Forward Map Research Articles

Imaging Anisotropic Conductivities from Current Densities

In this paper, we propose and analyze a reconstruction algorithm for imaging an anisotropic conductivity tensor in a second-order elliptic PDE with a nonzero Dirichlet boundary condition from internal current densities. It is based on a regularized output least-squares formulation with the standard $L^2(\Omega)^{d,d}$ penalty, which is then discretized by the standard Galerkin finite element method. We establish the continuity and differentiability of the forward map with respect to the conductivity tensor in the $L^p(\Omega)^{d,d}$-norms, the existence of minimizers and optimality systems of the regularized formulation using the concept of H-convergence. Further, we provide a detailed analysis of the discretized problem, especially the convergence of the discrete approximations with respect to the mesh size, using the discrete counterpart of H-convergence. In addition, we develop a projected Newton algorithm for solving the first-order optimality system. We present extensive two-dimensional numerical examples to show the efficiency of the proposed method.

Series reversion in Calderón’s problem

This work derives explicit series reversions for the solution of Calderón’s problem. The governing elliptic partial differential equation is ∇ ⋅ ( A ∇ u ) = 0 \nabla \cdot (A\nabla u)=0 in a bounded Lipschitz domain and with a matrix-valued coefficient. The corresponding forward map sends A A to a projected version of a local Neumann-to-Dirichlet operator, allowing for the use of partial boundary data and finitely many measurements. It is first shown that the forward map is analytic, and subsequently reversions of its Taylor series up to specified orders lead to a family of numerical methods for solving the inverse problem with increasing accuracy. The convergence of these methods is shown under conditions that ensure the invertibility of the Fréchet derivative of the forward map. The introduced numerical methods are of the same computational complexity as solving the linearised inverse problem. The analogous results are also presented for the smoothened complete electrode model.

Iterated Kalman methodology for inverse problems

This paper is focused on the optimization approach to the solution of inverse problems. We introduce a stochastic dynamical system in which the parameter-to-data map is embedded, with the goal of employing techniques from nonlinear Kalman filtering to estimate the parameter given the data. The extended Kalman filter (which we refer to as ExKI in the context of inverse problems) can be effective for some inverse problems approached this way, but is impractical when the forward map is not readily differentiable and is given as a black box, and also for high dimensional parameter spaces because of the need to propagate large covariance matrices. Application of ensemble Kalman filters, for example use of the ensemble Kalman inversion (EKI) algorithm, has emerged as a useful tool which overcomes both of these issues: it is derivative free and works with a low-rank covariance approximation formed from the ensemble. In this paper, we work with the ExKI, EKI, and a variant on EKI which we term unscented Kalman inversion (UKI).The paper contains two main contributions. Firstly, we identify a novel stochastic dynamical system in which the parameter-to-data map is embedded. We present theory in the linear case to show exponential convergence of the mean of the filtering distribution to the solution of a regularized least squares problem. This is in contrast to previous work in which the EKI has been employed where the dynamical system used leads to algebraic convergence to an unregularized problem. Secondly, we show that the application of the UKI to this novel stochastic dynamical system yields improved inversion results, in comparison with the application of EKI to the same novel stochastic dynamical system.The numerical experiments include proof-of-concept linear examples and various applied nonlinear inverse problems: learning of permeability parameters in subsurface flow; learning the damage field from structure deformation; learning the Navier-Stokes initial condition from solution data at positive times; learning subgrid-scale parameters in a general circulation model (GCM) from time-averaged statistics.

Neural Fields in Visual Computing and Beyond

AbstractRecent advances in machine learning have led to increased interest in solving visual computing problems using methods that employ coordinate‐based neural networks. These methods, which we callneural fields, parameterize physical properties of scenes or objects across space and time. They have seen widespread success in problems such as 3D shape and image synthesis, animation of human bodies, 3D reconstruction, and pose estimation. Rapid progress has led to numerous papers, but a consolidation of the discovered knowledge has not yet emerged. We provide context, mathematical grounding, and a review of over 250 papers in the literature on neural fields. InPart I, we focus on neural field techniques by identifying common components of neural field methods, including different conditioning, representation, forward map, architecture, and manipulation methods. InPart II, we focus on applications of neural fields to different problems in visual computing, and beyond (e.g., robotics, audio). Our review shows the breadth of topics already covered in visual computing, both historically and in current incarnations, and highlights the improved quality, flexibility, and capability brought by neural field methods. Finally, we present a companion website that acts as a living database that can be continually updated by the community.

Deep neural-network based optimization for the design of a multi-element surface magnet for MRI applications

We present a combination of a CNN-based encoder with an analytical forward map for solving inverse problems. We call it an encoder-analytic (EA) hybrid model. It does not require a dedicated training dataset and can train itself from the connected forward map in a direct learning fashion. A separate regularization term is not required either, since the forward map also acts as a regularizer. As it is not a generalization model it does not suffer from overfitting. We further show that the model can be customized to either find a specific target solution or one that follows a given heuristic. As an example, we apply this approach to the design of a multi-element surface magnet for low-field magnetic resonance imaging (MRI). We further show that the EA model can outperform the benchmark genetic algorithm model currently used for magnet design in MRI, obtaining almost 10 times better results.

Solving inverse wave scattering with deep learning

This paper proposes a neural network approach for solving two classical problems in the two-dimensional inverse wave scattering: far field pattern problem and seismic imaging. The mathematical problem of inverse wave scattering is to recover the scatterer field of a medium based on the boundary measurement of the scattered wave from the medium, which is high-dimensional and nonlinear. For the far field pattern problem under the circular experimental setup, 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 introduced BCR-Net along with the standard convolution layers. Analogously for the seismic imaging problem, we propose a similar neural network architecture under the rectangular domain setup with a depth-dependent background velocity. Numerical results demonstrate the efficiency of the proposed neural networks.

A new approach to calculating fiber fields in 2D vessel cross sections using conformal maps.

An arterial vessel has three layers, namely, the intima, the media and the adventitia. Each of these layers is modeled to have two families of strain-stiffening collagen fibers that are transversely helical. In an unloaded configuration, these fibers are coiled up. In the case of a pressurized lumen, these fibers stretch and start to resist further outward expansion. As the fibers elongate, they stiffen, affecting the mechanical response. Having a mathematical model of vessel expansion is crucial in cardiovascular applications such as predicting stenosis and simulating hemodynamics. Thus, to study the mechanics of the vessel wall under loading, it is important to calculate the fiber configurations in the unloaded configuration. The aim of this paper is to introduce a new technique of using conformal maps to numerically calculate the fiber field in a general arterial cross-section. The technique relies on finding a rational approximation of the conformal map. First, points on the physical cross section are mapped to points on a reference annulus using a rational approximation of the forward conformal map. Next, we find the angular unit vectors at the mapped points, and finally a rational approximation of the inverse conformal map is used to map the angular unit vectors back to vectors on the physical cross section. We have used MATLAB software packages to achieve these goals.

Research on fast video mapping method of borehole in front view

Camera borehole technology is widely used in borehole engineering, which can obtain the structure of deep-buried rock mass. However, when people recognise these images, there will be problems such as heavy workload, low efficiency and easy recognition errors. In order to solve these problems, this paper proposes a fast unfolding mosaic and fusion method of looped image in the forward view video. This paper proposes a borehole image expansion method with autonomous positioning of the borehole centre and adaptive adjustment of the expansion radius to achieve rapid and automatic image expansion. On this basis, an adaptive adjustment algorithm of image stitching points based on grey difference registration is proposed, which solves the defect of determining stitching points by artificial experience. The practical results show that the image mosaic technology is simple, the processing is flexible and efficient and the forward view borehole expansion map obtained has an ideal effect.

Determining the nonlinearity in an acoustic wave equation

We consider an undetermined coefficient inverse problem for a nonlinear partial differential equation describing high‐intensity ultrasound propagation as widely used in medical imaging and therapy. The usual nonlinear term in the standard model using the Westervelt equation in pressure formulation is of the form ppt. However, this should be considered as a low‐order approximation to a more complex physical model where higher order terms will be required. Here we assume a more general case where the form taken is f(p) pt and f is unknown and must be recovered from data measurements. Corresponding to the typical measurement setup, the overposed data consist of time trace observations of the acoustic pressure at a single point or on a one‐dimensional set Σ representing the receiving transducer array at a fixed time. Additionally to an analysis of well‐posedness of the resulting pde, we show injectivity of the linearized forward map from f to the overposed data and use this as motivation for several iterative schemes to recover f. Numerical simulations will also be shown to illustrate the efficiency of the methods.

Stochastic Modeling and identification of material parameters on structures produced by additive manufacturing

A methodology enabling the representation, sampling, and identification of spatially-dependent stochastic material parameters on complex structures produced by additive manufacturing is presented. The modeling component builds upon earlier works by the authors and relies on the combination of two ingredients. First, a fractional stochastic partial differential equation is introduced and parameterized in order to automatically capture the complex features of additively manufactured parts. Information-theoretic transport maps are subsequently introduced with the aim of ensuring well-posedness in the forward propagation problem. The identification of stochastic elasticity tensors on titanium scaffolds produced by laser powder bed fusion is then discussed. To this end, we consider an isotropic approximation at a mesoscale where fluctuations are aggregated over several layers, and address both the calibration and validation of the probabilistic model by using different sets of physical structural experiments. Despite the high sensitivity of the forward map to applied boundary conditions, geometrical parameters, and structural porosity, it is shown that the calibrated stochastic model can generate non-vanishing probability levels for all experimental observations.

Stability of the Non-abelian X-ray Transform in Dimension ge 3

Non-abelian X-ray tomography seeks to recover a matrix potential Phi :Mrightarrow {mathbb {C}}^{mtimes m} in a domain M from measurements of its so-called scattering data C_Phi at partial M. For dim Mge 3 (and under appropriate convexity and regularity conditions), injectivity of the forward map Phi mapsto C_Phi was established in (Paternain et al. in Am J Math 141(6):1707–1750, 2019). The present article extends this result by proving a Hölder-type stability estimate. As an application, a statistical consistency result for dim M =2 (Monard et al. in Commun Pure Appl Math, 2019) is generalised to higher dimensions. The injectivity proof in (Paternain et al. in Am J Math 141(6):1707–1750, 2019) relies on a novel method by Uhlmann and Vasy (Invent Math 205(1):83–120, 2016), which first establishes injectivity in a shallow layer below partial M and then globalises this by a layer stripping argument. The main technical contribution of this paper is a more quantitative version of these arguments, in particular, proving uniform bounds on layer depth and stability constants.

On Hölder solutions to the spiral winding problem

The winding problem concerns understanding the regularity of functions which map a line segment onto a spiral. This problem has relevance in fluid dynamics and conformal welding theory, where spirals arise naturally. Here we interpret ‘regularity’ in terms of Hölder exponents and establish sharp results for spirals with polynomial winding rates, observing that the sharp Hölder exponent of the forward map and its inverse satisfy a formula reminiscent of Sobolev conjugates. We also investigate the dimension theory of these spirals, in particular, the Assouad dimension, Assouad spectrum and box dimensions. The aim here is to compare the bounds on the Hölder exponents in the winding problem coming directly from knowledge of dimension (and how dimension distorts under Hölder image) with the sharp results. We find that the Assouad spectrum provides the best information, but that even this is not sharp. We also find that the Assouad spectrum is the only ‘dimension’ which distinguishes between spirals with different polynomial winding rates in the superlinear regime.

Error Control of the Numerical Posterior with Bayes Factors in Bayesian Uncertainty Quantification

In this paper, we address the numerical posterior error control problem for the Bayesian approach to inverse problems or recently known as Bayesian Uncertainty Quantification (UQ). We generalize the results of Capistrán et al. (2016) to (a priori) expected Bayes factors (BF) and in a more general, infinite-dimensional setting. In this inverse problem, the unavoidable numerical approximation of the Forward Map (FM, i.e., the regressor function), arising from the numerical solution of a system of differential equations, demands error estimates of the corresponding approximate numerical posterior distribution. Our approach is to make such comparisons in the setting of Bayesian model selection and BFs. The main result of this paper is a bound on the absolute global error tolerated by the numerical solver of the FM in order to keep the BF of the numerical versus the theoretical posterior near one. For two examples, we provide a detailed analysis of the computation and implementation of the introduced bound. Furthermore, we show that the resulting numerical posterior turns out to be nearly identical from the theoretical posterior, given the control of the BF near one.

ST-LBAGAN: Spatio-temporal learnable bidirectional attention generative adversarial networks for missing traffic data imputation

Real-time, accurate and comprehensive traffic flow data is the key of intelligent transportation systems to provide efficient services for urban transportation. In the process of collecting data, there are many factors causing data loss, which needs to be supplemented and repaired to reduce the instability, and improve the precision of system application in the intelligent transportation system. This paper proposes a Spatio-Temporal Learnable Bidirectional Attention Generative Adversarial Networks (ST-LBAGAN) for missing traffic data imputation. First, we take external factors, historical observations, incomplete data, and masked image as the input of generator, and obtain the missing data imputation by using binary classification as the output of the discriminator. Secondly, the encoder and decoder of generator are constructed on the basis of the U-Net. The forward attention map and the reverse attention map of learnable bidirectional attention correspond to the encoder and the decoder respectively to effectively obtain the spatial–temporal random characteristics of traffic flow. Thirdly, high-level and low-level features, in the encoder and decoder, are combined by multiple skip connections. Furthermore, a new objective function is optimized by combining masked reconstruction loss, perceptual loss, discriminative loss and adversarial loss to improve the data imputation ability. Finally, our model is well-adapted on the Beijing taxi GPS dataset. The experimental results show that an improved state-of-the-art performance is achieved on various standard benchmarks.

Markov Chain Generative Adversarial Neural Networks for Solving Bayesian Inverse Problems in Physics Applications

In the context of solving inverse problems for physics applications within a Bayesian framework, we present a new approach, Markov Chain Generative Adversarial Neural Networks (MCGANs), to alleviate the computational costs associated with solving the Bayesian inference problem. GANs pose a very suitable framework to aid in the solution of Bayesian inference problems, as they are designed to generate samples from complicated high-dimensional distributions. By training a GAN to sample from a low-dimensional latent space and then embedding it in a Markov Chain Monte Carlo method, we can highly efficiently sample from the posterior, by replacing both the high-dimensional prior and the expensive forward map. We prove that the proposed methodology converges to the true posterior in the Wasserstein-1 distance and that sampling from the latent space is equivalent to sampling in the high-dimensional space in a weak sense. The method is showcased on three test cases where we perform both state and parameter estimation simultaneously. The approach is shown to be up to two orders of magnitude more accurate than alternative approaches while also being up to an order of magnitude computationally faster, in several test cases, including the important engineering setting of detecting leaks in pipelines.

Deep Learning Model-Aware Regulatization With Applications to Inverse Problems

There are various inverse problems – including reconstruction problems arising in medical imaging - where one is often aware of the forward operator that maps variables of interest to the observations. It is therefore natural to ask whether such knowledge of the forward operator can be exploited in deep learning approaches increasingly used to solve inverse problems. In this paper, we provide one such way via an analysis of the generalisation error of deep learning approaches to inverse problems. In particular, by building on the algorithmic robustness framework, we offer a generalisation error bound that encapsulates key ingredients associated with the learning problem such as the complexity of the data space, the size of the training set, the Jacobian of the deep neural network and the Jacobian of the composition of the forward operator with the neural network. We then propose a ‘plug-and-play’ regulariser that leverages the knowledge of the forward map to improve the generalization of the network. We likewise also use a new method allowing us to tightly upper bound the Jacobians of the relevant operators that is much more computationally efficient than existing ones. We demonstrate the efficacy of our model-aware regularised deep learning algorithms against other state-of-the-art approaches on inverse problems involving various sub-sampling operators such as those used in classical compressed sensing tasks, image super-resolution problems and accelerated Magnetic Resonance Imaging (MRI) setups.

ERROR CONTROL IN THE NUMERICAL POSTERIOR DISTRIBUTION IN THE BAYESIAN UQ ANALYSIS OF A SEMILINEAR EVOLUTION PDE

We elaborate on results obtained in \cite{christen2018} for controlling the numerical posterior error for Bayesian UQ problems, now considering forward maps arising from the solution of a semilinear evolution partial differential equation. Results in \cite{christen2018} demand an estimate for the absolute global error (AGE) of the numeric forward map. Our contribution is a numerical method for computing the AGE for semilinear evolution PDEs and shows the potential applicability of \cite{christen2018} in this important wide range family of PDEs. Numerical examples are given to illustrate the efficiency of the proposed method, obtaining numerical posterior distributions for unknown parameters that are nearly identical to the corresponding theoretical posterior, by keeping their Bayes factor close to 1.

Ensemble Kalman Inversion for nonlinear problems: Weights, consistency, and variance bounds

<p style='text-indent:20px;'>Ensemble Kalman Inversion (EnKI) [<xref ref-type="bibr" rid="b23">23</xref>] and Ensemble Square Root Filter (EnSRF) [<xref ref-type="bibr" rid="b36">36</xref>] are popular sampling methods for obtaining a target posterior distribution. They can be seem as one step (the analysis step) in the data assimilation method Ensemble Kalman Filter [<xref ref-type="bibr" rid="b17">17</xref>,<xref ref-type="bibr" rid="b3">3</xref>]. Despite their popularity, they are, however, not unbiased when the forward map is nonlinear [<xref ref-type="bibr" rid="b12">12</xref>,<xref ref-type="bibr" rid="b16">16</xref>,<xref ref-type="bibr" rid="b25">25</xref>]. Important Sampling (IS), on the other hand, obtains the unbiased sampling at the expense of large variance of weights, leading to slow convergence of high moments.</p><p style='text-indent:20px;'>We propose WEnKI and WEnSRF, the weighted versions of EnKI and EnSRF in this paper. It follows the same gradient flow as that of EnKI/EnSRF with weight corrections. Compared to the classical methods, the new methods are unbiased, and compared with IS, the method has bounded weight variance. Both properties will be proved rigorously in this paper. We further discuss the stability of the underlying Fokker-Planck equation. This partially explains why EnKI, despite being inconsistent, performs well occasionally in nonlinear settings. Numerical evidence will be demonstrated at the end.</p>

Randomized reduced forward models for efficient Metropolis–Hastings MCMC, with application to subsurface fluid flow and capacitance tomography

Bayesian modelling and computational inference by Markov chain Monte Carlo (MCMC) is a principled framework for large-scale uncertainty quantification, though is limited in practice by computational cost when implemented in the simplest form that requires simulating an accurate computer model at each iteration of the MCMC. The delayed acceptance Metropolis–Hastings MCMC leverages a reduced model for the forward map to lower the compute cost per iteration, though necessarily reduces statistical efficiency that can, without care, lead to no reduction in the computational cost of computing estimates to a desired accuracy. Randomizing the reduced model for the forward map can dramatically improve computational efficiency, by maintaining the low cost per iteration but also avoiding appreciable loss of statistical efficiency. Randomized maps are constructed by a posteriori adaptive tuning of a randomized and locally-corrected deterministic reduced model. Equivalently, the approximated posterior distribution may be viewed as induced by a modified likelihood function for use with the reduced map, with parameters tuned to optimize the quality of the approximation to the correct posterior distribution. Conditions for adaptive MCMC algorithms allow practical approximations and algorithms that have guaranteed ergodicity for the target distribution. Good statistical and computational efficiencies are demonstrated in examples of calibration of large-scale numerical models of geothermal reservoirs and electrical capacitance tomography.

Structure of the Singular Manifold of Protein Backbone and Its Applications

This paper aims to study the structure of the singular manifold of a fragment of protein backbone using the method of robot kinematics. By modeling chemical bonds as rigid links, and torsional motion around chemical bonds as revolute joints, a fragment of protein backbone turns into a kinematic linkage. The singular manifold of the forward kinematic map of this linkage contains important information about the structure of its inverse kinematics map, and the results also apply to the study of the conformation space of protein loops. By splitting the forward kinematic map into an orientation and position map, we derive equations of the singular manifolds for both maps and analyze their geometric and combinatorial structures. We discuss the application of singular manifolds in several problems of computational biology, including minimal parametrization of protein conformation space and conformation space sampling of protein loops.