. In this paper, we develop a novel UNet-based neural network model for inversion in elasticity (El-UNet) to infer the spatial distributions of mechanical parameters from strain maps as input images, normal stress boundary conditions, and domain physics information. We show superior performance – both in terms of accuracy and computational cost – by El-UNet compared to fully-connected physics-informed neural networks in estimating unknown parameters and stress distributions for isotropic linear elasticity. We characterize different variations of El-UNet and propose a self-adaptive spatial loss weighting approach. To validate our inversion models, we performed various finite-element simulations of isotropic domains with heterogenous distributions of material parameters to generate synthetic data. El-UNet is faster and more accurate than the fully-connected physics-informed implementation in resolving the distribution of unknown fields. Among the tested models, the self-adaptive spatially weighted models had the most accurate reconstructions in equal computation times. The learned spatial weighting distribution visibly corresponded to regions that the unweighted models were resolving inaccurately. Our work demonstrates a computationally efficient inversion algorithm for elasticity imaging using convolutional neural networks and presents a potential fast framework for three-dimensional inverse elasticity problems that have proven unachievable through previously proposed methods.

Novel and flexible parameter estimation methods for data-consistent inversion in mechanistic modelling.

Predictions for physical systems often rely upon knowledge acquired from ensembles of entities, e.g. ensembles of cells in biological sciences. For qualitative and quantitative analysis, these ensembles are simulated with parametric families of mechanistic models (MMs). Two classes of methodologies, based on Bayesian inference and population of models, currently prevail in parameter estimation for physical systems. However, in Bayesian analysis, uninformative priors for MM parameters introduce undesirable bias. Here, we propose how to infer parameters within the framework of stochastic inverse problems (SIPs), also termed data-consistent inversion, wherein the prior targets only uncertainties that arise due to MM non-invertibility. To demonstrate, we introduce new methods to solve SIPs based on rejection sampling, Markov chain Monte Carlo, and generative adversarial networks (GANs). In addition, to overcome limitations of SIPs, we reformulate SIPs based on constrained optimization and present a novel GAN to solve the constrained optimization problem.

Bilevel inverse problems in neuromorphic imaging

Event or neuromorphic cameras are novel biologically inspired sensors that record data based on the change in light intensity at each pixel asynchronously. They have a temporal resolution of microseconds. This is useful for scenes with fast moving objects that can cause motion blur in traditional cameras, which record the average light intensity over an exposure time for each pixel synchronously. This paper presents a bilevel inverse problem framework for neuromorphic imaging. Existence of solution to the inverse problem is established. Second order sufficient conditions are derived under special situations for this nonconvex problem. A second order Newton type solver is derived to solve the problem. The efficacy of the approach is shown on several examples.

A differentiable, physics-informed ecosystem modeling and learning framework for large-scale inverse problems: demonstration with photosynthesis simulations

Abstract. Photosynthesis plays an important role in carbon, nitrogen, and water cycles. Ecosystem models for photosynthesis are characterized by many parameters that are obtained from limited in situ measurements and applied to the same plant types. Previous site-by-site calibration approaches could not leverage big data and faced issues like overfitting or parameter non-uniqueness. Here we developed an end-to-end programmatically differentiable (meaning gradients of outputs to variables used in the model can be obtained efficiently and accurately) version of the photosynthesis process representation within the Functionally Assembled Terrestrial Ecosystem Simulator (FATES) model. As a genre of physics-informed machine learning (ML), differentiable models couple physics-based formulations to neural networks (NNs) that learn parameterizations (and potentially processes) from observations, here photosynthesis rates. We first demonstrated that the framework was able to correctly recover multiple assumed parameter values concurrently using synthetic training data. Then, using a real-world dataset consisting of many different plant functional types (PFTs), we learned parameters that performed substantially better and greatly reduced biases compared to literature values. Further, the framework allowed us to gain insights at a large scale. Our results showed that the carboxylation rate at 25 ∘C (Vc,max25) was more impactful than a factor representing water limitation, although tuning both was helpful in addressing biases with the default values. This framework could potentially enable substantial improvement in our capability to learn parameters and reduce biases for ecosystem modeling at large scales.

A differentiable, physics-informed ecosystem modeling and learning framework for large-scale inverse problems

A differentiable, physics-informed ecosystem modeling and learning framework for large-scale inverse problems

SBFEM and Bayesian inference for efficient multiple flaw detection in structures

Bayesian inference is a powerful technique for damage/flaw detection in critical structures. This paper explores the application of Bayesian inference to identify the flaws (voids/inclusions) and its parameters, assuming no information about the flaws is known a priori. Multiple flaws are represented by a parameter vector that contains the locations and the geometric information. In the inverse problem framework, the Scaled Boundary Finite Element Method (SBFEM) with quadtree decomposition is used to solve the forward problem. The flaw parameters are statistically quantified using the Bayesian inference. The likelihood function and prior information update the joint distribution of the number of flaws and their corresponding flaw parameters. The sampling to estimate the posterior distribution of the flaw parameters is based on the trans-dimensional Reversible Jump Markov Chain Monte Carlo (RJMCMC). Also, the impact of additive Gaussian noise on the observed data is investigated. Numerical examples include identifying multiple voids of different shapes such as circle and ellipse and identifying voids and inclusions, using the proposed approach with input sensor data at different noise levels.

A differentiable, physics-informed ecosystem modeling and learning framework for large-scale inverse problems

A differentiable, physics-informed ecosystem modeling and learning framework for large-scale inverse problems

A new regularized stochastic approximation framework for stochastic inverse problems

We study the nonlinear inverse problem of estimating stochastic parameters in the fourth-order partial differential equation with random data. The primary focus is on developing a novel stochastic approximation framework for inverse problems consisting of three key components. As a first step, we reformulate the inverse problem into a stochastic convex optimization problem. The second step includes developing a new regularized stochastic extragradient framework for a nonlinear variational inequality, which subsumes the optimality conditions for the optimization formulation of the inverse problem. The third step involves modeling random variables by a Karhunen–Loève type finite-dimensional noise representation, allowing the direct and the inverse problems to be conveniently discretized. We show that the regularized extragradient methods are strongly convergent in a Hilbert space setting, and we also provide several auxiliary results for the inverse problem, including Lipschitz continuity and a derivative characterization of the solution map. We provide the outcome of computational experiments to estimate stochastic and deterministic parameters. The numerical results demonstrate the feasibility and effectiveness of the developed framework and validate stochastic approximation as an effective method for stochastic inverse problems.

Damage Detection in Reinforced Concrete Member Using Local Time-Frequency Transform Applied to Vibration Measurements

Signal processing and analysis of structural vibration measurements are key components of structural damage detection (SDD) in structural health monitoring (SHM). The goal of signal processing is to extract subtle changes in the measured signals, which can be used to infer changes in structural parameters and damage. Time-frequency analysis is one of the most popular characterization methods for studying non-stationary vibration signals. In this article, the local time-frequency transform (LTFT) is applied and evaluated to calculate the time-domain signals because of its excellent time-frequency energy distribution properties. The LTFT matches the input data by the Fourier basis in an inverse problem framework and uses the least squares method to solve the time-varying Fourier coefficients. Subsequently, it defines the time-frequency spectrum as the calculated time-varying Fourier coefficients. While the LTFT has been used in the field of geophysics for seismic data processing, its application to structural vibration signals is novel. Both synthetic signals as well as signals collected from a large-scale laboratory test of a reinforced concrete girder were processed with the LTFT and compared with Rényi entropy for quantifying the time-frequency spectrum, the time-frequency resolution abilities of short time Fourier transform (STFT), and S transform (ST). The results show that the LTFT is superior to the traditional time-frequency analysis schemes, in that it is more effective in identifying the energy changes in the time-frequency spectrum before and after structural damage in the form of cracking has occurred. At the same time, it provides high-precision time-frequency resolution and excellent noise suppression abilities. The effectiveness and feasibility of the LTFT applied to the synthetic and experimental signals are verified.

Inverse design of anisotropic spinodoid materials with prescribed diffusivity

The three-dimensional microstructure of functional materials determines its effective properties, like the mass transport properties of a porous material. Hence, it is desirable to be able to tune the properties by tuning the microstructure accordingly. In this work, we study a class of spinodoid i.e. spinodal decomposition-like structures with tunable anisotropy, based on Gaussian random fields. These are realistic yet computationally efficient models for bicontinuous porous materials. We use a convolutional neural network for predicting effective diffusivity in all three directions. We demonstrate that by incorporating the predictions of the neural network in an approximate Bayesian computation framework for inverse problems, we can in a computationally efficient manner design microstructures with prescribed diffusivity in all three directions.

Convergence of a spectral regularization of a time-reversed reaction-diffusion problem with high-order Sobolev-Gevrey smoothness

The present paper analyzes a spectral regularization of a time-reversed reaction-diffusion problem with globally and locally Lipschitz nonlinearities. This type of inverse and ill-posed problems arises in a variety of real-world applications concerning heat conduction and tumour source localization. In accordance with the weak solvability result for the forward problem, we focus on the inverse problem with the high-order Sobolev-Gevrey smoothness and with Sobolev measurements. As expected from the well-known results for the linear case, we prove that this nonlinear spectral regularization possesses a logarithmic rate of convergence in the high-order Sobolev norm. The proof can be done by the verification of variational source condition; this way interestingly validates such a fine strategy in the framework of inverse problems for nonlinear partial differential equations. Ultimately, we design an iteration-based version of the proposed method for a class of reaction-diffusion problems with non-degenerate nonlinearity, taking into account the conventional time discretization. The convergence of this iterative scheme is also investigated.

Information Field Theory and Artificial Intelligence.

Information field theory (IFT), the information theory for fields, is a mathematical framework for signal reconstruction and non-parametric inverse problems. Artificial intelligence (AI) and machine learning (ML) aim at generating intelligent systems, including such for perception, cognition, and learning. This overlaps with IFT, which is designed to address perception, reasoning, and inference tasks. Here, the relation between concepts and tools in IFT and those in AI and ML research are discussed. In the context of IFT, fields denote physical quantities that change continuously as a function of space (and time) and information theory refers to Bayesian probabilistic logic equipped with the associated entropic information measures. Reconstructing a signal with IFT is a computational problem similar to training a generative neural network (GNN) in ML. In this paper, the process of inference in IFT is reformulated in terms of GNN training. In contrast to classical neural networks, IFT based GNNs can operate without pre-training thanks to incorporating expert knowledge into their architecture. Furthermore, the cross-fertilization of variational inference methods used in IFT and ML are discussed. These discussions suggest that IFT is well suited to address many problems in AI and ML research and application.