Ill-posed Research Articles

Local exponential stabilisation of the delayed Fisher's equation

Exponential stabilisation is undoubtedly an essential property to study for any control system. This property is often difficult to establish for nonlinear distributed systems and even more difficult if the state of such a system is subject to a time-delay. In this context, this paper is devoted to the study of the well-posedness and exponential stabilisation problems for the one-dimensional delayed Fisher's partial differential equation (PDE) posed in a bounded interval, with homogeneous Dirichlet condition at the left boundary and controlled at the right boundary by Dirichlet condition. Based on the infinite dimensional backstepping method for the linear and undelayed corresponding case, a feedback law is built. Under this feedback, we prove that the closed-loop system is well-posed in L 2 ( 0 , 1 ) and locally exponentially stable in the topology of the L 2 ( 0 , 1 ) -norm by the use of semigroup theory relating to differential difference equations and Lyapunov–Razumikhin analysis, respectively. A numerical example that reflects the effectiveness of the proposed approach is given.

A Learned-SVD Approach to the Electromagnetic Inverse Source Problem.

We propose an artificial intelligence approach based on deep neural networks to tackle a canonical 2D scalar inverse source problem. The learned singular value decomposition (L-SVD) based on hybrid autoencoding is considered. We compare the reconstruction performance of L-SVD to the Truncated SVD (TSVD) regularized inversion, which is a canonical regularization scheme, to solve an ill-posed linear inverse problem. Numerical tests referring to far-field acquisitions show that L-SVD provides, with proper training on a well-organized dataset, superior performance in terms of reconstruction errors as compared to TSVD, allowing for the retrieval of faster spatial variations of the source. Indeed, L-SVD accommodates a priori information on the set of relevant unknown current distributions. Different from TSVD, which performs linear processing on a linear problem, L-SVD operates non-linearly on the data. A numerical analysis also underlines how the performance of the L-SVD degrades when the unknown source does not match the training dataset.

An accelerated inexact Newton regularization scheme with a learned feature-selection rule for non-linear inverse problems

With computational inverse problems, it is desirable to develop an efficient inversion algorithm to find a solution from measurement data through a mathematical model connecting the unknown solution and measurable quantity based on the first principles. However, most of mathematical models represent only a few aspects of the physical quantity of interest, and some of them are even incomplete in the sense that one measurement corresponds to many solutions satisfying the forward model. In this paper, in light of the recently developed iNETT method in (2023 Inverse Problems 39 055002), we propose a novel iterative regularization method for efficiently solving non-linear ill-posed inverse problems with potentially non-injective forward mappings and (locally) non-stable inversion mappings. Our approach integrates the inexact Newton iteration, the non-stationary iterated Tikhonov regularization, the two-point gradient acceleration method, and the structure-free feature-selection rule. The main difficulty in the regularization technique is how to design an appropriate regularization penalty, capturing the key feature of the unknown solution. To overcome this difficulty, we replace the traditional regularization penalty with a deep neural network, which is structure-free and can identify the correct solution in a huge null space. A comprehensive convergence analysis of the proposed algorithm is performed under standard assumptions of regularization theory. Numerical experiments with comparisons with other state-of-the-art methods for two model problems are presented to show the efficiency of the proposed approach.

Spectral reconstruction using neural networks in filter-array-based chip-size spectrometers

Abstract Spectral reconstruction in filter-based miniature spectrometers remains challenging due to the ill-posed nature of identifying stable solutions. Even minor deviations in sensor data can cause misleading reconstruction outcomes, particularly in the absence of proper regularization techniques. While previous research has attempted to mitigate this instability by incorporating neural networks into the reconstruction pipeline to denoise the data before reconstruction or correct it after reconstruction, these approaches have not fully resolved the underlying issue. This work functions as a proof-of-concept for data-driven reconstruction that relies exclusively on neural networks, thereby circumventing the need to address the ill-posed inverse problem. We curate a dataset holding transmission spectra from various colored foils, commonly used in theatrical, and train five distinct neural networks optimized for spectral reconstruction. Subsequently, we benchmark these networks against each other and compare their reconstruction capabilities with a linear reconstruction model to show the applicability of cognitive sensors to the problem of spectral reconstruction. In our testing, we discovered that (i) spectral reconstruction can be achieved using neural networks with an end-to-end approach, and (ii) while a classic linear model can perform equal to neural networks under optimal conditions, the latter can be considered more robust against data deviations.

Hierarchical disentangled representation for image denoising and beyond

Image denoising is a typical ill-posed problem due to complex degradation. Leading methods based on normalizing flows have tried to solve this problem with an invertible transformation instead of a deterministic mapping. However, it is difficult to construct feasible bijective mapping to remove spatial-variant noise while recovering fine texture and structure details due to latent ambiguity in inverse problems. Inspired by a common observation that noise tends to appear in the high-frequency part of the image, we propose a fully invertible denoising method that injects the idea of disentangled learning into a general invertible architecture to split noise from the high-frequency part. More specifically, we decompose the noisy image into clean low-frequency and hybrid high-frequency parts with an invertible transformation and then disentangle case-specific noise and high-frequency components in the latent space. In this way, denoising is made tractable by inversely merging noiseless low and high-frequency parts. Furthermore, we construct a flexible hierarchical disentangling framework, which aims to decompose most of the low-frequency image information while disentangling noise from the high-frequency part in a coarse-to-fine manner. Extensive experiments on real image denoising, JPEG compressed artifact removal, and medical low-dose CT image restoration have demonstrated that the proposed method achieves competitive performance on both quantitative metrics and visual quality, with significantly less computational cost.

Single-shot lensless masked imaging with enhanced self-calibrated phase retrieval.

Single-shot lensless imaging with a binary amplitude mask enables a low-cost and miniaturized configuration for wave field recovery. However, the mask only allows a part of the wave field to be captured, and thus the inverse decoding process becomes a highly ill-posed problem. Here we propose an enhanced self-calibrated phase retrieval (eSCPR) method to realize single-shot joint recovery of mask distribution and the sample's wavefront. In our method, a sparse regularized phase retrieval (SrPR) algorithm is designed to calibrate the mask distribution. Then, a denoising regularized phase retrieval (DrPR) algorithm is constructed to reconstruct the wavefront of the sample. Compared to conventional single-shot methods, our method shows robust and flexible image recovery. Experimental results of different samples are given to demonstrate the superiority of our method.

Digging into Depth and Color Spaces: A Mapping Constraint Network for Depth Super-Resolution

Scene depth super-resolution (DSR) poses an inherently ill-posed problem due to the extremely large space of one-to-many mapping functions from a given low-resolution (LR) depth map, which possesses limited depth information, to multiple plausible high-resolution (HR) depth maps. This characteristic renders the task highly challenging, as identifying an optimal solution becomes significantly intricate amidst this multitude of potential mappings. While simplistic constraints have been proposed to address the DSR task, the relationship between LR and HR depth maps and the color image has not been thoroughly investigated. In this paper, we introduce a novel mapping constraint network (MCNet) that incorporates additional constraints derived from both LR depth maps and color images. This integration aims to optimize the space of mapping functions and enhance the performance of DSR. Specifically, alongside the primary DSR network (DSRNet) dedicated to learning LR-to-HR mapping, we have developed an auxiliary degradation network (ADNet) that operates in reverse, generating the LR depth map from the reconstructed HR depth map to obtain depth features in LR space. To enhance the learning process of DSRNet in LR-to-HR mapping, we introduce two mapping constraints in LR space: 1) the cycle-consistent constraint, which offers additional supervision by establishing a closed loop between LR-to-HR and HR-to-LR mappings, and 2) the region-level contrastive constraint, aimed at reinforcing region-specific HR representations by explicitly modeling the consistency between LR and HR spaces. To leverage the color image effectively, we introduce a feature screening module (FSM) to adaptively fuse color features at different layers, which can simultaneously maintain strong structural context and suppress texture distraction through subspace generation and image projection. Comprehensive experimental results across synthetic and real-world benchmark datasets unequivocally demonstrate the superiority of our proposed method over state-of-the-art DSR methods. Our MCNet achieves an average MAD reduction of 3.7% and 7.5% over state-of-the-art DSR method for \(\times\) 8 and \(\times\) 16 cases on Milddleburry dataset, respectively, without incurring additional costs during inference.

A Bayesian Approach To Locally Varying Regularization In Optical Flow Velocimetry

A recent method for determining accurate, dense velocity fields from a pair of particle images is to solve the optical flow problem, but the ill-posed inverse problem of optical flow velocimetry (OFV) generally entails minimizing a weighted sum of two terms--fidelity and regularization--and the weights in the sum are parameters that require manual tuning based on the flow properties and particle images. This tuning of the weighting parameters makes the experimental applicability of the method much more challenging as the calculated velocity field has demonstrated sensitivity to the value of the weights. This work proposes a hierarchical model for the weighting parameters in the framework of maximum a posteriori (MAP)-based Bayesian optimization approaches. The resulting method is tested on three different synthetic datasets and on experimental particle images. The method is found to be capable of self-adjusting the local weights of the optimization process in real-time while simultaneously determining the velocity field, leading to a more accurate estimate of the velocity field without requiring any dataset-specific manual tuning of parameters.

Generalizing event-based HDR imaging to various exposures

Single-exposure High Dynamic Range Imaging (HDRI), as a typical ill-posed problem, has attracted extensive attention from researchers. However, restoration in real-world scenarios has always been an intractable task due to various exposures and noise artifacts. This work proposes an event-based HDRI framework that generalizes to scenes under various exposures by exploiting the high dynamic range of events. To address the challenge of processing diverse exposures, we propose an exposure-aware network incorporating the exposure attention fusion module, which facilitates the adaptive fusion of SDR image and event features. Moreover, the problem of noise in extremely under-exposed regions and events is effectively alleviated by introducing a self-supervised loss, namely EDDN, which effectively enhances the details of saturated areas while simultaneously decreasing noise. We conduct novel event-based HDRI datasets to evaluate our proposed method for benchmarking with diverse exposed images. Comprehensive experiments have demonstrated that our method outperforms the state-of-the-art.

Instruct Pix-to-3D: Instructional 3D object generation from a single image

We study a challenging task, creating the instructional 3D object based on a single image and human instruction. Although existing methods utilize pre-trained text-to-image diffusion models to optimize Neural Radiance Fields (NeRF) and achieve remarkable results, they are inherently challenged in addressing this task due to two inherent obstacles. First, the written instruction is mismatched or uncorrelated with target images in the distribution of pre-trained text-to-image diffusion models, which leads to insufficient supervision. Another is editing 3D content starting from a single image, which is an ill-posed problem as the image only provides partial information. To tackle the difficulties, we propose an Instruct Pixel-to-3D framework, which exploits the supervisory signals from the given instructions and integrates the domain-specific knowledge drawn from 2D and 3D diffusion models. Specifically, to address the first challenge, we propose instruction guidance to facilitate meticulous image editing using fine-grained instructions and regulate the novel views of 3D objects using positional prompts. To tackle the second difficulty, we propose diffusion guidance for 3D generation, which optimizes the NeRF employing dual guidances: a 2D diffusion guidance derived from the score distillation sampling (SDS) loss and a 3D diffusion guidance embodied by the Zero-1-to-3 loss. Moreover, Instruct Pix-to-3D adopts a coarse-to-fine training strategy aimed at mitigating the complexities inherent in reconstructing scenes from sparse viewpoints. Qualitative and quantitative experiments on the DTU dataset and in-the-wild images demonstrate that our approach could synthesize high-quality and high-fidelity 3D content based on a single image and human instruction.

Low rank approximation in the computation of first kind integral equations with TauToolbox

Tau Toolbox is a mathematical library for the solution of integro-differential problems, based on the spectral Lanczos' Tau method. Over the past few years, a class within the library, called polynomial, has been developed for approximating functions by classical orthogonal polynomials and it is intended to be an easy-to-use yet efficient object-oriented framework.In this work we discuss how this class has been designed to solve linear ill-posed problems and we provide a description of the available methods, Tikhonov regularization and truncated singular value expansion. For the solution of the Fredholm integral equation of the first kind, which is built from a low-rank approximation of the kernel followed by a numerical truncated singular value expansion, an error estimate is given.Numerical experiments illustrate that this approach is capable of efficiently compute good approximations of linear discrete ill-posed problems, even facing perturbed available data function, with no programming effort. Several test problems are used to evaluate the performance and reliability of the solvers.The final product of this paper is the numerical solution of a first-kind integral equation, which is constructed using only two inputs from the user: the kernel and the right-hand side.

Validating and speeding up x-ray tomographic inversions in tokamak plasmas

X-ray tomography is a precious tool in tokamaks that provides rich information about the core plasma, such as local impurity concentration, electron temperature and density as well as magnetic equilibrium (ME) and magnetohydrodynamic activity. Nevertheless, inferring the local plasma emissivity from a sparse set of line-integrated measurements is an ill-posed problem that requires dedicated regularization and validation methods. Besides, speeding up the inversion algorithm in order to be compatible with real-time control systems is a challenging task with traditional approaches. In this contribution, in a first part we introduce tools aiming at validating and speeding up the x-ray tomographic inversions based on Tikhonov regularization, including ME constraint and parameter optimization, taking the WEST geometry as an example. In a second part, an alternative approach compatible with real-time, based on a set of neural networks is proposed and compared with the Tikhonov approach for an experimental case.

Neural Networks to Find the Optimal Forcing for Offsetting the Anthropogenic Climate Change Effects

Abstract Of great relevance to climate engineering is the systematic relationship between the radiative forcing to the climate system and the response of the system, a relationship often represented by the linear response function (LRF) of the system. However, estimating the LRF often becomes an ill-posed inverse problem due to high-dimensionality and nonunique relationships between the forcing and response. Recent advances in machine learning make it possible to address the ill-posed inverse problem through regularization and sparse system fitting. Here, we develop a convolutional neural network (CNN) for regularized inversion. The CNN is trained using the surface temperature responses from a set of Green’s function perturbation experiments as imagery input data together with data sample densification. The resulting CNN model can infer the forcing pattern responsible for the temperature response from out-of-sample forcing scenarios. This promising proof of concept suggests a possible strategy for estimating the optimal forcing to negate certain undesirable effects of climate change. The limited success of this effort underscores the challenges of solving an inverse problem for a climate system with inherent nonlinearity. Significance Statement Predicting the climate response for a given climate forcing is a direct problem, while inferring the forcing for a given desired climate response is often an inverse, ill-posed, problem, posing a new challenge to the climate community. This study makes the first attempt to infer the radiative forcing for a given target pattern of global surface temperature response using a deep learning approach. The resulting deeply trained convolutional neural network inversion model shows promise in capturing the forcing pattern corresponding to a given surface temperature response, with a significant implication on the design of an optimal solar radiation management strategy for curbing global warming. This study also highlights the technical challenges that future research should prioritize in seeking feasible solutions to the inverse climate problem.

Tensor nonconvex unified prior for tensor recovery

Tensor data, such as hyperspectral images and multi-frame videos, have gained significant attention in practical applications. However, the inherent degradation phenomena during data acquisition, including noise and missing pixels, give rise to a series of ill-posed inverse problems that need to be addressed. Currently, the rational exploration of prior knowledge for tensor recovery, including global low-rankness and local smoothness, has emerged as a common concern. Inspired by recent notable works, this paper proposes a novel tensor non-convex unified prior term, which employs weighted tensor Schatten p-norm as a rank surrogate function in the gradient domain. The new prior can yield a regularizer that effectively captures low-rankness and smoothness, and is applied to tensor completion and tensor robust principal component analysis models. An efficient algorithm is developed by using the alternating direction method of multipliers and its convergence analysis is also provided. Extensive experimental results demonstrate that the proposed method outperforms the state-of-the-art methods, particularly in cases of high missing rates and strong noise levels.

TRIQS/Nevanlinna: Implementation of the Nevanlinna Analytic Continuation method for noise-free data

We present the TRIQS/Nevanlinna analytic continuation package, an efficient implementation of the methods proposed by J. Fei et al. (2021) [53] and (2021) [55]. TRIQS/Nevanlinna strives to provide a high quality open source (distributed under the GNU General Public License version 3) alternative to the more widely adopted Maximum Entropy based analytic continuation programs. With the additional Hardy functions optimization procedure, it allows for an accurate resolution of wide band and sharp features in the spectral function. Those problems can be formulated in terms of imaginary time or Matsubara frequency response functions. The application is based on the TRIQS C++/Python framework, which allows for easy interoperability with other TRIQS-based applications, electronic band structure codes and visualization tools. Similar to other TRIQS packages, it comes with a convenient Python interface. Program summaryProgram Title:TRIQS/NevanlinnaCPC Library link to program files:https://doi.org/10.17632/4cbzfy5rds.1Developer's repository link:https://github.com/TRIQS/NevanlinnaLicensing provisions: GPLv3Programming language:C++/PythonExternal routines/libraries:TRIQS 3.2[1], Boost >= 1.76.0, Eigen >= 3.4.0, cmake >= 3.20.Nature of problem: Finite-temperature field theories are widely used to study quantum many-body effects and electronic structure of correlated materials. Obtaining physically relevant spectral functions from results in the imaginary time/Matsubara frequency domains requires solution of an ill-posed analytic continuation problem as a post-processing step.Solution method: We present an efficient C++/Python open-source implementation of the Nevanlinna/Caratheodory analytic continuation.

The spectral conjugate gradient method in variational adjoint assimilation for model terrain correction III: convergence analysis

Abstract In the former parts of this paper, the spectral conjugate gradient (SCG) algorithm was proposed to identify the optimal bottom terrain used for predicting the meteorological fields. In this part, the convergence analysis of the SCG method is presented to verify its validity when solving a typical ill-posed inverse problem. The scaled cost function, the scaled gradient norm, the linear convergence rate, and the Hessian approximation condition number are evaluated for convergence analysis. The SCG method reduces the cost functional value and the gradient norm sufficiently. It always performs linear rates of convergence with a quickly reduced condition number of the Hessian matrix. Sometimes it provides very fast linear convergence rates. The SCG code with optimal step size and Andrei’s equation is effective on ill-posed test problems.

Well-posedness and inverse problems for semilinear nonlocal wave equations

This article is devoted to forward and inverse problems associated with time-independent semilinear nonlocal wave equations. We first establish comprehensive well-posedness results for some semilinear nonlocal wave equations. The main challenge is due to the low regularity of the solutions of linear nonlocal wave equations. We then turn to an inverse problem of recovering the nonlinearity of the equation. More precisely, we show that the exterior Dirichlet-to-Neumann map uniquely determines homogeneous nonlinearities of the form f(x,u) under certain growth conditions. On the other hand, we also prove that initial data can be determined by using passive measurements under certain nonlinearity conditions. The main tools used for the inverse problem are the unique continuation principle of the fractional Laplacian and a Runge approximation property. The results hold for any spatial dimension n∈N.

Prior frequency guided diffusion model for limited angle (LA)-CBCT reconstruction

Objective. Cone-beam computed tomography (CBCT) is widely used in image-guided radiotherapy. Reconstructing CBCTs from limited-angle acquisitions (LA-CBCT) is highly desired for improved imaging efficiency, dose reduction, and better mechanical clearance. LA-CBCT reconstruction, however, suffers from severe under-sampling artifacts, making it a highly ill-posed inverse problem. Diffusion models can generate data/images by reversing a data-noising process through learned data distributions; and can be incorporated as a denoiser/regularizer in LA-CBCT reconstruction. In this study, we developed a diffusion model-based framework, prior frequency-guided diffusion model (PFGDM), for robust and structure-preserving LA-CBCT reconstruction. Approach. PFGDM uses a conditioned diffusion model as a regularizer for LA-CBCT reconstruction, and the condition is based on high-frequency information extracted from patient-specific prior CT scans which provides a strong anatomical prior for LA-CBCT reconstruction. Specifically, we developed two variants of PFGDM (PFGDM-A and PFGDM-B) with different conditioning schemes. PFGDM-A applies the high-frequency CT information condition until a pre-optimized iteration step, and drops it afterwards to enable both similar and differing CT/CBCT anatomies to be reconstructed. PFGDM-B, on the other hand, continuously applies the prior CT information condition in every reconstruction step, while with a decaying mechanism, to gradually phase out the reconstruction guidance from the prior CT scans. The two variants of PFGDM were tested and compared with current available LA-CBCT reconstruction solutions, via metrics including peak signal-to-noise ratio (PSNR) and structural similarity index measure (SSIM). Main results. PFGDM outperformed all traditional and diffusion model-based methods. The mean(s.d.) PSNR/SSIM were 27.97(3.10)/0.949(0.027), 26.63(2.79)/0.937(0.029), and 23.81(2.25)/0.896(0.036) for PFGDM-A, and 28.20(1.28)/0.954(0.011), 26.68(1.04)/0.941(0.014), and 23.72(1.19)/0.894(0.034) for PFGDM-B, based on 120°, 90°, and 30° orthogonal-view scan angles respectively. In contrast, the PSNR/SSIM was 19.61(2.47)/0.807(0.048) for 30° for DiffusionMBIR, a diffusion-based method without prior CT conditioning. Significance. PFGDM reconstructs high-quality LA-CBCTs under very-limited gantry angles, allowing faster and more flexible CBCT scans with dose reductions.

Error estimates for simplified Levenberg–Marquardt method for nonlinear ill-posed operator equations in Hilbert Spaces

Abstract In this paper, we consider the simplified Levenberg–Marquardt method for nonlinear ill-posed inverse problems in Hilbert spaces for obtaining stable approximations of solutions to the ill-posed nonlinear equations of the form F ⁢ ( u ) = y {F(u)=y} , where F : 𝒟 ⁢ ( F ) ⊂ 𝖴 → 𝖸 {F:\mathcal{D}(F)\subset\mathsf{U}\to\mathsf{Y}} is a nonlinear operator between Hilbert spaces 𝖴 {\mathsf{U}} and 𝖸 {\mathsf{Y}} . The method is defined as follows: u n + 1 δ = u n δ - ( T 0 ∗ ⁢ T 0 + α n ⁢ I ) - 1 ⁢ T 0 ∗ ⁢ ( F ⁢ ( u n δ ) - y δ ) , u_{n+1}^{\delta}=u_{n}^{\delta}-(T_{0}^{\ast}T_{0}+\alpha_{n}I)^{-1}T_{0}^{% \ast}(F(u_{n}^{\delta})-y^{\delta}), where T 0 = F ′ ⁢ ( u 0 ) {T_{0}=F^{\prime}(u_{0})} and T 0 ∗ = F ′ ⁢ ( u 0 ) ∗ {T_{0}^{\ast}=F^{\prime}(u_{0})^{\ast}} . Here F ′ ⁢ ( u 0 ) {F^{\prime}(u_{0})} denotes the Frèchet derivative of F at an initial guess u 0 ∈ 𝒟 ⁢ ( F ) {u_{0}\in\mathcal{D}(F)} for the exact solution u † {u^{\dagger}} , F ′ ⁢ ( u 0 ) ∗ {F^{\prime}(u_{0})^{\ast}} is the adjoint of F ′ ⁢ ( u 0 ) {F^{\prime}(u_{0})} and { α n } {\{\alpha_{n}\}} is an a priori chosen sequence of non-negative real numbers satisfying suitable properties. We use Morozov-type stopping rule to terminate the iterations. Under suitable non-linearity conditions on operator F, we show convergence of the method and also obtain a convergence rate result under a Hölder-type source condition on the element u 0 - u † {u_{0}-u^{\dagger}} . Furthermore, we derive convergence of the method for the case when no source conditions are used and the study concludes with numerical examples which validate the theoretical conclusions.

Learning a stable approximation of an existing but unknown inverse mapping: application to the half-time circular Radon transform

Supervised deep learning-based methods have inspired a new wave of image reconstruction methods that implicitly learn effective regularization strategies from a set of training data. While they hold potential for improving image quality, they have also raised concerns regarding their robustness. Instabilities can manifest when learned methods are applied to find approximate solutions to ill-posed image reconstruction problems for which a unique and stable inverse mapping does not exist, which is a typical use case. In this study, we investigate the performance of supervised deep learning-based image reconstruction in an alternate use case in which a stable inverse mapping is known to exist but is not yet analytically available in closed form. For such problems, a deep learning-based method can learn a stable approximation of the unknown inverse mapping that generalizes well to data that differ significantly from the training set. The learned approximation of the inverse mapping eliminates the need to employ an implicit (optimization-based) reconstruction method and can potentially yield insights into the unknown analytic inverse formula. The specific problem addressed is image reconstruction from a particular case of radially truncated circular Radon transform (CRT) data, referred to as ‘half-time’ measurement data. For the half-time image reconstruction problem, we develop and investigate a learned filtered backprojection method that employs a convolutional neural network to approximate the unknown filtering operation. We demonstrate that this method behaves stably and readily generalizes to data that differ significantly from training data. The developed method may find application to wave-based imaging modalities that include photoacoustic computed tomography.