General Inverse Problem Research Articles

General Inverse Problem Solution for Two-Level Systems and Its Application to Charge Transfer

General Inverse Problem Solution for Two-Level Systems and Its Application to Charge Transfer

Noise-Generating and Imaging Mechanism Inspired Implicit Regularization Learning Network for Low Dose CT Reconstrution.

Low-dose computed tomography (LDCT) helps to reduce radiation risks in CT scanning while maintaining image quality, which involves a consistent pursuit of lower incident rays and higher reconstruction performance. Although deep learning approaches have achieved encouraging success in LDCT reconstruction, most of them treat the task as a general inverse problem in either the image domain or the dual (sinogram and image) domains. Such frameworks have not considered the original noise generation of the projection data and suffer from limited performance improvement for the LDCT task. In this paper, we propose a novel reconstruction model based on noise-generating and imaging mechanism in full-domain, which fully considers the statistical properties of intrinsic noises in LDCT and prior information in sinogram and image domains. To solve the model, we propose an optimization algorithm based on the proximal gradient technique. Specifically, we derive the approximate solutions of the integer programming problem on the projection data theoretically. Instead of hand-crafting the sinogram and image regularizers, we propose to unroll the optimization algorithm to be a deep network. The network implicitly learns the proximal operators of sinogram and image regularizers with two deep neural networks, providing a more interpretable and effective reconstruction procedure. Numerical results demonstrate our proposed method improvements of > 2.9 dB in peak signal to noise ratio, > 1.4% promotion in structural similarity metric, and > 9 HU decrements in root mean square error over current state-of-the-art LDCT methods.

Reconstruction techniques for quantum trees

The inverse problem of recovery of a potential on a quantum tree graph from the Weyl matrix given at a number of points is considered. A method for its numerical solution is proposed. The overall approach is based on the leaf peeling method combined with Neumann series of Bessel functions (NSBF) representations for solutions of Sturm–Liouville equations. In each step, the solution of the arising inverse problems reduces to dealing with the NSBF coefficients. The leaf peeling method allows one to localize the general inverse problem to local problems on sheaves, while the approach based on the NSBF representations leads to splitting the local problems into two‐spectrum inverse problems on separate edges and reduces them to systems of linear algebraic equations for the NSBF coefficients. Moreover, the potential on each edge is recovered from the very first NSBF coefficient. The proposed method leads to an efficient numerical algorithm that is illustrated by numerical tests.

A review of regularization strategies and solution techniques for ill-posed inverse problems, with application to inverse heat transfer problems

With the presence of a large number of inversion algorithms for inverse heat transfer problems (IHTPs) and non-IHTPs, a need for review to have a holistic view is seen. An exhaustive literature review, with the motivation of selecting the inversion technique best fit for a given problem, was made for a general inverse problem. For ill-posedness, a classification of available regularization algorithms namely Tikhonov’s regularization, Bayesian regularization, mollification method, Beck’s sequential approach and Alifanov’s iterative approach, has been provided. Inversion methods like singular value decomposition, truncated singular value decomposition, Tikhonov regularization and total variation regularization are explained. Optimization methods namely steepest descent method, conjugate gradient method, Newton method, Levenberg–Marquardt method, Lagrange method, adjoint method, function specification method, genetic algorithm, differential evolution and particle swarm optimization (PSO) are reviewed. Further, a technique based on neural networks is studied, and wavelet methods like shrinking and wavelet vaguelette decomposition are reviewed. Associated literature has also been listed, highlighting the gaps. The usability of various algorithms in IHTP, starting from the golden section search method, for retrieval of a single parameter, to the regularized versions of the inversion technique, for retrieval of multiple parameters with uncertainty, demonstrates real-life applications to fins in IHTP. An inversion algorithm capable to handle every kind of nonlinearity is sought in literature, whose absence raises the research question, “Is there a technique that works globally for every inverse problem?”, is asked prior to, “What if the available techniques were not utilized to an extent that they should?” is posed. In lieu of this gap, a general comparative framework is developed, such that an efficient technique is selected, based on the total minimum error, which can be used in any field of interest.

Ensemble Kalman Inversion for upstream parameter estimation and indirect streamflow correction: A simulation study

Data assimilation (DA) techniques such as the Ensemble Kalman filter (EnKF) and its extensions allow for real-time corrections of state-space models and model parameters based on an assumption of Gaussian error. The hydrological DA literature primarily documents applications of the EnKF to solve sequential state estimation problems. Recent advances in the DA literature demonstrate the potential of applying EnKF-based methods as efficient, derivative-free algorithms to solve various general Bayesian inverse problems, such as parameter estimation, while simultaneously providing Uncertainty Quantification (UQ). In this paper, the authors employ the Ensemble Kalman Inversion (EKI) algorithm to infer the distribution of a set of routing parameters. Through this correction, we improve streamflow at locations upstream of the gauged site in a virtual catchment setting. The algorithm enables learning spatially distributed routing parameters with observations available only at the outlet. The study reveals that this method sufficiently improves model performance throughout the basin. The performance of this method is demonstrated in a virtual catchment for three different model/data configurations. Favorable results, even with model misspecification, indicate that this method holds promise for operational application and more general hydrologic parameter estimation problems.

Noise Level Free Regularization of General Linear Inverse Problems under Unconstrained White Noise

In this note we solve a general statistical inverse problem under absence of knowledge of both the noise level and the noise distribution via application of the (modified) heuristic discrepancy principle. Hereby the unbounded (non-Gaussian) noise is controlled via introducing an auxiliary discretization dimension and choosing it in an adaptive fashion. We first show convergence for completely arbitrary compact forward operator and ground solution. Then the uncertainty of reaching the optimal convergence rate is quantified in a specific Bayesian-like environment. We conclude with numerical experiments.

A Real Time Method Based on Deep Learning for Reconstructing Holographic Acoustic Fields from Phased Transducer Arrays.

Phased transducer arrays (PTA) can control ultrasonic waves to produce a holographic acoustic field. However, obtaining the phase of the corresponding PTA from a given holographic acoustic field is an inverse propagation problem, which is a mathematically unsolvable nonlinear system. Most of the existing methods use iterative methods, which are complex and time-consuming. To better solve this problem, this paper proposed a novel method based on deep learning to reconstruct the holographic sound field from PTA. For the imbalance and randomness of the focal point distribution in the holographic acoustic field, we constructed a novel neural network structure incorporating attention mechanisms to focus on useful focal point information in the holographic sound field. The results showed that the transducer phase distribution obtained from the neural network fully supports the PTA to generate the corresponding holographic sound field, and the simulated holographic sound field can be reconstructed with high efficiency and quality. The method proposed in this paper has the advantage of real-time performance that is difficult to achieve by traditional iterative methods and has the advantage of higher accuracy compared with the novel AcousNet methods.

Specific Cases of One General Inverse Problem for Abstract Differential Equations

Specific Cases of One General Inverse Problem for Abstract Differential Equations

Deep learning tomographic reconstruction through hierarchical decomposition of domain transforms.

Deep learning (DL) has shown unprecedented performance for many image analysis and image enhancement tasks. Yet, solving large-scale inverse problems like tomographic reconstruction remains challenging for DL. These problems involve non-local and space-variant integral transforms between the input and output domains, for which no efficient neural network models are readily available. A prior attempt to solve tomographic reconstruction problems with supervised learning relied on a brute-force fully connected network and only allowed reconstruction with a 1284 system matrix size. This cannot practically scale to realistic data sizes such as 5124 and 5126 for three-dimensional datasets. Here we present a novel framework to solve such problems with DL by casting the original problem as a continuum of intermediate representations between the input and output domains. The original problem is broken down into a sequence of simpler transformations that can be well mapped onto an efficient hierarchical network architecture, with exponentially fewer parameters than a fully connected network would need. We applied the approach to computed tomography (CT) image reconstruction for a 5124 system matrix size. This work introduces a new kind of data-driven DL solver for full-size CT reconstruction without relying on the structure of direct (analytical) or iterative (numerical) inversion techniques. This work presents a feasibility demonstration of full-scale learnt reconstruction, whereas more developments will be needed to demonstrate superiority relative to traditional reconstruction approaches. The proposed approach is also extendable to other imaging problems such as emission and magnetic resonance reconstruction. More broadly, hierarchical DL opens the door to a new class of solvers for general inverse problems, which could potentially lead to improved signal-to-noise ratio, spatial resolution and computational efficiency in various areas.

Recovery of Inhomogeneity from Output Boundary Data

We consider the Sturm–Liouville equation on a finite interval with a real-valued integrable potential and propose a method for solving the following general inverse problem. We recover the potential from a given set of the output boundary values of a solution satisfying some known initial conditions for a set of values of the spectral parameter. Special cases of this problem include the recovery of the potential from the Weyl function, the inverse two-spectra Sturm–Liouville problem, as well as the recovery of the potential from the output boundary values of a plane wave that interacted with the potential. The method is based on the special Neumann series of Bessel functions representations for solutions of Sturm–Liouville equations. With their aid, the problem is reduced to the classical inverse Sturm–Liouville problem of recovering the potential from two spectra, which is solved again with the help of the same representations. The overall approach leads to an efficient numerical algorithm for solving the inverse problem. Its numerical efficiency is illustrated by several examples.

Finite-Time Solution of Time-Varying Tensor Inversion by a Novel Dynamic-Parameter Zeroing Neural-Network

Time-varying tensor inversion (TVTI) problem is a kind of general time-varying inversion problem in mathematics because scalars, vectors and matrices can all be represented by tensors. The TVTI problem is based on a novel tensor product [termed the TensorFlow (TF) product], which is extracted from the TensorFlow. For solving such a prevalent problem, the matricization of the TF product is defined, and a novel dynamicparameter zeroing neural-network (DP-ZNN) model is proposed by combining a zeroing neural-network design formula and a dynamic-parameter. The global convergence and the upper bound of finite-time convergence of the DP-ZNN model are analyzed theoretically. For highlighting the superior convergence performance and excellent efficiency of the DP-ZNN model in solving the TVTI problem, three comparative experiments are presented in this paper. Experimental results show that the DP-ZNN model has remarkable convergent speciality.

Проекционный метод решения дискретизированных обратных задач в области антенных измерений

In the article, the problems of determining and monitoring antenna parameters, solved using measurements in the near field, are presented in a unified formulation corresponding to the gene-ralized inverse radiation problem in the form of an integral equation of the first kind. Within the framework of the numerical approach to the solution, the equation is discretized, and then the solution is sought approximately as a solution to the normal system of equations. It is shown that an approximate solution to the discretized problem that satisfies the normal system is also a solution to an equation with an orthogonal projector on its right-hand side. On the basis of this formulation, a new method of iterative projections is proposed. The efficiency of the method is demonstrated by the example of solving the problem of reconstructing the radiation pattern of a horn antenna from near-field data calculated for the electrodynamic model of the antenna in the HFSS program. The results of the proposed method are compared with the results of solving the problem by the method of decomposition in singular numbers.

Adaptive neural network surrogate model for solving the implied volatility of time-dependent American option via Bayesian inference

<abstract><p>In this paper, we propose an adaptive neural network surrogate method to solve the implied volatility of American put options, respectively. For the forward problem, we give the linear complementarity problem of the American put option, which can be transformed into several standard American put option problems by variable substitution and discretization in the temporal direction. Thus, the price of the option can be solved by primal-dual active-set method using numerical transformation and finite element discretization in spatial direction. For the inverse problem, we give the framework of the general Bayesian inverse problem, and adopt the direct Metropolis-Hastings sampling method and adaptive neural network surrogate method, respectively. We perform some simulations of volatility in the forward model with one- and four-dimension to compare the point estimates and posterior density distributions of two sampling methods. The superiority of adaptive surrogate method in solving the implied volatility of time-dependent American options are verified.</p></abstract>

From Penrose Equations to Zhang Neural Network, Getz–Marsden Dynamic System, and DDD (Direct Derivative Dynamics) Using Substitution Technique

The time-variant matrix inversion (TVMI) problem solving is the hotspot of current research because of its frequent appearance and application in scientific research and industrial production. The generalized inverse problem of singular square matrix and nonsquare matrix can be related to Penrose equations (PEs). The PEs implicitly define the generalized inverse of a known matrix, which is of fundamental theoretical significance. Therefore, the in-depth study of PEs might enlighten problem solving of TVMI in a foreseeable way. For the first time, we construct three different matrix error-monitoring functions based on PEs and propose the corresponding models for TVMI problem solving by using the substitution technique and ZNN design formula. In order to facilitate computer simulation, the obtained continuous-time models are discretized by using ZTD (Zhang time discretization) formulas. Furthermore, the feasibility and effectiveness of the novel Zhang neural network (ZNN) multiple-multiplication model for matrix inverse (ZMMMI) and the PEs-based Getz–Marsden dynamic system (PGMDS) model in solving the problem of TVMI are investigated and shown via theoretical derivation and computer simulation. Computer experiment results also illustrate that the direct derivative dynamics model for TVMI is less effective and feasible.

Deep denoiser prior based deep analytic network for lensless image restoration.

Mask based lensless imagers have huge application prospects due to their ultra-thin body. However, the visual perception of the restored images is poor due to the ill conditioned nature of the system. In this work, we proposed a deep analytic network by imitating the traditional optimization process as an end-to-end network. Our network combines analytic updates with a deep denoiser prior to progressively improve lensless image quality over a few iterations. The convergence is proven mathematically and verified in the results. In addition, our method is universal in non-blind restoration. We detailed the solution for the general inverse problem and conducted five groups of deblurring experiments as examples. Both experimental results demonstrate that our method achieves superior performance against the existing state-of-the-art methods.

ЧИСЛЕННОЕ РЕШЕНИЕ ОБРАТНОЙ ЗАДАЧИ ДЛЯ АКУСТИЧЕСКОГО УРАВНЕНИЯ

This article presents the continuation problem for one-dimensional equations of acoustics. Currently, one of the most difficult areas of research in applied mathematics is the reverse calculation of wave propagation, which has an important area of application in geophysics and medicine. The propagation of an acoustic wave in an inhomogeneous medium and the measurement of the return of some part of the sound by special devices leads to the calculation of the initial-boundary for the partial derivative equation. Most of these problems are ill-posed. One such ill-posed problem is the continuation problem. The continuation problem is based on finding the value of the desired function in the rest of the boundary using additional data in a certain part of the boundary. In this regard, we present the continuation problems to the inverse boundary value problem. There are many methods for solving general boundary value inverse problems, for example, the gradient method. In this paper, we construct a finite-difference scheme for this inverse problem and find an unknown function on the characteristic from this difference equation by inverting the difference scheme. The effectiveness of this method lies in a simple and accurate fast calculation algorithm. To test the correctness of any method or algorithm, we first solve a direct problem by specifying a exact function. Then we find additional information and solve the inverse problem. Let us compare the inverse problem with solving a problem with an exact function. However, in practice it often happens that additional information is provided with certain errors. Therefore, the correctness of the algorithm was verified by adding different error levels to the additional information. Quantitative data and graphs of these quantitative results are compared.

Derivation of Heat Conductivity from Temperature and Heat Flux Measurements in Soil

The general inverse problem formulation for a heat conductance equation is adopted for the types of measurement routinely carried out in the soil active layer. The problem solution delivers a constant thermal diffusivity coefficient a0 (in general, different from true value a) and respective heat conductivity λ0 for the layer, located between two temperature sensors and equipped with a temperature or heat flux sensor in the middle. We estimated the error of solution corresponding to systematic shifts in sensor readings and mislocation of sensors in the soil column. This estimation was carried out by a series of numerical experiments using boundary conditions from observations on Mukhrino wetland (Western Siberia, Russia), performed in summer, 2019. Numerical results were corroborated by analytical estimates of inverse problem solution sensitivity derived from classical Fourier law. The main finding states that heat conductivity error due to systematic shifts in temperature measurements become negligible when using long temperature series, whereas the relative error of a is approximately twice the relative error of sensor depth. The error a0−a induced by heat flux plate displacement from expected depth is 3–5 times less than the same displacement of thermometers, which makes the requirements for heat flux installation less rigid. However, the relative errors of heat flux observation typical for modern sensors (±15%) cause the uncertainty of a above 15% in absolute value. Comparison of the inverse problem solution to a estimated from in situ moss sampling on Mukhrino wetland proves the feasibility of the method and corroborates the conclusions of the error sensitivity study.

Intrinsic non-uniqueness of the acoustic full waveform inverse problem

SUMMARYIn the context of seismic imaging, full waveform inversion (FWI) is increasingly popular. Because of its lower numerical cost, the acoustic approximation is often used, especially at the exploration geophysics scale, both for tests and for real data. Moreover, some research domains such as helioseismology face true acoustic media for which FWI can be useful. In this work, an argument that combines particle relabelling and homogenization is used to show that the general acoustic inverse problem based on band-limited data is intrinsically non-unique. It follows that the results of such inversions should be interpreted with caution. To illustrate these ideas, we consider 2-D numerical FWI examples based on a Gauss–Newton iterative inversion scheme and demonstrate effects of this non-uniqueness in the local optimization context.

Hierarchical and Multi-resolution Neuron Network Computing for Complex Inverse Problems-An Approach Based on Brain-inspired Computing and Learning Method

Background: Complex inverse problems such as Radar Imaging and CT/EIT imaging are well investigated in mathematical algorithms with various regularization methods. However it is difficult to obtain stable inverse solutions with fast convergence and high accuracy at the same time due to the ill-posed property and non-linear property. Objective: In this paper, we propose a hierarchical and multi-resolution scalable method from both algorithm perspective and hardware perspective to achieve fast and accurate solu-tions for inverse problems by taking radar and EIT imaging as examples. Method: We present an extension of discussion on neuromorphic computing as brain-inspired computing method and the learning/training algorithm to design a series of problem specific AI “brains” (with different memristive values) to solve a general complex ill-posed inverse problems that are traditionally solved by mathematical regular operators. We design a hierarchical and multi-resolution scalable method and an algorithm framework to train AI deep learning neuron network and map into the memristive circuit so that the memristive val-ues are optimally obtained. We propose FPGA as an emulation implementation for neuro-morphic circuit as well. Result: We compared the methodology between our approach and traditional regulariza-tion method. In particular we use Electrical Impedance Tomography (EIT) and Radar imaging as typical examples to compare how to design an AI deep learning neuron network architec-tures to solve inverse problems. Conclusion: With EIT imaging as a typical example, we show that any moderate complex inverse problem, as long as it can be described as combinational problem, AI deep learning neuron network is a practical alternative approach to try to solve the inverse problems with any given expected resolution accuracy, as long as the neuron network width is large enough and computational power is strong enough for all combination samples training purpose.

A General Robot Inverse Kinematics Solution Method Based on Improved PSO Algorithm

Robots whose geometric structure does not meet the Pieper criterion are called general robots. For the inverse kinematic operation of general robots, the closed solution method cannot be solved, and the numerical solution calculation amount is too large and the singular position cannot be calculated. To solve this problem, this paper proposes an inverse kinematics calculation method based on improved particle swarm optimization (PSO) algorithm and applicable to general robots. In order to avoid the particle update rate not adapting to each stage of the optimization process, a nonlinear dynamic inertia weight adjustment method based on the concept of similarity is introduced, so that the search process is more robust; in addition, to overcome the problem of local optimal solution At the same time, multiple populations are introduced to perform optimization search at the same time, and the immigration operator is proposed to increase the diversity of the particle population in the iteration. This article uses Comau NJ-220 robot for test verification, compared with the original PSO and multi-subswarm algorithm, the results show that the proposed improved PSO has higher algorithm stability for general robot kinematics inverse solution problems, and can greatly improve the convergence accuracy and speed. This method provides a new solution to the field of robot inverse kinematics, and provides a more efficient and stable kinematics foundation for robot motion planning.