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This paper aims to discuss the inverse problems that arise in various practical heat transfer processes. The purpose of this paper is to provide an identification method for predicting the internal boundary conditions for thermal analysis of mechanical structure. A few examples of heat transfer systems are given to illustrate the applicability of the method and the challenges that must be addressed in solving the inverse problem.

In this paper, the thermal network method and the finite difference method are used to model the two-dimensional heat conduction inverse problem of the tube structure, and the heat balance equation is arranged into an explicit form for heat load prediction. To solve the matrix ill-conditioned problem in the process of solving the inverse problem, a Tikhonov regularization parameter selection method based on the inverse computation-contrast-adjustment-approach was proposed.

The applicability of the proposed method is illustrated by numerical examples for different dynamically varying heat source functions. It is proved that the method can predict dynamic heat source with different complexity.

The modeling calculation method described in this paper can be used to predict the boundary conditions for the inner wall of the heat transfer tube, where the temperature sensor cannot be placed.

This paper presents a general method for the direct prediction of heat sources or boundary conditions in mechanical structure. It can directly obtain the time-varying heat flux load and thtemperature field of the machine structure.

The purpose of this study is to estimate inputs (outputs) and flexible measures when outputs (inputs) are changed provided that the relative efficiency values remain without change.

A novel inverse data envelopment analysis (DEA) approach with flexible measures is proposed in this research to assess inputs (outputs) and flexible measures when outputs (inputs) are perturbed on condition that the relative efficiency scores remain unchanged. Furthermore, flexible inverse DEA approaches proposed in this study are used for a numerical example from the literature and an application of Iranian banking industry to clarify and validate them.

The findings show that including flexible measures into the investigation effects on the changes of performance measures estimated and leads to more reasonable achievements.

The traditional inverse DEA models usually investigate the changes of some determinate input-output factors for the changes of other given input-output indicators assuming that the efficiency values are preserved. However, there are situations that the changes of performance measures should be tackled while some measures, called flexible measures, can play either input or output roles. Accordingly, inverse DEA optimization models with flexible measures are rendered in this paper to address these issues.

Assessing inputs and outputs is a significant aspect of taking decisions while there are complex and multistage processes in many examinations. Due to the presence of interval performance measures in various real-world studies, the purpose of this study is to address the changes of interval inputs of two-stage processes for the perturbations of interval outputs of two-stage systems, given that the overall efficiency scores are maintained.

Actually, an interval inverse two-stage data envelopment analysis (DEA) model is proposed to plan resources. To illustrate, an interval two-stage network DEA model with external interval inputs and outputs and also its inverse problem are suggested to estimate the upper and lower bounds of the entire efficiency and the stages efficiency along with the variations of interval inputs.

An example from the literature and a real case study of the banking industry are applied to demonstrate the introduced approach. The results show the proposed approach is suitable to estimate the resources of two-stage systems when interval measures are presented.

To the best of the authors’ knowledge, there is no study to estimate the fluctuation of imprecise inputs related to network structures for the changes of imprecise outputs while the interval efficiency of network processes is maintained. Accordingly, this paper considers the resource planning problem when there are imprecise and interval measures in two-stage networks.

This chapter proposes a nonparametric estimator of the risk neutral density (RND) based on cross-sectional European option prices. The authors recast the arbitrage-free equation for option pricing as a functional linear regression model where the regressor is a curve and the independent variable is a scalar corresponding to the option price. Then, the authors show that the RND can be viewed as the solution of an ill-posed integral equation. To estimate the RND, the authors use an iterative method called Landweber-Fridman (LF). Then, the authors establish the consistency and asymptotic normality of the estimated RND. These results can be used to construct a confidence interval around the curve. Finally, some Monte Carlo simulations and application to the S&P 500 options show that this method performs well compared to alternative methods.

Two-dimensional (2D) problems are governed by unsteady anisotropic modified-Helmholtz equation of time–space dependent coefficients are considered. The problems are transformed into a boundary-only integral equation which can be solved numerically using a standard boundary element method (BEM). Some examples are solved to show the validity of the analysis and examine the accuracy of the numerical method.

The 2D problems which are governed by unsteady anisotropic modified-Helmholtz equation of time–space dependent coefficients are solved using a combined BEM and Laplace transform. The time–space dependent coefficient equation is reduced to a time-dependent coefficient equation using an analytical transformation. Then, the time-dependent coefficient equation is Laplace transformed to get a constant coefficient equation, which can be written as a boundary-only integral equation. By utilizing a BEM, this integral equation is solved to find numerical solutions to the problems in the frame of the Laplace transform. These solutions are then inversely transformed numerically to obtain solutions in the original time–space frame.

The main finding of this research is the derivation of a boundary-only integral equation for the solutions of initial-boundary value problems governed by a modified-Helmholtz equation of time–space dependent coefficients for anisotropic functionally graded materials with time-dependent properties.

The originality of the research lies on the time dependency of properties of the functionally graded material under consideration.

The purpose of this paper is to study large deviations for the solution processes of a stochastic equation incorporated with the effects of nonlocal condition.

A weak convergence approach is adopted to establish the Laplace principle, which is same as the large deviation principle in a Polish space. The sufficient condition for any family of solutions to satisfy the Laplace principle formulated by Budhiraja and Dupuis is used in this work.

Freidlin–Wentzell type large deviation principle holds good for the solution processes of the stochastic functional integral equation with nonlocal condition.

The asymptotic exponential decay rate of the solution processes of the considered equation towards its deterministic counterpart can be estimated using the established results.

The classical advection-dispersion equation (ADE) model cannot accurately depict the gas transport process in natural geological formations. This paper aims to study the behavior of CO₂ transport in fractal porous media by using an effective Hausdorff fractal derivative advection-dispersion equation (HFDADE) model.

Anomalous dispersion behaviors of CO₂ transport are effectively characterized by the investigation of time and space Hausdorff derivatives on non-Euclidean fractal metrics. The numerical simulation has been performed with different Hausdorff fractal dimensions to reveal characteristics of the developed fractal ADE in fractal porous media. Numerical experiments focus on the influence of the time and space fractal dimensions on flow velocity and dispersion coefficient.

The physical mechanisms of parameters in the Hausdorff fractal derivative model are analyzed clearly. Numerical results demonstrate that the proposed model can well fit the history of gas production data and it can be a powerful technique for depicting the early arrival and long-tailed phenomenon by incorporating a fractal dimension.

To the best of the authors’ knowledge, first time these results are presented.

This research is aimed to mainly be applicable to expediting engineering projects, uses the method of inverse optimization and the double-layer nested genetic algorithm combined with nonlinear programming algorithm, study how to schedule the number of labor in each process at the minimum cost to achieve an extremely short construction period goal.

The method of inverse optimization is mainly used in this study. In the first phase, establish a positive optimization model, according to the existing labor constraints, aiming at the shortest construction period. In the second phase, under the condition that the expected shortest construction period is known, on the basis of the positive optimization model, the inverse optimization method is used to establish the inverse optimization model aiming at the minimum change of the number of workers, and finally the optimal labor allocation scheme that meets the conditions is obtained. Finally, use algorithm to solve and prove with a case.

The case study shows that this method can effectively achieve the extremely short duration goal of the engineering project at the minimum cost, and provide the basis for the decision-making of the engineering project.

The contribution of this paper to the existing knowledge is to carry out a preliminary study on the relatively blank field of the current engineering project with a very short construction period, and provide a path for the vast number of engineering projects with strict requirements on the construction period to achieve a very short construction period, and apply the inverse optimization method to the engineering field. Furthermore, a double-nested genetic algorithm and nonlinear programming algorithm are designed. It can effectively solve various optimization problems.

The conventional design method relies on a priori knowledge, which limits the rapid and efficient development of electronic packaging structures. The purpose of this study is to propose a hybrid method of data-driven inverse design, which couples adaptive surrogate model technology with optimization algorithm to to enable an efficient and accurate inverse design of electronic packaging structures.

The multisurrogate accumulative local error-based ensemble forward prediction model is proposed to predict the performance properties of the packaging structure. As the forward prediction model is adaptive, it can identify respond to sensitive regions of design space and sample more design points in those regions, getting the trade-off between accuracy and computation resources. In addition, the forward prediction model uses the average ensemble method to mitigate the accuracy degradation caused by poor individual surrogate performance. The Particle Swarm Optimization algorithm is then coupled with the forward prediction model for the inverse design of the electronic packaging structure.

Benchmark testing demonstrated the superior approximate performance of the proposed ensemble model. Two engineering cases have shown that using the proposed method for inverse design has significant computational savings while ensuring design accuracy. In addition, the proposed method is capable of outputting multiple structure parameters according to the expected performance and can design the packaging structure based on its extreme performance.

Because of its data-driven nature, the inverse design method proposed also has potential applications in other scientific fields related to optimization and inverse design.

This paper aims to reconstruct the spatially varying orthotropic conductivity based on a two-dimensional inverse heat conduction problem described by a partial differential equation (PDE) model with mixed boundary conditions. The proposed discretization uses a highly accurate technique and allows simple implementations. Also, the authors solve the related inverse problem in such a way that smoothness is enforced on the iterations, showing promising results in synthetic examples and real problems with moving heat source.

The discretization procedure applied to the model for the direct problem uses a pseudospectral collocation strategy in the spatial variables and Crank–Nicolson method for the time-dependent variable. Then, the related inverse problem of recovering the conductivity from temperature measurements is solved by a modified version of Levenberg–Marquardt method (LMM) which uses singular scaling matrices. Problems where data availability is limited are also considered, motivated by a face milling operation problem. Numerical examples are presented to indicate the accuracy and efficiency of the proposed method.

The paper presents a discretization for the PDEs model aiming on simple implementations and numerical performance. The modified version of LMM introduced using singular scaling matrices shows the capabilities on recovering quantities with precision at a low number of iterations. Numerical results showed good fit between exact and approximate solutions for synthetic noisy data and quite acceptable inverse solutions when experimental data are inverted.

The paper is significant because of the pseudospectral approach, known for its high precision and easy implementation, and usage of singular regularization matrices on LMM iterations, unlike classic implementations of the method, impacting positively on the reconstruction process.