Forward Map Research Articles

Derivative-Free Iterative One-Step Reconstruction for Multispectral CT.

Image reconstruction in multispectral computed tomography (MSCT) requires solving a challenging nonlinear inverse problem, commonly tackled via iterative optimization algorithms. Existing methods necessitate computing the derivative of the forward map and potentially its regularized inverse. In this work, we present a simple yet highly effective algorithm for MSCT image reconstruction, utilizing iterative update mechanisms that leverage the full forward model in the forward step and a derivative-free adjoint problem. Our approach demonstrates both fast convergence and superior performance compared to existing algorithms, making it an interesting candidate for future work. We also discuss further generalizations of our method and its combination with additional regularization and other data discrepancy terms.

Effective Document Image Rectification via a Deep Learning Framework

This paper proposes an efficient method for rectifying distorted document images via deep learning, ultimately improving the legibility of graphics and text in documents. The framework comprises two interconnected UNets, working in tandem to predict a 3D coordinate map and a forward map for the input distorted document image, respectively. At the beginning of the process, a page mask is predicted and used as input to both U-Nets to help improve the performance of their tasks. In the last step, the predicted forward map is transformed into a corresponding backward map, which is utilized to rectify the distorted image. The experimental results not only reveal that the predicted page masks and 3D coordinate maps significantly enhance the accuracy of predicting forward maps for subsequent rectification but also demonstrate satisfactory results both globally and locally.

Are almost all trajectories dense in a chaotic set?

Chaos is the most complex dynamic behavior a system can show. Yet, it is produced by two simple, though nonlinear, deterministic mechanisms, hence the more proper name Deterministic Chaos. The first mechanism, stretching, acts locally and linearly, expanding small sets in the system's state space in (at least) one direction and contracting in the others. The second mechanism, folding, acts globally and is the nonlinear operation of bending the elongated sets to form the forward image of the original sets. These two simple operations are invertible and form a chaotic set for the system's forward and backward map, containing an uncountable infinity of astonishingly complex system's trajectories. The simplest explanation of chaos I ever heard is James A. Yorke “pizza machine” (plenary lecture at the SIAM DS, Snowbird, 2003): kneading the pizza dough is nothing but stretching and folding. The mathematical formalization is Stephen Smale's horseshoe map. Using this nice tool, I revisit the hallmarks of chaos, one of which (in the title) still misses a constructive proof.

Physics constrained unsupervised deep learning for rapid, high resolution scanning coherent diffraction reconstruction

By circumventing the resolution limitations of optics, coherent diffractive imaging (CDI) and ptychography are making their way into scientific fields ranging from X-ray imaging to astronomy. Yet, the need for time consuming iterative phase recovery hampers real-time imaging. While supervised deep learning strategies have increased reconstruction speed, they sacrifice image quality. Furthermore, these methods’ demand for extensive labeled training data is experimentally burdensome. Here, we propose an unsupervised physics-informed neural network reconstruction method, PtychoPINN, that retains the factor of 100-to-1000 speedup of deep learning-based reconstruction while improving reconstruction quality by combining the diffraction forward map with real-space constraints from overlapping measurements. In particular, PtychoPINN gains a factor of 4 in linear resolution and an 8 dB improvement in PSNR while also accruing improvements in generalizability and robustness. This blend of performance and computational efficiency offers exciting prospects for high-resolution real-time imaging in high-throughput environments such as X-ray free electron lasers (XFELs) and diffraction-limited light sources.

Reduced-order model-based variational inference with normalizing flows for Bayesian elliptic inverse problems

We propose a reduced-order model-based variational inference method with normalizing flows for Bayesian elliptic inverse problems. The coefficient of the elliptic PDE is represented by a finite number of parameters. We aim to estimate the posterior distributions of these parameters in the framework of variational inference, and the approximation of the posterior distribution is constructed through a normalizing flow. Moreover, as data of inputs and outputs of the forward problem are accumulated during the training of the normalizing flow, we can naturally exploit the low-dimensional intrinsic structure of the forward elliptic PDE using reduced-order models based on these data, in which information of the true posterior is incorporated. We construct a low-dimensional set of data-driven basis functions in the solution space using proper orthogonal decomposition (POD) and train a neural network that maps the parameters to the coefficients of these data-driven basis functions. The surrogate forward map, which is the combination of the reduced-order model and the parameter-to-coefficient neural network, is then applied to the Bayesian elliptic inverse problem, where in each iteration the forward problem is evaluated in the space spanned by the POD basis functions using the surrogate forward map. Thus the inversion will be significantly accelerated as the surrogate forward map is essentially a low-dimensional model. We present numerical examples to show the accuracy and efficiency of the proposed method.

A statistical framework for domain shape estimation in Stokes flows

We develop and implement a Bayesian approach for the estimation of the shape of a two dimensional annular domain enclosing a Stokes flow from sparse and noisy observations of the enclosed fluid. Our setup includes the case of direct observations of the flow field as well as the measurement of concentrations of a solute passively advected by and diffusing within the flow. Adopting a statistical approach provides estimates of uncertainty in the shape due both to the non-invertibility of the forward map and to error in the measurements. When the shape represents a design problem of attempting to match desired target outcomes, this ‘uncertainty’ can be interpreted as identifying remaining degrees of freedom available to the designer. We demonstrate the viability of our framework on three concrete test problems. These problems illustrate the promise of our framework for applications while providing a collection of test cases for recently developed Markov chain Monte Carlo algorithms designed to resolve infinite-dimensional statistical quantities.

Series reversion for electrical impedance tomography with modeling errors *

This work extends the results of Garde and Hyvönen (2022 Math. Comput. 91 1925–1953) on series reversion for Calderón’s problem to the case of realistic electrode measurements, with both the internal admittivity of the investigated body and the contact admittivity at the electrode-object interfaces treated as unknowns. The forward operator, sending the internal and contact admittivities to the linear electrode current-to-potential map, is first proven to be analytic. A reversion of the corresponding Taylor series yields a family of numerical methods of different orders for solving the inverse problem of electrical impedance tomography, with the possibility to employ different parametrizations for the unknown internal and boundary admittivities. The functionality and convergence of the methods is established only if the employed finite-dimensional parametrization of the unknowns allows the Fréchet derivative of the forward map to be injective, but we also heuristically extend the methods to more general settings by resorting to regularization motivated by Bayesian inversion. The performance of this regularized approach is tested via three-dimensional numerical examples based on simulated data. The effect of modeling errors related to electrode shapes and contact admittances is a focal point of the numerical studies.

Neural network control design for solid composite materials

An innovative numerical method based on a neural network approach is used to solve inverse problems involving the Dirichlet eigenfrequencies for different partial differential operators in bounded domains filled with solid composite materials. The inhomogeneity of the investigated materials is characterized by a vector that is designed to control the constituent mixture of solid homogeneous materials that compose these materials. Using the finite element method, we create a training set for a forward artificial neural network, solving the forward problem. A forward nonlinear map of the Dirichlet eigenfrequencies as a function of the vector design parameter is then obtained. This forward relationship is inverted and applied to obtain a training set for an inverse radial basis neural network, solving the aforementioned inverse problem. Numerical examples that demonstrate the applicability of this methodology are presented.

On the Spatial Response of Electromagnetic Flowmeters

This paper investigates the spatial response of electromagnetic flowmeters, which are commonly used to measure bulk flow in various industrial applications. While most flowmeters focus on measuring the overall flow rate, this paper explores the response of a flowmeter to the spatial distribution of the flow and develops a new dipole form of the forward map. By writing the Neumann Green’s function and using the dipole form, we investigate reconstruction of the distribution of flow within the pipe. The paper presents examples of flow tomography using a truncated singular value reconstruction, which enables accurate visualization of the flow field. By examining the spatial response of electromagnetic flowmeters and presenting a new dipole form of the forward map, this paper contributes to the ongoing effort to improve the accuracy and informativeness of flow measurements in industrial settings.

Convergence Rates for Learning Linear Operators from Noisy Data

This paper studies the learning of linear operators between infinite-dimensional Hilbert spaces. The training data comprises pairs of random input vectors in a Hilbert space and their noisy images under an unknown self-adjoint linear operator. Assuming that the operator is diagonalizable in a known basis, this work solves the equivalent inverse problem of estimating the operator's eigenvalues given the data. Adopting a Bayesian approach, the theoretical analysis establishes posterior contraction rates in the infinite data limit with Gaussian priors that are not directly linked to the forward map of the inverse problem. The main results also include learning-theoretic generalization error guarantees for a wide range of distribution shifts. These convergence rates quantify the effects of data smoothness and true eigenvalue decay or growth, for compact or unbounded operators, respectively, on sample complexity. Numerical evidence supports the theory in diagonal and non-diagonal settings.

Solving Traveltime Tomography with Deep Learning

This paper introduces a neural network approach for solving two-dimensional traveltime tomography (TT) problems based on the eikonal equation. The mathematical problem of TT is to recover the slowness field of a medium based on the boundary measurement of the traveltimes of waves going through the medium. This inverse map is high-dimensional and nonlinear. For the circular tomography geometry, a perturbative analysis shows that the forward map can be approximated by a vectorized convolution operator in the angular direction. Motivated by this and filtered back-projection, we propose an effective neural network architecture for the inverse map using the recently proposed BCR-Net, with weights learned from training datasets. Numerical results demonstrate the efficiency of the proposed neural networks.

Local recovery of a piecewise constant anisotropic conductivity in EIT on domains with exposed corners

We study the local recovery of an unknown piecewise constant anisotropic conductivity in electric impedance tomography on certain bounded Lipschitz domains Ω in with corners. The measurement is conducted on a connected open subset of the boundary of Ω containing corners and is given as a localized Neumann-to-Dirichlet map. The above unknown conductivity is defined via a decomposition of Ω into polygonal cells. Specifically, we consider a parallelogram-based decomposition and a trapezoid-based decomposition. We assume that the decomposition is known, but the conductivity on each cell is unknown. We prove that the local recovery is almost surely true near a known piecewise constant anisotropic conductivity γ 0. We do so by proving that the injectivity of the Fréchet derivative of the forward map F, say, at γ 0 is almost surely true. The proof presented, here, involves defining different classes of decompositions for γ 0 and a perturbation or contrast H in a proper way so that we can find in the interior of a cell for γ 0 exposed single or double corners of a cell of suppH for the former decomposition and latter decomposition, respectively. Then, by adapting the usual proof near such corners, we establish the aforementioned injectivity.

On the Activation Space of ReLU equipped Deep Neural Networks

Modern Deep Neural Networks are getting wider and deeper in their architecture design. However, with an increasing number of parameters the decision mechanisms becomes more opaque. Therefore, there is a need for understanding the structures arising in the hidden layers of deep neural networks.In this work, we present a new mathematical framework for describing the canonical polyhedral decomposition in the input space, and in addition, we introduce the notions of collapsing- and preserving patches, pertinent to understanding the forward map and the activation space they induce.The activation space can be seen as the output of a layer and, in the particular case of ReLU activations, we prove that this output has the structure of a polyhedral complex.

A Bernstein–von-Mises theorem for the Calderón problem with piecewise constant conductivities

This note considers a finite dimensional statistical model for the Calderón problem with piecewise constant conductivities. In this setting it is shown that injectivity of the forward map and its linearisation suffice to prove the invertibility of the information operator, resulting in a Bernstein–von-Mises theorem and optimality guarantees for estimation by Bayesian posterior means.

Change Alignment-Based Image Transformation for Unsupervised Heterogeneous Change Detection

Change detection (CD) with heterogeneous images is currently attracting extensive attention in remote sensing. In order to make heterogeneous images comparable, the image transformation methods transform one image into the domain of another image, which can simultaneously obtain a forward difference map (FDM) and backward difference map (BDM). However, previous methods only fuse the FDM and BDM in the post-processing stage, which cannot fundamentally improve the performance of CD. In this paper, a change alignment-based change detection (CACD) framework for unsupervised heterogeneous CD is proposed to deeply utilize the complementary information of the FDM and BDM in the image transformation process, which enhances the effect of domain transformation, thus improving CD performance. To reduce the dependence of the transformation network on labeled samples, we propose a graph structure-based strategy of generating prior masks to guide the network, which can reduce the influence of changing regions on the transformation network in an unsupervised way. More importantly, based on the fact that the FDM and BDM are representing the same change event, we perform change alignment during the image transformation, which can enhance the image transformation effect and enable FDM and BDM to effectively indicate the real change region. Comparative experiments are conducted with six state-of-the-art methods on five heterogeneous CD datasets, showing that the proposed CACD achieves the best performance with an average overall accuracy (OA) of 95.9% on different datasets and at least 6.8% improvement in the kappa coefficient.

Recovery of a space-time-dependent diffusion coefficient in subdiffusion: stability, approximation and error analysis

Abstract In this work we study an inverse problem of recovering a space-time-dependent diffusion coefficient in the subdiffusion model from the distributed observation, where the mathematical model involves a Djrbashian–Caputo fractional derivative of order $\alpha \in (0,1)$ in time. The main technical challenges of both theoretical and numerical analyses lie in the limited smoothing properties due to the fractional differential operator and high degree of nonlinearity of the forward map from the unknown diffusion coefficient to the distributed observation. We establish two conditional stability results using a novel test function, which leads to a stability bound in $L^2(0,T;L^2(\varOmega ))$ under a suitable positivity condition. The positivity condition is verified for a large class of problem data. Numerically, we develop a rigorous procedure for recovering the diffusion coefficient based on a regularized least-squares formulation, which is then discretized by the standard Galerkin method with continuous piecewise linear elements in space and backward Euler convolution quadrature in time. We provide a complete error analysis of the fully discrete formulation, by combining several new error estimates for the direct problem (optimal in terms of data regularity), a discrete version of fractional maximal $L^p$ regularity and a nonstandard energy argument. Under the positivity condition, we obtain a standard $\ell ^2(L^2(\varOmega ))$ error estimate consistent with the conditional stability. Further, we illustrate the analysis with some numerical examples.

Fast inverse estimation of hydraulic conductivity field based on a deep convolutional-cycle generative adversarial neural network

This study proposes a novel deep convolutional-cycle generative adversarial neural network (DC-CGAN) for estimating the hydraulic conductivity field K from the observed head field H. The estimation of the heterogeneous hydraulic conductivity is apparently a high-dimensional inverse problem. Although the indirect approach is popularly utilized to solve the inverse problem, when compared with the indirect approach, the direct approach can conduct the inversion task faster without any iteration process. The key of the direct approach is to establish the inverse operator which maps the observed head to hydraulic conductivity. However, the complex inverse mapping relationship and high-dimensional estimation have always challenged the direct approach. Owing to the potential ability to handle high-dimensional images, six residual blocks with convolutional kernels have been designed as the generators of DC-CGAN to capture the potential distribution for enhancing the generation of estimation images similar to the real hydraulic conductivity. However, the traditional adversarial pattern of one-way generator and discriminator might suffer from the equifinality from different parameters. Therefore, to alleviate the equifinality, a cycle adversarial training pattern involving two generators and discriminators has been employed to supervise the map. Particularly, this pattern performs a forward map from H to K and then the reconstructed K back to H. The reconstruction process has been a supervision to the forward map process. A hypothetical case with 40×30 unknown K values has been designed to evaluate the performance of DC-CGAN, and the results indicate that DC-CGAN has achieved a desired generalization accuracy of MRE of 9.25 % and SSIM of 0.85 with the swift running speed of 0.2 s, which outperforms indirect approach. The application of the proposed DC-CGAN shows that it can provide a reliable and fast high-dimensional estimation task for the heterogenous hydraulic conductivity.

A fourth-order unfitted characteristic finite element method for free-boundary problems

A fourth-order unfitted characteristic finite element method (UCFEM) is proposed to solve free-boundary problems of time-dependent partial differential equations (PDEs). The boundary of the domain is implicitly driven by the solution of the PDE, while the PDE is proposed on the time-varying unknown domain. In this way, they form a coupled nonlinear system. The key ingredient of the UCFEM is to trace the domain explicitly with a fourth-order forward flow map and discretize the PDE in time with a fourth-order backward flow map. The PDE is solved with a fourth-order unfitted finite element method on a fixed mesh. Since the boundary of the domain is expressed explicitly with cubic spline functions, quadrature on cut elements can be done accurately and efficiently even the domain undergoes severe deformations. The UCFEM provides a framework of designing high-order numerical methods for free-boundary problems. We apply the UCFEM to a two-dimensional convection-diffusion equation and Navier-Stokes equations, and obtain the overall fourth order of accuracy. With extensive numerical experiments, we show that the proposed method can achieve the fourth-order convergence even on severely deformed domains.

Variational inference for nonlinear inverse problems via neural net kernels: Comparison to Bayesian neural networks, application to topology optimization

Inverse problems and, in particular, inferring unknown or latent parameters from data are ubiquitous in engineering simulations. A predominant viewpoint in identifying unknown parameters is Bayesian inference where both prior information about the parameters and the information from the observations via likelihood evaluations are incorporated into the inference process. In this paper, we adopt a similar viewpoint with a slightly different numerical procedure from standard inference approaches to provide insight about the localized behavior of unknown underlying parameters. We present a variational inference approach which mainly incorporates the observation data in a point-wise manner, i.e. we invert a limited number of observation data leveraging the gradient information of the forward map with respect to parameters, and find true individual samples of the latent parameters when the forward map is noise-free and one-to-one. For statistical calculations (as the ultimate goal in simulations), a large number of samples are generated from a trained neural network which serves as a transport map from the prior to posterior latent parameters. Our neural network machinery, developed as part of the inference framework and referred to as Neural Net Kernels (NNK), is based on hierarchical (deep) kernels which provide greater flexibility for training compared to standard neural networks. We showcase the effectiveness of our inference procedure in identifying bimodal and irregular distributions compared to a number of approaches including a maximum a posteriori probability (MAP)-based approach, Markov Chain Monte Carlo sampling with both Metropolis–Hastings and Hamiltonian Monte Carlo algorithms, and a Bayesian neural network approach, namely the widely-known Bayes by Backprop algorithm via a pedagogical example. We further apply our inference procedure to two and three dimensional topology optimization problems where we identify the latent parameters in the random field elastic modulus, modeled as a Karhunen–Loéve expansion, considering the high dimensional design iterate-displacement pair as training data. As future research, we discuss the application of this inference approach in identifying constitutive models in nonlinear elasticity and development of fast linear algebraic solvers for large scale similar, i.e. finite element-based inverse problems calculations.

Parametric derivatives in inverse conductivity problems with total variation regularization

The focus of this paper is an improved differentiability result for the forward map in inverse problems involving elliptic partial differential equations, and examination of its significance in the context of the electrical impedance tomography (EIT) problem with total variation (TV) regularization. We base our analysis on the Fréchet derivative of the mapping which takes a given conductivity function (spatially varying) in an electrostatic model to a corresponding elliptic PDE solution, and we develop the implications of a certain compactness property of the parameter space. By following this approach, we show Fréchet differentiability with a weaker norm (the L1 norm) for the parameter space than is usually used (the L∞ norm), thus improving the Fréchet differentiability result. The EIT problem with TV regularization is well studied in the literature, and several authors have addressed the Fréchet differentiability question. However, to the best of our knowledge and as we argue, our result is the strongest analytical result in this context. Many derivative-based methods such as Gauss–Newton and Levenburg-Marquardt lie at the heart of many proposed methods for EIT, and the results described herein for these derivative calculations provide a firm theoretical footing for them.