Class Of Inverse Problems Research Articles

Identification of plastic constitutive parameters at large deformations from three dimensional displacement fields

The aim of this paper is to provide a general procedure to extract the constitutive parameters of a plasticity model starting from displacement measurements and using the Virtual Fields Method. This is a classical inverse problem which has been already investigated in the literature, however several new features are developed here. First of all the procedure applies to a general three-dimensional displacement field which leads to large plastic deformations, no assumptions are made such as plane stress or plane strain although only pressure-independent plasticity is considered. Moreover the equilibrium equation is written in terms of the deviatoric stress tensor that can be directly computed from the strain field without iterations. Thanks to this, the identification routine is much faster compared to other inverse methods such as finite element updating. The proposed method can be a valid tool to study complex phenomena which involve severe plastic deformation and where the state of stress is completely triaxial, e.g. strain localization or necking occurrence. The procedure has been validated using a three dimensional displacement field obtained from a simulated experiment. The main potentialities as well as a first sensitivity study on the influence of measurement errors are illustrated.

Conductivity imaging with interior potential measurements

In this article, we present two reconstruction methods intended to be used for conductivity imaging with data obtained from a planar electrical impedance tomography device for breast cancer detection. The inverse problem to solve is different from the classical inverse conductivity problem. We reconstruct the electrical conductivity of a two-dimensional domain from boundary measurements of currents and interior measurements of the potential. One reconstruction algorithm is based on a discrete resistor model; the other one is an integral equation approach for smooth conductivity distributions.

Inverse problem for Sturm-Liouville operators on hedgehog-type graphs

We study the inverse spectral problem for Sturm-Liouville differential operators on hedgehog-type graphs with a cycle and with standard matching conditions at interior vertices. We prove a uniqueness theorem and obtain a constructive solution for this class of inverse problems.

Determining Individual Stresses Throughout a Pinned Aluminum Joint by Reflective Photoelasticity

Pinned (bolted) joints are an extremely important, but difficult to analyze, structural or mechanical element. They are a class of inverse problems in which the stresses at the pin/hole interface are typically unknown. Moreover, stresses vary non-linearly with applied load. Failures of mechanical or structural systems frequently initiate at connections. Although almost always present, many stress analyses of such mechanical connections ignore friction for simplicity. The stresses are evaluated here in an aluminum connector using a series solution of an Airy stress function, the coefficients being evaluated from known boundary tractions (near, but not including the contact region on the hole) and photoelastically measured data obtained from a bonded birefringent coating. Both friction and pin/hole clearance are accounted for, and individual stresses are evaluated full-field, including on the contact boundary of the hole.

On some classes of inverse problems for parabolic and elliptic equations

We study solvability of inverse problems of finding the right-hand side together with a solution itself for vector-valued parabolic and elliptic equations. The usual boundary conditions are supplemented with the overdetermination conditions that are the values of a solution on some system of surfaces.

Geodesic Active Fields—A Geometric Framework for Image Registration

In this paper we present a novel geometric framework called geodesic active fields for general image registration. In image registration, one looks for the underlying deformation field that best maps one image onto another. This is a classic ill-posed inverse problem, which is usually solved by adding a regularization term. Here, we propose a multiplicative coupling between the registration term and the regularization term, which turns out to be equivalent to embed the deformation field in a weighted minimal surface problem. Then, the deformation field is driven by a minimization flow toward a harmonic map corresponding to the solution of the registration problem. This proposed approach for registration shares close similarities with the well-known geodesic active contours model in image segmentation, where the segmentation term (the edge detector function) is coupled with the regularization term (the length functional) via multiplication as well. As a matter of fact, our proposed geometric model is actually the exact mathematical generalization to vector fields of the weighted length problem for curves and surfaces introduced by Caselles-Kimmel-Sapiro. The energy of the deformation field is measured with the Polyakov energy weighted by a suitable image distance, borrowed from standard registration models. We investigate three different weighting functions, the squared error and the approximated absolute error for monomodal images, and the local joint entropy for multimodal images. As compared to specialized state-of-the-art methods tailored for specific applications, our geometric framework involves important contributions. Firstly, our general formulation for registration works on any parametrizable, smooth and differentiable surface, including nonflat and multiscale images. In the latter case, multiscale images are registered at all scales simultaneously, and the relations between space and scale are intrinsically being accounted for. Second, this method is, to the best of our knowledge, the first reparametrization invariant registration method introduced in the literature. Thirdly, the multiplicative coupling between the registration term, i.e. local image discrepancy, and the regularization term naturally results in a data-dependent tuning of the regularization strength. Finally, by choosing the metric on the deformation field one can freely interpolate between classic Gaussian and more interesting anisotropic, TV-like regularization.

Numerical solution of an inverse medium scattering problem with a stochastic source

This paper is concerned with the inverse medium scattering problem with a stochastic source, the reconstruction of the refractive index of an inhomogeneous medium from the boundary measurements of the scattered field. As an inverse problem, there are two major difficulties in addition to being highly nonlinear: the ill-posedness and the presence of many local minima. To overcome these difficulties, a stable and efficient recursive linearization method has been recently developed for solving the inverse medium scattering problem with a deterministic source. Compared to classical inverse problems, stochastic inverse problems, referred to as inverse problems involving uncertainties, have substantially more difficulties due to randomness and uncertainties. Based on the Wiener chaos expansion, the stochastic problem is converted into a set of decoupled deterministic problems. The strategy developed is a new hybrid method combining the WCE with the recursive linearization method for solving the inverse medium problem with a stochastic source. Numerical experiments are reported to demonstrate the effectiveness of the proposed approach.

Planning foot placements for a humanoid robot: A problem of inverse kinematics

We present a novel approach to plan foot placements for a humanoid robot according to kinematic tasks. In this approach, the foot placements are determined by the continuous deformation of a robot motion including a locomotion phase according to the desired tasks. We propose to represent the motion by a virtual kinematic tree composed of a kinematic model of the robot and articulated foot placements. This representation allows us to formulate the motion deformation problem as a classical inverse kinematics problem on a kinematic tree. We first provide details of the basic scheme where the number of footsteps is given in advance and illustrate it with scenarios on the robot HRP-2. Then we propose a general criterion and an algorithm to adapt the number of footsteps progressively to the kinematic goal. The limits and possible extensions of this approach are discussed last.

Solution of the Initial Inverse Problems in the Heat Equation Using the Finite Difference Method with Positivity-Preserving Padé Schemes

The classical inverse problem of recovering the initial temperature distribution from the final temperature distribution is extremely ill-posed. We propose a class of numerical schemes based on positivity-preserving Padé approximations to solve initial inverse problems in the heat equation. We also utilize a partial fraction decomposition technique to solve the problem more efficiently when higher order Padé approximations are used. We apply the proposed numerical schemes on the parabolic heat equation. Our aim is to model the problem as a direct problem and use our numerical schemes to recover the initial profile in a stable and efficient way.

Inverse mode problems for real and symmetric quadratic models

Many natural phenomena can be modeled by a second-order dynamical system , where stands for an appropriate state variable and M, C, K are time-invariant, real and symmetric matrices. In contrast to the classical inverse vibration problem where a model is to be determined from natural frequencies corresponding to various boundary conditions, the inverse mode problem concerns the reconstruction of the coefficient matrices (M, C, K) from a prescribed or observed subset of natural modes. This paper set forth a mathematical framework for the inverse mode problem and resolves some open questions raised in the literature. In particular, it shows that given merely the desirable structure of the spectrum, namely given the cardinalities of real or complex eigenvalues but not of the actual eigenvalues, the set of eigenvectors can be completed via solving an under-determined nonlinear system of equations. This completion suffices to construct symmetric coefficient matrices (M, C, K) whereas the underlying system can have arbitrary eigenvalues. Generic conditions under which the real symmetric quadratic inverse mode problem is solvable are discussed. Applications to important tasks such as updating models without spill-over or constructing models with positive semi-definite coefficient matrices are discussed.

Path-Weighted Diffusivity Functions for Parameterization of Heat Deposition Processes

General parameterizations are constructed for spatial modulation of heat diffusion patterns according to energy deposition characteristics occurring within a volume of material where there exist inhomogeneous or anisotropic thermal diffusivity. These parameterizations are formulated in terms of path-weighted diffusivity functions. The construction of a general parameterization of energy deposition processes where spatially dependent thermal diffusivity exists is necessary for their inverse analysis. The structure of such a parameterization follows from the concepts of model and data spaces that imply the existence of an optimal parametric representation for a given class of inverse problems. Accordingly, the optimal parametric representation is determined by the characteristics of the available data, which, in principle, can contain both experimental measurements and numerical simulation data. Parameterizations for spatial modulation of heat diffusion follow from the observation that many different types of energy deposition processes can be represented by weighted sums of basis functions whose general forms are that of spatially modulated diffusion. The parameterizations presented are constructed according to a specific definition of the inverse heat deposition problem that provides a rigorous foundation for a highly flexible and general parameterization of energy deposition processes, which is essential for their inverse analysis.

Background adjustment of cDNA microarray images by Maximum Entropy distributions

Many empirical studies have demonstrated the exquisite sensitivity of both traditional and novel statistical and machine intelligence algorithms to the method of background adjustment used to analyze microarray datasets. In this paper we develop a statistical framework that approaches background adjustment as a classic stochastic inverse problem, whose noise characteristics are given in terms of Maximum Entropy distributions. We derive analytic closed form approximations to the combined problem of estimating the magnitude of the background in microarray images and adjusting for its presence. The proposed method reduces standardized measures of log expression variability across replicates in situations of known differential and non-differential gene expression without increasing the bias. Additionally, it results in computationally efficient procedures for estimation and learning based on sufficient statistics and can filter out spot measures with intensities that are numerically close to the background level resulting in a noise reduction of about 7%.

Spatial Modulation and Filtering of Diffusion Patterns for Inverse Analysis of Heat Deposition

General parameterizations are constructed for spatial modulation and filtering of heat diffusion patterns according to general energy deposition characteristics occurring within a volume of material resulting from a volumetrically coupled energy source. These parameterizations include previously constructed models of energy deposition as special cases. The construction of a general parameterization of energy deposition processes is necessary for their inverse analysis. The structure of such a parameterization follows from the concepts of model and data spaces that imply the existence of an optimal parametric representation for a given class of inverse problems. Accordingly, the optimal parametric representation is determined by the characteristics of the available data, which in principle can contain both experimental measurements and numerical simulation data. Parameterizations for spatial modulation and filtering of heat diffusion follow from the observation that many different types of energy deposition processes can be represented by weighted sums of basis functions whose general forms are that of spatially modulated or filtered diffusion. A significant aspect of the parameterizations presented is that the definition of the inverse heat deposition problem, which is adopted for their construction, provides a rigorous foundation for a highly flexible and general parameterization of energy deposition processes, which is essential for their inverse analysis. A preliminary proof is presented that shows the significance of these parameterizations for the application of similarity transformations to the inverse analysis of energy deposition processes. The applicability of similarity transforms to the inverse analysis of heat deposition is another property that follows from the specific definition of the inverse heat deposition problem considered here.

A Remark on Inverse Problems for Sturm–Liouville Operators on Graphs

We compare some formulations of the inverse spectral problems for differential operators on graphs with the classical inverse problems on an interval.

General Basis Functions for Parametric Representation of Energy Deposition Processes

General basis functions for parametric representation of energy deposition processes are constructed according to the general physical characteristics of energy deposition within a volume of material from a beam energy source. These basis functions include previously constructed source functions as special cases. The construction of a general parameterization of energy deposition processes, e.g., welding and rapid prototyping, is necessary for inverse analysis of such processes. The structure of such a parameterization follows from the concepts of model and data spaces that imply the existence of an optimal parametric representation for a given class of inverse problems. Accordingly, the optimal parametric representation lying within the model space is determined by the characteristics of the available data, i.e., data space, which contain both experimental measurements and numerical simulation data. Experimental measurements include solidification cross-sections, thermocouple measurements, and microstructural changes. Numerical simulation data include general temperature field trend characteristics, response characteristics of materials to volumetric energy deposition, and the relative sensitivity of temperature field characteristics to phenomena occurring on different space and time scales. A general procedure is described for using basis functions with the available experimental and numerical data to construct a multidimensional field representation of a large class of energy deposition processes.

Radon transform inversion using the shearlet representation

The inversion of the Radon transform is a classical ill-posed inverse problem where some method of regularization must be applied in order to accurately recover the objects of interest from the observable data. A well-known consequence of the traditional regularization methods is that some important features to be recovered are lost, as evident in imaging applications where the regularized reconstructions are blurred versions of the original. In this paper, we show that the affine-like system of functions known as the shearlet system can be applied to obtain a highly effective reconstruction algorithm which provides near-optimal rate of convergence in estimating a large class of images from noisy Radon data. This is achieved by introducing a shearlet-based decomposition of the Radon operator and applying a thresholding scheme on the noisy shearlet transform coefficients. For a given noise level ε, the proposed shearlet shrinkage method can be tuned so that the estimator will attain the essentially optimal mean square error O ( log ( ε − 1 ) ε 4 / 5 ) , as ε → 0 . Several numerical demonstrations show that its performance improves upon similar competitive strategies based on wavelets and curvelets.

Regularization of the solution of one class of inverse problems of remote atmospheric measurements

A regularized inversion method for a certain narrow class of atmospheric measurement problems is proposed. An example of the use of the method to determine the components of a multicomponent continuum in the medium ir-band is given.

STRINGS, WAVES, DRUMS: SPECTRA AND INVERSE PROBLEMS

This survey treats a number of interconnected topics related in one way or another to the famous paper of Mark Kac, "Can one hear the shape of a drum?": wave motion, classical and quantum inverse problems, integrable systems, and the relations between spectra and geometry. We sketch the history and some of the principal developments from the vibrating string to quantum inverse problems, the KdV equation and integrable systems, spectral geometry and the index problem.

Numerical Identification Procedure for a Phenomenological Vector Hysteresis Model

This paper presents an identification procedure for a phenomenological vector hysteresis model, called the Della Torre, Pinzaglia and Cardelli Model, introduced as a general hysteresis vector model for magnetic materials. The identification of the distribution function is approached by using the classical deterministic inverse problem theory and a suitable regularization procedure. A discussion on the presented method and an identification process for a sample of commercial Fe-Si magnetic steel are provided.

A pseudo-dynamical systems approach to a class of inverse problems in engineering

A pseudo-dynamical approach for a class of inverse problems involving static measurements is proposed and explored. Following linearization of the minimizing functional associated with the underlying optimization problem, the new strategy results in a system of linearized ordinary differential equations (ODEs) whose steady-state solutions yield the desired reconstruction. We consider some explicit and implicit schemes for integrating the ODEs and thus establish a deterministic reconstruction strategy without an explicit use of regularization. A stochastic reconstruction strategy is then developed making use of an ensemble Kalman filter wherein these ODEs serve as the measurement model. Finally, we assess the numerical efficacy of the developed tools against a few linear and nonlinear inverse problems of engineering interest.