Class Of Inverse Problems Research Articles

An iterative method to reconstruct the refractive index of a medium from time-of-flight measurements**The authors dedicate this article to Prof Todd Quinto on the occasion of his 65th birthday.

The article deals with a classical inverse problem: the computation of the refractive index of a medium from ultrasound time-of-flight measurements. This problem is very popular in seismics but also for tomographic problems in inhomogeneous media. For example ultrasound vector field tomography needs a priori knowledge of the sound speed. According to Fermat’s principle ultrasound signals travel along geodesic curves of a Riemannian metric which is associated with the refractive index. The inverse problem thus consists of determining the index of refraction from integrals along geodesics curves associated with the integrand leading to a nonlinear problem. In this article we describe a numerical solver for this problem scheme based on an iterative minimization method for an appropriate Tikhonov functional. The outcome of the method is a stable approximation of the sought index of refraction as well as a corresponding set of geodesic curves. We prove some analytical convergence results for this method and demonstrate its performance by means of several numerical experiments. Another novelty in this article is the explicit representation of the backprojection operator for the ray transform in Riemannian geometry and its numerical realization relying on a corresponding phase function that is determined by the metric. This gives a natural extension of the conventional backprojection from 2D computerized tomography to inhomogeneous geometries.

A history of DNA sequence assembly

Abstract DNA sequence assembly is a rich combinatorial problem that arose with the first DNA sequencing projects in the early 80's. Here we give a short history of the progression of algorithmic ideas used to solve the de novo problem of inferring a genome given a large sampling of substrings covering it. This classic inverse problem is compounded by a variety of experimental features and artifacts that must be considered in any realistic solution. While current methods produce very good results, the perfect assembler has yet to be built.

Method of stress calculation in rock mass around Underground openings, considering unit weight

The practical calculation of rock mass deformation around an underground opening accounts for the unit weight of the rock mass by solving a complimentary problem on “weightless” rock mass. A domain with an opening is bounded by a plane with the preset zero vertical displacements, which enables taking into account difference of pressure along the height of the opening. This solution, with the adequately selected boundary conditions, is added with stress field of an intact rock mass and offers zero boundary conditions at the future opening perimeter, however, the issue on the validity of setting boundary conditions at the lower boundary of the calculation domain remains yet to be handled. This article presents a phenomenological model of rock mass deformation to answer the set question. It is taken into account that action of roof rock weight coincides with the orientation of tensile stresses at the opening perimeter and differs from it in the floor. The author thinks it is required to add the class of inverse problems of rock mechanics with the problems directly accounting for rock weight.

Identification of unknown spatial load distributions in a vibrating Euler–Bernoulli beam from limited measured data

Two types of inverse source problems of identifying asynchronously distributed spatial loads governed by the Euler–Bernoulli beam equation , with hinged–clamped ends ( ), are studied. Here are linearly independent functions, describing an asynchronous temporal loading, and are the spatial load distributions. In the first identification problem the values , of the deflection , are assumed to be known, as measured output data, in a neighbourhood of the finite set of points , corresponding to the internal points of a continuous beam, for all . In the second identification problem the values , of the slope , are assumed to be known, as measured output data in a neighbourhood of the same set of points P for all . These inverse source problems will be defined subsequently as the problems ISP1 and ISP2. The general purpose of this study is to develop mathematical concepts and tools that are capable of providing effective numerical algorithms for the numerical solution of the considered class of inverse problems. Note that both measured output data and contain random noise. In the first part of the study we prove that each measured output data and can uniquely determine the unknown functions . In the second part of the study we will introduce the input–output operators , and , , corresponding to the problems ISP1 and ISP2, and then reformulate these problems as the operator equations: and , where and . Since both measured output data contain random noise, we use the most prominent regularisation method, Tikhonov regularisation, introducing the regularised cost functionals and . Using a priori estimates for the weak solution of the direct problem and the Tikhonov regularisation method combined with the adjoint problem approach, we prove that the Fréchet gradients and of both cost functionals can explicitly be derived via the corresponding weak solutions of adjoint problems and the known temporal loads . Moreover, we show that these gradients are Lipschitz continuous, which allows the use of gradient type iteration convergent algorithms. Two applications of the proposed theory are presented. It is shown that solvability results for inverse source problems related to the synchronous loading case, with a single interior measured data, are special cases of the obtained results for asynchronously distributed spatial load cases.

On estimation of internal state by an optimal control approach for elastoplastic material

After a general formulation of the evolution of an elastoplastic body using duality based on the constitutive behaviour, some classes of inverse problems (estimation of the internal state, determination of an unknown history, ...) for such materials are investigated. A general formulation based on optimal control is proposed, the control variables are related to the internal state. In each class of inverse problem, the solution is obtained by introducing a adjoin state and a suitable cost function.

Analysis and Approximation of Some Shape-from-Shading Models for Non-Lambertian Surfaces

The reconstruction of a 3D object or a scene is a classical inverse problem in Computer Vision. In the case of a single image this is called the Shape-from-Shading (SfS) problem and it is known to be ill-posed even in a simplified version like the vertical light source case. A huge number of works deals with the orthographic SfS problem based on the Lambertian reflectance model, the most common and simplest model which leads to an eikonal-type equation when the light source is on the vertical axis. In this paper, we want to study non-Lambertian models since they are more realistic and suitable whenever one has to deal with different kind of surfaces, rough or specular. We will present a unified mathematical formulation of some popular orthographic non-Lambertian models, considering vertical and oblique light directions as well as different viewer positions. These models lead to more complex stationary non-linear partial differential equations of Hamilton---Jacobi type which can be regarded as the generalization of the classical eikonal equation corresponding to the Lambertian case. However, all the equations corresponding to the models considered here (Oren---Nayar and Phong) have a similar structure so we can look for weak solutions to this class in the viscosity solution framework. Via this unified approach, we are able to develop a semi-Lagrangian approximation scheme for the Oren---Nayar and the Phong model and to prove a general convergence result. Numerical simulations on synthetic and real images will illustrate the effectiveness of this approach and the main features of the scheme, also comparing the results with previous results in the literature.

Digital image correlation involves an inverse problem: A regularization scheme based on subset size constraint

Digital image correlation (DIC) is essentially implicated in a class of inverse problem. Here, a regularization scheme is developed for the subset-based DIC technique to effectively inhibit potential ill-posedness that likely arises in actual deformation calculations and hence enhance numerical stability, accuracy and precision of correlation measurement. With the aid of a parameterized two-dimensional Butterworth window, a regularized subpixel registration strategy is established, in which the amount of speckle information introduced to correlation calculations may be weighted through equivalent subset size constraint. The optimal regularization parameter associated with each individual sampling point is determined in a self-adaptive way by numerically investigating the curve of 2-norm condition number of coefficient matrix versus the corresponding equivalent subset size, based on which the regularized solution can eventually be obtained. Numerical results deriving from both synthetic speckle images and actual experimental images demonstrate the feasibility and effectiveness of the set of newly-proposed regularized DIC algorithms.

A Multiscale Strategy for Bayesian Inference Using Transport Maps

In many inverse problems, model parameters cannot be precisely determined from observational data. Bayesian inference provides a mechanism for capturing the resulting parameter uncertainty, but typically at a high computational cost. This work introduces a multiscale decomposition that exploits conditional independence across scales, when present in certain classes of inverse problems, to decouple Bayesian inference into two stages: (1) a computationally tractable coarse-scale inference problem; and (2) a mapping of the low-dimensional coarse-scale posterior distribution into the original high-dimensional parameter space. This decomposition relies on a characterization of the non-Gaussian joint distribution of coarse- and fine-scale quantities via optimal transport maps. We demonstrate our approach on a sequence of inverse problems arising in subsurface flow, using the multiscale finite element method to discretize the steady state pressure equation. We compare the multiscale strategy with full-dimensional Markov chain Monte Carlo on a problem of moderate dimension (100 parameters) and then use it to infer a conductivity field described by over 10,000 parameters.

A Novel Method for Force Identification Based on the Discrete Cosine Transform

Force identification is a classical inverse problem, in which the measured data and the mathematical models of mechanical structures are used to determine the applied force. However, the identified force may seriously diverge from the true solution due to the unknown noise contaminating the measured data and the inverse of the ill-posed transfer matrix characterizing the mechanical structure. In this paper, a novel method based on the discrete cosine transform (DCT) in the time domain is proposed for force identification, which overcomes the deficiency of the ill-posedness of the transfer matrix. The unknown force is expanded by a set of cosine basis functions and then the original governing equation is reformulated to find the coefficient of each cosine basis function. Furthermore, a modified generalized cross-validation (GCV) criterion for determining the regularization parameter is developed for the truncated singular value decomposition (TSVD), Chebyshev polynomial, and DCT solutions. Numerical simulation reveals that compared with the L-curve criterion, the modified GCV criterion is quite robust in the presence of noise. Finally, a clamped-free shell structure that is excited by an impact hammer is selected as an example to validate the performance of the proposed method. Experimental results demonstrate that compared with the TSVD-based and Chebyshev-based methods, the DCT-based method combined with the modified GCV criterion can reconstruct the force time history and identify the peak of impact force with high accuracy. Additionally, the identification of force location using the DCT-based method is also discussed.

Inverse Ablation Analysis and the Calibration Integral Equation Method

The calibration integral equation method has been demonstrated for resolving inverse heat conduction problems based on an invariant sample geometry. This paper proposes to extend its applicability to the investigation of ablation through an abstraction based on forming a fictive surface temperature poised at the nonrecessive origin. Resolving this time-dependent temperature history provides a boundary condition that can be used in a restricted space defined between the nonrecessive origin and an in-depth thermocouple. This fictive temperature (or heat flux) provides an equivalence-based formulation defined in the original spatial domain because the calibration integral equation method is based on conservation principles. The extraction of the fictive boundary condition does not require knowledge of the thermophysical properties or probe position(s) because it is based on calibration. The complexity of ablation requires a compromise between a fully calibrative technique and forward solving approaches. The second step of the two-step process now requires the specification of the thermophysical and geometrical parameters. For this preliminary study, the classical single-ablation temperature inverse problem is revisited but without any additional constraints being imposed. The recession, recession rate, and recession heat flux for a single-temperature ablative material can be solved for by direct means. This paper presents the first test study illustrating the concept and provides highly favorable results using simulated test data based on a Teflon sample with known ablation temperature while under significant thermal loading conditions.

Comparison of deterministic and probabilistic approaches to identify the dynamic moving load and damages of a reinforced concrete beam

Two classical civil engineering inverse problems are considered. The first deals with the determination of dynamic moving loads applied to a reinforced concrete beam. The second one corresponds to the monitoring and the damage assessment. The concrete damage due to overloading is modeled by a loss of the concrete Young’ modulus, whereas the steel bar damage due to corrosion effects is modeled by a reduction of the steel bar cross section. To identify the loading and damage parameters, deterministic and probabilistic model updating techniques are applied and compared. In the deterministic approach, a gradient descent technique based on the adjoint framework is used to minimize the data misfit functional with a Tikhonov regularization term. Then, a regularization by a means of Bayes’s rule is considered in a probabilistic approach. The estimation is of the minimum variance type achieved with the help of the transformed ensemble Kalman filter.

Zeta-invariants of the Steklov spectrum of a planar domain

The classical inverse problem of the determination of a smooth simply-connected planar domain by its Steklov spectrum [1] is equivalent to the problem of the reconstruction, up to conformal equivalence, a positive function $$a \in C^\infty (\mathbb{S})$$ on the unit circle $$(\mathbb{S}) = \left\{ {e^{i\theta } } \right\}$$ from the spectrum of the operator aΛ e , where Λ e = (−d 2/dθ 2)1/2. We introduce 2k-forms Z k (a) (k = 1, 2,... ) of the Fourier coefficients of a, called the zeta-invariants. These invariants are determined by the eigenvalues of aΛ e . We study some properties of the forms Z k (a); in particular, their invariance under the conformal group. A few open questions about zeta-invariants is posed at the end of the article.

Sequential Monte Carlo methods for Bayesian elliptic inverse problems

In this article, we consider a Bayesian inverse problem associated to elliptic partial differential equations in two and three dimensions. This class of inverse problems is important in applications such as hydrology, but the complexity of the link function between unknown field and measurements can make it difficult to draw inference from the associated posterior. We prove that for this inverse problem a basic sequential Monte Carlo (SMC) method has a Monte Carlo rate of convergence with constants which are independent of the dimension of the discretization of the problem; indeed convergence of the SMC method is established in a function space setting. We also develop an enhancement of the SMC methods for inverse problems which were introduced in Kantas et al. (SIAM/ASA J Uncertain Quantif 2:464---489, 2014); the enhancement is designed to deal with the additional complexity of this elliptic inverse problem. The efficacy of the methodology and its desirable theoretical properties, are demonstrated for numerical examples in both two and three dimensions.

The augmented Lagrangian method based on the APG strategy for an inverse damped gyroscopic eigenvalue problem

In this paper, we propose an augmented Lagrangian method based on the accelerated proximal gradient (APG) strategy for an inverse damped gyroscopic eigenvalue problem (IDGEP), which is a special case of the classical inverse quadratic eigenvalue problem. Under mild conditions, we show that the whole sequence of iterations generated by the proposed algorithm converges to the unique solution of the IDGEP. In view of the iteration-complexity, the proposed algorithm requires at most $$O(\log (\varepsilon ^{-1}))$$O(log(?-1)) outer iterations and at most $$O(\varepsilon ^{-1})$$O(?-1) APG calls to obtain an $$\varepsilon $$?-feasible and $$\varepsilon $$?-optimal solution of the IDGEP. Numerical results indicate that the proposed algorithm can solve the test problems efficiently.

Inverse problems in the mechanical characterization of elastic arteries

Abstract

Modeling Nonstationary Nonlinear-Viscous Liquid Flows Through a Pipeline

The process of nonstationary incompressible nonlinear-viscous liquid flow through a pipeline is considered. It is described by a parabolic-type one-dimensional nonlinear equation. The problem of determining the dependence of the pressure difference on time from the assigned volumetric flow rate in the given pipeline is formulated. Such a problem pertains to the class of inverse problems connected with the restoration of the dependence of the right-hand sides of parabolic equations on time. A computational algorithm for solving the problem has been suggested.

Trans-dimensional Monte Carlo sampling applied to the magnetotelluric inverse problem

The data required to build geological models of the subsurface are often unavailable from direct measurements or well logs. In order to image the subsurface geological structures several geophysical methods have been developed. The magnetotelluric (MT) method uses natural, time-varying electromagnetic (EM) fields as its source to measure the EM impedance of the subsurface. The interpretation of these data is routinely undertaken by solving inverse problems to produce 1D, 2D or 3D electrical conductivity models of the subsurface. In classical MT inverse problems the investigated models are parametrized using a fixed number of unknowns (i.e. fixed number of layers in a 1D model, or a fixed number of cells in a 2D model), and the non-uniqueness of the solution is handled by a regularization term added to the objective function. This study presents a different approach to the 1D MT inverse problem, by using a trans-dimensional Monte Carlo sampling algorithm, where trans-dimensionality implies that the number of unknown parameters is a parameter itself. This construction has been shown to have a built-in Occam razor, so that the regularization term is not required to produce a simple model. The influences of subjective choices in the interpretation process can therefore be sensibly reduced. The inverse problem is solved within a Bayesian framework, where posterior probability distribution of the investigated parameters are sought, rather than a single best-fit model, and uncertainties on the model parameters, and their correlation, can be easily measured.

A Posteriori Error Estimates for the Solution of Variational Inverse Problems

Dynamically data-driven application systems integrate computational simulations and physical measurements in a symbiotic feedback control system. Inverse problems in this framework use data from measurements along with a numerical model to estimate the parameters or state of a physical system of interest. Uncertainties in both the measurements and the computational model lead to inaccurate estimates. This work develops a methodology to estimate the impact of different errors on the variational solutions of inverse problems. The focus is on time evolving systems described by differential equations, and on a particular class of inverse problems, namely, data assimilation. The computational algorithm uses first order and second order adjoint models. In a deterministic setting the methodology provides a posteriori error estimates for the inverse solution. In a probabilistic setting it provides an a posteriori quantification of uncertainty in the inverse solution, given the uncertainties in the model and data....

Recovering the Density of a String from Only Lowest Frequency Data

A classical inverse problem is “can you hear the density of a string clamped at both ends''? The answer as narrowly stated is negative; one has to give an additional data sequence such as the energy in each mode (norming constants) or information on the string slope at the endpoints or, alternatively, change the boundary conditions and obtain the new frequencies. What if neither of these options is possible? One solution recently proposed was to add a known mass-density and remeasure the Dirichlet spectrum [W. Rundell and P. E. Sacks, SIAM J. Appl. Math., 73 (2013), pp. 1020--1037]. In this work we take a related approach: we are able to add any number of known mass-densities but we can only measure the fundamental frequency, that is, the lowest eigenvalue, in each case. We will show that our problem can be closely approximated by a related problem for which there is uniqueness and we discuss a suite of reconstruction methods.

Determining Upconversion Parameters and Invertibility of Nonlinear Dynamical Systems

ABSTRACTDetermining upconversion parameters is of high interest in laser material development. For many materials these parameters cannot be directly measured by experimental methods. These upconversion coefficients appear as unknown parameters in the laser rate equations, which are a system of coupled nonlinear differential equations that are used to model the dynamics of population densities in different energy levels. In this paper we propose the well-established system theoretic tools pertaining to the system inversion to be applied in this case. The unknown parameters can be considered as the inputs and the fluorescence signals can be considered as the outputs of the dynamical system. Therefore the determination of the unknown upconversion rates in the system equations from the output data is a classical system inversion problem. In this paper we demonstrate how to compute the unknown coefficients in the rate equations from the experimental emission data utilizing this method.