Analysis Of Inverse Problems Research Articles

On the finite element analysis of inverse problems in fracture mechanics

This paper presents a numerical study of inverse parameter identification problems in fracture mechanics. Inverse methodology is applied to the detection of subsurface cracks and to the study of propagating cracks. The procedure for detecting subsurface cracks combines the finite element method with a sequential quadratic programming algorithm to solve for the unknown geometric parameters associated with the internal flaw. The procedure utilizes finite element substructuring capabilities in order to minimize the processing and solution time for practical problems. The finite element method and non‐linear optimization are also used in determining the direction a crack will propagate in a heterogeneous planar domain. This procedure involves determining the direction that produces the maximum strain energy release for a given increment of crack growth. The procedure is applied to several numerical examples. The results of these numerical studies coincide with theoretical predictions and experimentally observed crack behavior.

Analysis of boundary inverse heat conduction problems using space marching with Savitzky-Gollay digital filter

This paper presents a method by which boundary inverse heat conduction problems can be analyzed. A space marching algorithm is used for formulating and solving parabolic and hyperbolic inverse heat conduction problems. The solution of numerical examples shows that a combination of the digital filter with the hyperbolic approximation of inverse heat conduction problem increases the stability of the results without loss of resolution. The validity of numerical solution for the inverse problem is examined by comparing the obtained results with the direct solution of the problem.

Probabilistic analysis of implicit inverse problems

Many inverse problems are most naturally formulated as implicit problems where data cannot be expressed in closed form as a function of the unknown model parameters. Notable examples are cases where the data satisfy a partial differential equation with model parameters as constants. When the problem can be recast as an ordinary explicit inverse problem the price to be paid is often that the functional relationship between data and model cannot be formulated in closed form. The consequence is typically that a computer time intensive numerical algorithm is needed to perform the forward calculation, thereby making an iterative solution of the problem unreasonably time consuming or impossible. In this paper we present a probabilistic procedure to solve rather general, implicit inverse problems through Monte Carlo sampling of feasible solutions. Our starting point is a probabilistic formulation of inverse problems, and our goal is to produce near-independent samples from the posterior distribution in model space. From these samples important information on the model and its resolution can be obtained. The proposed algorithm is applied to an implicit real-data problem involving analysis of the fluid motions at the Earth's core-mantle boundary from geomagnetic data.

Identification of Boundary Condition for Elastostatic Thin Plate Bending Problems. 1st Report, BEM Analysis of Inverse Problems for Boundary Values.

Identification of Boundary Condition for Elastostatic Thin Plate Bending Problems. 1st Report, BEM Analysis of Inverse Problems for Boundary Values.

An inverse problem analysis for the determination of probalistic parameters of concrete behavior modeled by a statistical approach

When the Monte Carlo method is used to simulate the heterogeneous behavior of concrete in the framework of a finite element probabilistic analysis, N samples of the vector of random variables (tensile strength, Young's modulus, etc.) are generated from a specific probability density function. If the uncertainties of these material parameters are assumed to vary spatially following a normal distribution, the N samples corresponding to a simulation are function of the mean and the standard deviation that define the Gauss density function. The problem is that these statistical moments are not known,a priori, for the characteristic volume of the finite elements for which the problem has been discretized. In this paper an algorithm is proposed to evaluate the parameters characterizing the statistical distribution (e. g. for a normal distribution: the mean and the standard deviation) for a given response of the structure (for instance, a mean load-displacement curve) following an inverse analysis procedure. A very simple mechanical system is used to verify the feasibility of the procedure. By way of example, it is shown that this kind of inverse problem for the identification of statistical parameters is suitable for concrete.

Resolution analysis of general inverse problems through inverse Monte Carlo sampling

The general inverse problem is characterized by at least one of the following two complications: (1) data can only be computed from the model by means of a numerical algorithm, and (2) the a priori model constraints can only be expressed via numerical algorithms. For linear problems and the so-called `weakly nonlinear problems', which can be locally approximated by a linear problem, analytical methods can provide estimates of the best fitting model and measures of resolution (nonuniqueness and uncertainty of solutions). This is, however, not possible for general problems. The only way to proceed is to use sampling methods that collect information on the posterior probability density in the model space. One such method is the inverse Monte Carlo strategy for resolution analysis suggested by Mosegaard and Tarantola. This method allows sampling of the posterior probability density even in cases where prior information is only available as an algorithm that samples the prior probability density. Once a collection of models sampled according to the posterior is available, it is possible to estimate, not only posterior model parameter covariances, but also resolution measures that are more useful in many applications. For example, posterior probabilities of the existence of interesting Earth structures like discontinuities and flow patterns can be estimated. These extended possibilities for resolution analysis may also provide new insight into problems that are usually treated by means of analytical methods.

Partial Derivatives of Repeated Eigenvalues and Their Eigenvectors

The analysis of inverse problems in linear modeling often require the sensitivities of the eigenvalues and eigenvectors. The calculation of these sensitivities is mathematically related to the corresponding partial derivatives, which do not exist for any parameterization. Inasmuch as eigenvalues and eigenvectors are coupled by the constitutional equation of the general eigenvalue problem, their derivatives are coupled, too. Conditions on the parameterization are derived and formulated as theorems, which ensure the existence of the partial derivatives of the eigenvalues and eigenvectors with respect to these parameters. The application of the theorems is demonstrated by examples.

Trimming and procrastination as inversion techniques

By examining the processes of truncating and approximating the model space (trimming it), and by committing to neither the objectivist nor the subjectivist interpretation of probability (procrastinating), we construct a formal scheme for solving linear and non-linear geophysical inverse problems. The necessary prior information about the correct model x E can be either a collection of inequalities or a probability measure describing where x E was likely to be in the model space X before the data vector y 0 was measured. The results of the inversion are (1) a vector z 0 that estimates some numerical properties z E of x E ; (2) an estimate of the error δz = z 0 − z E . As y 0 is finite dimensional, so is z 0, and hence in principle inversion cannot describe all of x E . The error δz is studied under successively more specialized assumptions about the inverse problem, culminating in a complete analysis of the linear inverse problem with a prior quadratic bound on x E . Our formalism appears to encompass and provide error estimates for many of the inversion schemes current in geomagnetism, and would be equally applicable in geodesy and seismology if adequate prior information were available there. As an idealized example we study the magnetic field at the core-mantle boundary, using satellite measurements of field elements at sites assumed to be almost uniformly distributed on a single spherical surface. Magnetospheric currents are neglected and the crustal field is idealized as a random process with rotationally invariant statistics. We find that an appropriate data compression diagonalizes the variance matrix of the crustal signal and permits an analytic trimming of the idealized problem.

Ensemble inference in terms of empirical orthogonal functions

SUMMARY Many geophysical problems involve inverting data in order to obtain meaningful descriptions of the Earth‘s interior. One of the basic characteristics of these inverse problems is their non-uniqueness. Since computation power has increased enormously in the last few years, it has become possible to deal with this non-uniqueness by generating and selecting a number of models that all fit the data up to a certain tolerance. In this way a solution space with acceptable models is created. The remaining task is then to infer the common robust properties of all the models in the ensemble. In this paper these properties are determined using empirical orthogonal function (EOF) analysis. This analysis provides a method to search for subspaces in the solution space (ensemble) that correspond to the patterns of minimum variability. In order to show the effectiveness of this method, two synthetic tests are presented. To verify the applicability of the analysis in geophysical inverse problems, the method is applied to an ensemble generated by a Monte Carlo search technique which inverts group-velocity dispersion data produced by using vertical-component, long-period synthetic seismograms of the fundamental Rayleigh mode. The result shows that EOF analysis successfully determines the well-constrained parts of the models and in effect reduces the variability present in the original ensemble while still recovering the earth model used to generate the synthetic seismograms. Finally, an application of the method to examine the contrast in upper-mantle S-wave velocity across the Tornquist-Tesseyre Zone is presented, indicating a significant change in S-wave velocity in the upper mantle beneath this zone bordering the East European Platform and Tectonic Europe, and a significantly thicker crust beneath the East European Platform.

Inverse problem in analysis of the diffusion battery data

Inverse problem in analysis of the diffusion battery data

Inverse Problem Analysis for Evaluation of Surface Tension of Liquid.

A function which represents the surface tension of a liquid based on the configuration of a droplet is obtained theoretically. Behavior of a droplet of liquid is governed by a nonlinear differential equation which can be solved numerically by setting its initial condition and surface tension. In order to solve the inverse problem of this equation, a formal solution is suggested as a series function, and a norm on the function space of droplet configurations is defined. Under the measure of this norm, a sequence of coefficients of the formal solution is determined from the observed droplet configuration. The inverse function of which variables are the sequence of coefficients that represents a drop configuration is obtained by using potential theory on the basis of the analytical representation of droplet configuration. It is also shown that convergence of the inverse function of finite series is sufficiently good for practical use.

Monte Carlo sampling of solutions to inverse problems

Probabilistic formulation of inverse problems leads to the definition of a probability distribution in the model space. This probability distribution combines a priori information with new information obtained by measuring some observable parameters (data). As, in the general case, the theory linking data with model parameters is nonlinear, the a posteriori probability in the model space may not be easy to describe (it may be multimodal, some moments may not be defined, etc.). When analysing an inverse problem, obtaining a maximum likelihood model is usually not sufficient, as we normally also wish to have information on the resolution power of the data. In the general case we may have a large number of model parameters, and an inspection of the marginal probability densities of interest may be impractical, or even useless. But it is possible to pseudorandomly generate a large collection of models according to the posterior probability distribution and to analyse and display the models in such a way that information on the relative likelihoods of model properties is conveyed to the spectator. This can be accomplished by means of an efficient Monte Carlo method, even in cases where no explicit formula for the a priori distribution is available. The most well known importance sampling method, the Metropolis algorithm, can be generalized, and this gives a method that allows analysis of (possibly highly nonlinear) inverse problems with complex a priori information and data with an arbitrary noise distribution.

On the velocity and acceleration snalyses of general parallel robotic manipulators

AbstractThis article presents a unified method for the velocity and acceleration analyses of general parallel robotic manipulators. In this method, the governing velocity and acceleration equations of the robot are formulated in the joint space by considering it as a tree‐type mechanism subjected to Cartesian velocity and acceleration constraints. The procedures for solving the direct and the inverse analysis problems are established based on the generalized coordinate partitioning technique and the full pivoting Gaussian elimination method. These procedures take into account the overconstrained kinematic loops and redundant joints, and are applicable to either fully parallel robots or hybrid series‐parallel robots without any modifications. In addition, a set of generalized recursive formulas for computing the forward Cartesian velocity and acceleration transformations are developed. These formulas are also useful for efficient evaluating the complex nonlinear terms of the governing acceleration equations. © 1995 John Wiley & Sons, Inc.

Distribution of Heat-Transfer Coefficients from Small Surfaces Cooled with Submerged Jets of Fluorocarbon Liquid Determined with Inverse-Problem Analysis of Heat Conduction.

The method of inverse-problem analysis was used to determine the distribution of heat-transfer coefficients of a silicon heater cooled with submerged jets of fluorocarbon liquid. The silicon heater was made specifically to investigate high-performance cooling systems of electronic devices. The heater is constructed of many cells which can independently generate heat and measure the temperature on one side of the heater, and it is colled from the other side of the heater. Inverse-problem analysis was used to correct for heat conduction in the heater and to determine the distribution of heat transfer coefficients on the cooling surface. The heat-transfer coefficients were also measured by independent techniques to confirm the values determined with the inverse-problem analysis. One of these independent techniques was to control the heater power to produce a uniform temperature. Another technique was to use a SUS film heater. The distribution of heat transfer coefficients determined with these three techniques were in reasonable agreement.

An analysis of the time-domain electromagnetic inverse problem of determining the susceptibility kernel for a stratified dispersive slab

Article An analysis of the time-domain electromagnetic inverse problem of determining the susceptibility kernel for a stratified dispersive slab was published on January 1, 1995 in the journal Journal of Inverse and Ill-posed Problems (volume 3, issue 6).

Mathematical Structure of Numerical Analysis and Regularization of an Inverse Boundary Value Problem in Laplace Field.

Inverse boundary value problems deal with the estimation of boundary values on incompletely prescribed boundaries. Numerical analysis of inverse boundary value problems using discretization methods can be reduced to solving simultaneous equations, which are ill-conditioned due to the ill-posedness of the problems. Errors included in prescribed boundary values are therefore magnified remarkably by inverse calculation without regularization. In this study, the mathematical structure of this error magnification behavior was studied theoretically for inverse boundary value problems of the Laplace field. Singular value decomposition was applied to evaluate the magnification amplitude called the condition number, which was given as the ratio of the maximum singular value to the nonzero minimum singular value of the coefficient matrix. The condition number was found to represent the magnification amplitude of the highest frequency fluctuation mode of variables. A regularization method using effective pseudoinverse was introduced, in which the rank of the coefficient matrix was reduced to effective rank and therefore small singular values were ignored. The effectiveness of the use of the effective pseudoinverse was explained using the singular values and the right singular vectors. An equation was proposed for evaluating the condition number. The admissible condition number method was proposed for determining the effective rank. Numerical simulations showed that the proposed equation and method were useful for estimating the condition number and the effective rank, and obtaining good estimates of boundary values.

Numerical analysis of inverse problems

Numerical analysis of inverse problems

Some aspects of inverse problem analysis for a layered halfspace

The present paper discusses the questions associated with the nondestructive testing of flexible pavements. The problem of how to determine layer moduli from the measured surface deformation is solved using technique based on the theory of ill-posed problems. The proposed iterative solution process exploits the capacity to compute effectively the derivatives of the error functional with respect to elastic moduli. The properties of the procedure are analysed and tested by numerically solved example problems. It is shown that by simulated computer experiments based on the solution procedure properties, valuable information on the measuring instrumentation setup may be obtained.

Spherulitic crystallization: An analysis of inverse stefan problems in cartesian, cylindrical, and spherical coordinate systems

AbstractSpherulitic crystallization presents an inverse Stefan problem, which is solved by numerical methods using the DuFort‐Frankel scheme and Lagrangian coordinate systems. The numerical solution shows excellent agreement with the analytical solution in the Cartesian coordinate system. A systematic error in the numerical solution for the spherical and cylindrical coordinate systems is examined by a material balance for the system. The systematic error was found to decrease rapidly as the dimensionless time increased and the grid spacing decreased.

静磁界系逆問題における一解法とその高速化

静磁界系逆問題における一解法とその高速化