On the solution of an inverse natural convection problem using various conjugate gradient methods

The inverse problem of determining the time‐varying strength of a heat source, which causes natural convection in a two‐dimensional cavity, is considered. The Boussinesq equation is used to model the natural convection induced by the heat source. The inverse natural convection problem is solved through the minimization of a performance function utilizing the conjugate gradient method. The gradient of the performance function needed in the minimization procedure of the conjugate gradient method is obtained by employing either the adjoint variable method or the direct differentiation method. The accuracy and efficiency of these two methods are compared, and a new method is suggested that exploits the advantageous aspects of both methods while avoiding the shortcomings of them. Copyright © 2000 John Wiley & Sons, Ltd.     

The inverse moment treatment from center of mass depth and four dimensionless shape factors

The inverse moment problem is considered for a homogeneous body with a vertical symmetry axis. One calculates the depth of the center of mass and the four dimensionless shape coefficients independent of the mass and density of the body from the linear and angular radii in models. __________ Translated from Izmeritel’naya Tekhnika, No. 2, pp. 10–14, February, 2008.     

The method of fundamental solution for the inverse source problem for the space-fractional diffusion equation

In this article, a meshless numerical method for solving the inverse source problem of the space-fractional diffusion equation is proposed. The numerical solution is approximated using the fundamental solution of the space-fractional diffusion equation as a basis function. Since the resulting matrix equation is extremely ill-conditioned, a regularized solution is obtained by adopting the Tikhonov regularization scheme, in which the choice of the regularization parameter is based on generalized cross-validation criterion. Two typical numerical examples are given to verify the efficiency and accuracy of the proposed method.     

Uncertainty reduction in residual stress measurements by an optimised inverse solution using nonconsecutive polynomials

Many destructive methods for measuring residual stresses such as the slitting method require an inverse analysis to solve the problem. The accuracy of the result as well as an uncertainty component (the model uncertainty) depends on the basis functions used in the inverse solution. The use of a series expansion as the basis functions for the inverse solution was analysed in a previous work for the particular case where functions orders grew consecutively. The present work presents a new estimation of the model uncertainty and a new improved methodology to select the final basis functions for the case where the basis is composed of polynomials. Including nonconsecutive polynomial orders in the basis generates a larger space of possible solutions to be evaluated and allows the possibility to include higher-order polynomials. The paper includes a comparison with two other inverse analyses methodologies applied to synthetically generated data. With the new methodology, the final error is reduced and the uncertainty estimation improved.     

Photoelastic inverse problem solution for a biaxially loaded infinite plate with a hole

This paper presents an inverse problem solution for the case of photoelastic isochromatic fringes around the hole of a biaxially uniformly loaded infinite plate for two problems. Problem 1 relates to an infinite plate in which the circular hole is drilled first and then the loads are applied. Problem 2 is the residual stress problem in which the hole is drilled after the biaxial load is applied to the infinite plate. This "hole method" solution may be used for all cases of biaxial, far field uniform loading conditions.     

Detection of material interfaces using a regularized level set method in piezoelectric structures

An algorithm to solve the inverse problem of detecting inclusion interfaces in a piezoelectric structure is proposed. The material interfaces are implicitly represented by level sets which are identified by applying regularization using total variation penalty terms. The inverse problem is solved iteratively and the extended finite element method is used for the analysis of the structure in each iteration. The formulation is presented for three-dimensional structures and inclusions made of different materials are detected by using multiple level sets. The results obtained prove that the iterative procedure proposed can determine the location and approximate shape of material sub-domains in the presence of higher noise levels.     

使用点源求解脉动球的声辐射逆问题时的精度分析

文章使用点源取代球谐函数作为独立函数，来推广赫姆霍兹方程-最小二乘法（HELS），之后对这种新方法求解脉动球的声辐射逆问题的精度问题进行了讨论，得出了一些有用的结论。     

Numerical solution of inverse nodal problem with an eigenvalue in the boundary condition

In this study, we consider Sturm–Liouville problem with a boundary condition depending on spectral parameter. We apply the nodal points as input data and calculate the approximate solution of the inverse nodal problem using Chebyshev polynomials of the first kind. Finally, we illustrate the numerical results by providing several examples.     

Low‐frequency assessment of the in situ acoustic absorption of materials in rooms: an inverse problem approach using evolutionary optimization

The in situ assessment of the acoustic absorption of materials is often a necessity. The need to cover the whole frequency range of interest for the building engineer has led the authors to an approach involving two frequency‐complementary measurement methods. This paper deals with the part dedicated to low frequencies. The measurement is defined here as a boundary inverse interior problem. A numerical model of the room under investigation, allowing for the computation of the pressure field in the volume, given impedance boundary conditions and a point source, is combined to a global optimization algorithm. The algorithm explores the set of possible boundary conditions in order to minimize the difference between the computed pressure values and the one observed at a few measurement points, leading to the determination of all the boundary conditions at a time. In practice, the finite element method (FEM) or the finite difference method (FDM) is used here to model the room and an Evolution Strategy as the optimization tool. After describing the ES operators, a numerical study is carried out on simulated measurements, both on problem‐ and algorithm‐specific parameters, in the case of an academic two‐dimensional room geometry. The method is then applied to a three‐dimensional room with promising results. Copyright © 2002 John Wiley & Sons, Ltd.     

Iterative solution of the random eigenvalue problem with application to spectral stochastic finite element systems

A new algorithm for the computation of the spectral expansion of the eigenvalues and eigenvectors of a random non‐symmetric matrix is proposed. The algorithm extends the deterministic inverse power method using a spectral discretization approach. The convergence and accuracy of the algorithm is studied for both symmetric and non‐symmetric matrices. The method turns out to be efficient and robust compared to existing methods for the computation of the spectral expansion of random eigenvalues and eigenvectors. Copyright © 2006 John Wiley & Sons, Ltd.     

Mathematical analysis and solution methodology for an inverse spectral problem arising in the design of optical waveguides

We analyse mathematically the problem of determining refractive index profiles from some desired/measured guided waves propagating in optical fibres. We establish the uniqueness of the solution of this inverse spectral problem assuming that only one guided mode is known. We then propose an iterative computational procedure for solving numerically the considered inverse spectral problem. Numerical results are presented to illustrate the potential of the proposed regularized Newton algorithm to efficiently and accurately retrieve the refractive index profiles even when the guided mode measurements are highly noisy.     

Proper generalized decomposition solution of the parameterized Helmholtz problem: application to inverse geophysical problems

The identification of the geological structure from seismic data is formulated as an inverse problem. The properties and the shape of the rock formations in the subsoil are described by material and geometric parameters, which are taken as input data for a predictive model. Here, the model is based on the Helmholtz equation, describing the acoustic response of the system for a given wave length. Thus, the inverse problem consists in identifying the values of these parameters such that the output of the model agrees the best with observations. This optimization algorithm requires multiple queries to the model with different values of the parameters. Reduced order models are especially well suited to significantly reduce the computational overhead of the multiple evaluations of the model. In particular, the proper generalized decomposition produces a solution explicitly stating the parametric dependence, where the parameters play the same role as the physical coordinates. A proper generalized decomposition solver is devised to inexpensively explore the parametric space along the iterative process. This exploration of the parametric space is in fact seen as a post‐process of the generalized solution. The approach adopted demonstrates its viability when tested in two illustrative examples. Copyright © 2016 John Wiley & Sons, Ltd.     

Adjoint‐weighted variational formulation for the direct solution of inverse problems of general linear elasticity with full interior data

We describe a novel variational formulation of inverse elasticity problems given interior data. The class of problems considered is rather general and includes, as special cases, plane deformations, compressibility and incompressiblity in isotropic materials, 3D deformations, and anisotropy. The strong form of this problem is governed by equations of pure advective transport. The variational formulation is based on a generalization of the adjoint‐weighted variational equation (AWE) formulation, originally developed for flow of a passive scalar. We describe how to apply AWE to various cases, and prove several properties. We prove that the Galerkin discretization of the AWE formulation leads to a stable, convergent numerical method, and prove optimal rates of convergence. The numerical examples demonstrate optimal convergence of the method with mesh refinement for multiple unknown material parameters, graceful performance in the presence of noise, and robust behavior of the method when the target solution is C^∞, C⁰, or discontinuous. Copyright © 2009 John Wiley & Sons, Ltd.     

Computational algorithms for solving the coefficient inverse problem for parabolic equations

Among inverse problems for partial differential equations, we distinguish coefficient inverse problems, which are associated with the identification of coefficients and/or the right-hand side of an equation using some additional information. When considering time-dependent problems, the identification of the coefficient dependences on space and on time is usually separated into individual problems. In some cases, we have linear inverse problems (e.g. identification problems for the right-hand side of an equation); this situation essentially simplify their study. This work deals with the problem of determining in a multidimensional parabolic equation the lower coefficient that depends on time only. To solve numerically a non-linear inverse problem, linearized approximations in time are constructed using standard finite difference approximations in space. The computational algorithm is based on a special decomposition, where the transition to a new time level is implemented via solving two standard elliptic problems.     

The linear sampling method for a mixed scattering problem

This paper is concerned with the scattering problem of time-harmonic acoustic plane waves by a mixed scatterer, which is a combination of an open crack and an impenetrable obstacle. Firstly, the well-posedness of the solution to the direct scattering problem is established using the variational method. Then, a uniqueness result for the inverse scattering problem is proved, that is, both of the crack and the impenetrable obstacle can be uniquely determined simultaneously by the knowledge of the far-field pattern. Furthermore, a mathematical basis is given to reconstruct the shape of the crack and the impenetrable obstacle using the linear sampling method, and some numerical examples are given to establish the viability of our approach.     

Numerical solution of the variation boundary integral equation for inverse problems

In this paper a procedure to solve the identification inverse problems for two‐dimensional potential fields is presented. The procedure relies on a boundary integral equation (BIE) for the variations of the potential, flux, and geometry. This equation is a linearization of the regular BIE for small changes in the geometry. The aim in the identification inverse problems is to find an unknown part of the boundary of the domain, usually an internal flaw, using experimental measurements as additional information. In this paper this problem is solved without resorting to a minimization of a functional, but by an iterative algorithm which alternately solves the regular BIE and the variation BIE. The variation of the geometry of the flaw is modelled by a virtual strainfield, which allows for greater flexibility in the shape of the assumed flaw. Several numerical examples demonstrate the effectiveness and reliability of the proposed approach. Copyright © 2000 John Wiley & Sons, Ltd.     

A contact algorithm for the Signorini problem using space–time finite elements

The most common approach in the finite‐element modelling of continuum systems over space and time is to employ the finite‐element discretization over the spatial domain to reduce the problem to a system of ordinary differential equations in time. The desired time integration scheme can then be used to step across the so‐called time slabs, mesh configurations in which every element shares the same degree of time refinement. These techniques may become inefficient when the nature of the initial boundary value problem is such that a high degree of time refinement is required only in specific spatial regions of the mesh. Ideally one would be able to increase the time refinement only in those necessary regions. We achieve this flexibility by employing space–time elements with independent interpolation functions in both space and time. Our method is used to examine the classic contact problem of Signorini and allows us to increase the time refinement only in the spatial region adjacent to the contact interface. We also develop an interface‐tracking algorithm that tracks the contact boundary through the space–time mesh and compare our results with those of Hertz contact theory. Copyright 2004 John Wiley & Sons, Ltd.     

模拟退火算法在线热源反问题数值求解中的应用

提出采用模拟退火算法(simulated annealing,SA)来数值求解线热源反问题.探讨了如何设计算法使之适合反问题求解,并给出了算法求解的伪代码;通过线源正问题的模拟数据,使用设计的SA算法进行反问题求解,以此来验证算法求解的准确性和可靠性,并对一组实测数据进行了计算.结果表明,该算法不但可以实现两个参数同时、快速反演,而且具有求解精度高,对初始条件依赖少,编制容易等优点.     

Regularization of the inverse heat conduction problem by the discrete Fourier transform

Two methods of solving the inverse heat conduction problem with employment of the discrete Fournier transform are presented in this article. The first one operates similarly to the SVD algorithm and consists in reducing the number of components of the discrete Fournier transform which are taken into account to determine the solution to the inverse problem. The second method is related to the regularization of the solution to the inverse problem in the discrete Fournier transform domain. Those methods were illustrated by numerical examples. In the first example, an influence of the boundary conditions disturbance by a random error on the solution to the inverse problem (its stability) was examined. In the second example, the temperature distribution on the inner boundary of the multiply connected domain was determined. Results of calculations made in both ways brought very good outcomes and confirm the usefulness of applying the discrete Fournier transform to solving inverse problems.     

Direct collocation method for identifying the initial conditions in the inverse wave problem using radial basis functions

A direct collocation method associated with explicit time integration using radial basis functions is proposed for identifying the initial conditions in the inverse problem of wave propagation. Optimum weights for the boundary conditions and additional condition are derived based on Lagrange’s multiplier method to achieve the prime convergence. Tikhonov regularization is introduced to improve the stability for the ill-posed system resulting from the noise, and the L-curve criterion is employed to select the optimum regularization parameter. No iteration scheme is required during the direct collocation computation which promotes the accuracy and stability for the solutions, while Galerkin-based methods demand the iteration procedure to deal with the inverse problems. High accuracy and good stability of the solution at very high noise level make this method a superior scheme for solving inverse problems.