Methods For Inverse Problems Research Articles

A progress in the inverse matrix method in QCD sum rules

In traditional QCD sum rules, the simple hadron spectral density model of the “delta-function-type ground state + theta-function-type continuous spectrum” determines that there is no perfect parameter selection. In recent years, inverse problem methods, particularly the inverse matrix method, have shown better handling of QCD sum rules. This work continues to develop the inverse matrix method. Considering that the narrow-width approximation may still be a good approximation, we separate the ground state from the spectral density. We then follow the general steps of the inverse matrix method to extract physical quantities such as decay constants that we may be more interested in.

An Experimental Study of Boiling Heat Transfer and Quench Front Propagation Velocity During Quenching of a Cylinder Rod in Subcooled Water

In this study, a quenching experiment was conducted at atmospheric pressure to investigate the flow and heat-transfer characteristics of cylindrical rods made from SS, FeCrAl, and Zr-4 under various subcooling degrees (ΔTsub). The inverse heat-conduction problem (IHCP) method and image-processing technique were utilized to determine the surface temperature and heat flux, vapor film thickness, and quench front propagation. The results show that smaller solid kρcp and larger ΔTsub result in relatively more efficient quenching boiling heat transfer, thinner vapor film thickness, and greater quench front propagation velocity. The quench front originates at the bottom of the test specimen and becomes progressively larger in velocity with time. It eventually converges with the downward-propagating quench front in the upper middle of the test specimen. Moreover, at the beginning of quench front propagation, the SS and FeCrAl test specimens have a constant velocity region. However, because the Zr-4 test specimen has a small kρcp, the velocities gradually increase from the onset of quench front generation. Furthermore, the measured average quench front velocities are consistent with the experimental datum from the literature. However, the predicted model proposed by Duffey underestimates the propagation velocity due to ignoring the cooling effect of film boiling.

Mathematical Model of the Process of Thermal Destruction of Polymer Binders under Arbitrary Heating Conditions

A mathematical model of the process of thermal destruction of polymer binders has been developed, taking into account experimentally established dependences of the decomposition depth on temperature and the amount of coke residue on the heating rate. A method for determining the kinetic parameters of the process by the method of inverse problems has been developed. Kinetic parameters of the model for six binders are determined.

Fourier method for inverse source problem using correlation of passive measurements*

Abstract We consider the inverse source problem for the time-dependent, constant-coefficient wave equation with Cauchy data and passive cross-correlation data.We propose to consider the cross-correlation as a wave equation itself and reconstruct the cross-correlation in the support of the source for the original Cauchy wave equation. Having access to the cross-correlation in the support of the source, we show that the cross-correlation solves a wave equation, and we reconstruct the cross-correlation from boundary data to recover the source in the original Cauchy wave equation. In addition, we show the inverse source problem is ill-posed and suffers from non-uniqueness when the mean of the source is zero and provide a uniqueness result and stability estimate in case of non-zero mean sources.

Integrating the probe and singular sources methods

Abstract The probe and singular sources methods are two well-known classical direct reconstruction methods in inverse obstacle problems governed by partial differential equations. In this paper, by considering an inverse obstacle problem governed by the Laplace equation in a bounded domain as a prototype case, an integrated theory of the probe and singular sources methods is proposed. The theory consists of three parts: (i) introducing the singular sources method combined with the notion of the probe method; (ii) finding a third indicator function whose two ways decomposition yields the indicator functions in the probe and singular sources methods; (iii) finding the completely integrated version of the probe and singular sources methods.

Retraction Note: A heat polynomials method for the two-phase inverse Stefan problem

Retraction Note: A heat polynomials method for the two-phase inverse Stefan problem

A Preconditioned Krylov Subspace Method for Linear Inverse Problems with General-Form Tikhonov Regularization

A Preconditioned Krylov Subspace Method for Linear Inverse Problems with General-Form Tikhonov Regularization

Inverse problem method application in building physics numerical optimization: new insight into ideal envelope properties determination

ABSTRACT Building thermal design optimization is pivotal for energy efficiency and built environment improvement. The optimal thermophysical properties for building walls remain ambiguous. This ambiguity stems from the constraints of traditional building physics, which: (1) disjoins the enhancement of wall thermal performance from the building service system management; and (2) relies on the forward method that fails to identify the ideal variable thermophysical property, the solution to which is inherently a function. This study elucidates the optimal temperature-dependent volumetric specific heat ( ρ c p ( t ) ) and thermal conductivity ( k ( t ) ) through the inverse problem and variation method. Illustrative example indicates that the optimal ρ c p ( t ) closely resembles a combination of two ‘ δ ’ functions, while the optimal k ( t ) mirrors a square wave function. Utilizing these ideal thermophysical properties, the building energy consumption can be significantly reduced by 77.1% with the optimized ρ c p ( t ) and 79.9% with the optimized k ( t ) . The ideal thermophysical properties for active buildings differ significantly from those recommended for passive buildings, underscoring that minimal energy consumption does not necessarily imply minimal discomfort. Therefore, the thermophysical properties of active and passive buildings should be optimized distinctly. This research offers valuable insights for defining ideal thermophysical properties of active building walls and construction materials.

An accelerated sequential function specification method based on LM gradient for transient inverse heat conduction problem

The sequential function specification method (SFSM) and the Levenberg-Marquardt method (LM) are two typical methods for inverse heat conduction problems (IHCP). However, these two methods have poor convergence and low convergence speed. Therefore, the combination of LM with SFSM is depended on to propose a modified Sequential Levenberg-Marquardt gradient method (SLM). The computation results of three shapes of heat fluxes with noise show the SLM is averagely improved over 16.3% more than SFSM in computation accuracy, but equally reduced over 39.9% less than in computation time. On other hand, when the emphasis is on the effect of the Aitken acceleration method, Tikhonov regularization and polynomial order on SLM, a neural network prediction model of the root mean square error is trained by regularization factor, polynomial order and the number of future time steps. Subsequently, the genetic algorithm optimization method is proposed for the minimum root mean square error so that it can improve the computation accuracy and efficiency of IHCP to the maximum extent. Finally, Second-order polynomial order and small order-of-magnitude regularization factor are highly recommended for parameter selection with multiple choices. The ASLM only needs to determine the optimal number of future time steps.

Adaptive Bregman–Kaczmarz: an approach to solve linear inverse problems with independent noise exactly

We consider the block Bregman–Kaczmarz method for finite dimensional linear inverse problems. The block Bregman–Kaczmarz method uses blocks of the linear system and performs iterative steps with these blocks only. We assume a noise model that we call independent noise, i.e. each time the method performs a step for some block, one obtains a noisy sample of the respective part of the right-hand side which is contaminated with new noise that is independent of all previous steps of the method. One can view these noise models as making a fresh noisy measurement of the respective block each time it is used. In this framework, we are able to show that a well-chosen adaptive stepsize of the block Bregman–Kaczmarz method is able to converge to the exact solution of the linear inverse problem. The plain form of this adaptive stepsize relies on unknown quantities (like the Bregman distance to the solution), but we show a way how these quantities can be estimated purely from given data. We illustrate the finding in numerical experiments and confirm that these heuristic estimates lead to effective stepsizes.

Neural‐network‐based regularization methods for inverse problems in imaging

AbstractThis review provides an introduction to—and overview of—the current state of the art in neural‐network based regularization methods for inverse problems in imaging. It aims to introduce readers with a solid knowledge in applied mathematics and a basic understanding of neural networks to different concepts of applying neural networks for regularizing inverse problems in imaging. Distinguishing features of this review are, among others, an easily accessible introduction to learned generators and learned priors, in particular diffusion models, for inverse problems, and a section focusing explicitly on existing results in function space analysis of neural‐network‐based approaches in this context.

A direct and analytical method for inverse problems under uncertainty in energy system design: combining inverse simulation and Polynomial Chaos theory

This article introduces and formalizes a novel stochastic method that combines inverse simulation with the theory of generalized Polynomial Chaos (gPC) to solve and study inverse problems under uncertainty in energy system design applications. The method is particularly relevant to design tasks where only a deterministic forward model of a physical system is available, in which a target design quantity is an input to the model that cannot be obtained directly, but can be quantified reversely via the outputs of the model. In this scenario, the proposed method offers an analytical and direct approach to invert such system models. The method puts emphasis on user-friendliness, as it enables its users to conduct the inverse simulation under uncertainty directly in the gPC domain by redefining basic algebra operations for computations. Moreover, the method incorporates an optimization-based approach to integrate supplementary constraints on stochastic quantities. This feature enables the solution of inverse problems bounding the statistical moments of stochastic system variables. The authors exemplify the application of the proposed method with proof-of-concept tests in energy system design, specifically performing uncertainty quantification and sensitivity analysis for a Multi-Energy System (MES). The findings demonstrate the high accuracy of the method as well as clear advantages over conventional sampling-based methods when dealing with a small number of stochastic variables in a system or model. However, the case studies also highlight the current limitations of the proposed method such as slow execution speed due to the optimization-based approach and the challenges associated with, for example, the curse of dimensionality in gPC.

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.

The meshless backward substitution method for inverse Cauchy problems in electroelastic piezoelectric structures

The backward substitution method is a recently proposed semi-analytical meshless collocation method. The main idea of the back substitution method is to obtain the boundary approximation from the boundary data, and this approximation does not need to satisfy the governing equation. Then, the traditional linear combination of basis functions is used to construct a correcting approximation that satisfies the homogeneous boundary conditions, the numerical solution is given by a linear combination system of boundary approximation and modified approximation, which satisfies the governing equation. In this way, the backward substitution method can be easily used to solve general problems without fundamental solutions or particular solutions. This paper makes the first attempt to extend the backward substitution method to the simulation of inverse Cauchy problems in piezoelectric structures. For the inverse Cauchy problems, the absence of data on some boundaries often leads to extreme instability of numerical solutions. A simple method is applied in this paper. For the points on the boundary without boundary information, the governing equation is enforced. Numerical results demonstrate that it will greatly increase the stability of the solution process.

Prediction of boundary heat flux in mold by inverse problem method

This study utilizes inverse heat transfer techniques to analyze heat flux distribution within a mold. By collecting temperature data from thermocouples, a two-dimensional heat transfer model of the mold’s cross section is developed to investigate the transverse heat flux behavior, temperature distribution, relationship between heat flux and height, and three-dimensional heat flux within the mold. The direct problem model is solved using the finite volume method, while the inverse problem is tackled with the Levenberg-Marquardt Method (LMM) in conjunction with the Broyden method (BM). The model’s validity is confirmed through experimentation, which includes assessing the impact of initial parameter estimations, thermocouple placements and quantities, and measurement inaccuracies.

Parametric investigation of flow and heat transfer on the reflooding quenching of a cylinder rod in forced convection

A reflooding quenching boiling experiment in forced convection was conducted to investigate the flow and heat transfer performance of a FeCrAl cylindrical rod under various subcooling degrees and flow rates. Employing image processing and inverse heat conduction problem (IHCP) methods, the research provides insights into vapor film thickness, quenching front propagation, wall temperature cooling rate and minimum film boiling temperature. Key observations include the presence of numerous extremely small bubbles dispersing around the rod and a third quenching front appears in the middle of the rod under higher subcooling condition at 25 °C, attributed to a thinner vapor film, smaller bubbles resulting from film rupture and the inertia of fluid flow. Within the experimental range, the impact of the subcooling degree on the fluid flow and heat transfer performance during the quenching boiling process is more pronounced than that of the coolant flow rate. The evolution of the vapor film indicates a substantial reduction of 78.76 % in film thickness and a significant increase of 185.71 % in propagation velocity with an increase in subcooling degree, while an elevation in flow rate lead to a 29.84 % decrease in film thickness and a 45 % increase in propagation velocity. Heat transfer performance benefits from increased subcooling degrees throughout the entire quenching boiling process, whereas flow rate impacts the heat transfer performance of film boiling alone. Furthermore, by incorporating the impact of flow velocity, a predictive formula for Tmin is established with an error margin within ±5 %.

Integration of negative-order modified Korteweg–de Vries equation with an integral source

In this paper, it is shown that the modified Korteweg–de Vries (mKdV) equation of negative order with an integral source can be integrated by the method of the inverse spectral problem. The main result of this work is the derivation of the evolution of the spectral data of the Dirac system with a periodic potential associated with the solution of the negative-order modified Korteweg–de Vries equation with an integral source. The obtained results allow us to apply the inverse problem method to solve the negative-order modified Korteweg–de Vries equation with an integral source.

Iterative methods for Navier-Stokes inverse problems.

Even when the partial differential equationunderlying a physical process can be evolved forward in time, the retrospective (backward in time) inverse problem often has its own challenges and applications. Direct adjoint looping (DAL) is the defacto approach for solving retrospective inverse problems, but it has not been applied to deterministic retrospective Navier-Stokes inverse problems in 2D or 3D. In this paper, we demonstrate that DAL is ill-suited for solving retrospective 2D Navier-Stokes inverse problems. Alongside DAL, we study two other iterative methods: simple backward integration (SBI) and the quasireversible method (QRM). As far as we know, our iterative SBI approach is novel, while iterative QRM has previously been used. Using these three iterative methods, we solve two retrospective inverse problems: 1D Korteweg-de Vries-Burgers (decaying nonlinear wave) and 2D Navier-Stokes (unstratified Kelvin-Helmholtz vortex). In both cases, SBI and QRM reproduce the target final states more accurately and in fewer iterations than DAL. We attribute this performance gap to additional terms present in SBI and QRM's respective backward integrations which are absent in DAL.

Integration of a sine-Gordon type equation with an additional term in the class of periodic infinite-gap functions

In this paper, the inverse spectral problem method is used to integrate a nonlinear sine-Gordon type equation with an additional term in the class of periodic infinite-gap functions. The solvability of the Cauchy problem for an infinite system of Dubrovin differential equations in the class of three times continuously differentiable periodic infinite-gap functions is proved. It is shown that the sum of a uniformly convergent functional series constructed by solving the system of Dubrovin equations and the first trace formula satisfies sine-Gordon-type equations with an additional term.

Оценка требований к точности сенсоров океанографических зондов при косвенном определении солености морской воды

The issues of measuring the mass of dissolved substances or absolute salinity under environmental conditions in situ using automated instruments still remain open. How these issues are likely to be resolved depends on the development of new oceanographic measurement technologies. Salinity is one of the seawater parameters that cannot yet be directly measured in situ. Until now, all known, existing or promising, methods for determining salinity are indirect. The purpose of this work is to find the maximum sensitivity coefficients in order to assess the uncertainty of sensors of input quantities and compare various methods for indirect determination of salinity. To solve this problem, the work uses methods of direct and inverse metrology problems. In this work, these problems were solved for four fundamentally different methods of indirect (calculated) determining of the salinity of sea water: 1) the relative electrical conductivity method (RCM), 2) the speed of sound method (SSM), 3) the density method (DM) and 4) the refractive index method (RIM). Depending on environmental conditions, the maximum values of sensitivity coefficients (13 coefficients in total) for the output value (salinity) were found for all input parameters of each method. By solving inverse problems for the accepted conditional base level of salinity uncertainty and at maximum sensitivity coefficients, estimates of the uncertainty of sensors of input parameters for each of the four methods were calculated. The results obtained make it possible to estimate the required accuracy of sensors of input quantities at any given level of uncertainty of the output quantity or the actual accuracy of the output quantity at any given level of uncertainty of sensors of input quantities. From the three promising methods for indirect salinity determination, MPP is the best candidate to replace MOE in the near future.