General Inverse Problem Research Articles

The representer method for state and parameter estimation in single-phase Darcy flow

The objective of this paper is to study the representer method for the inverse problem of combined parameter and state estimation in single-phase Darcy flow in porous media. A variational formulation is posed for the generalized inverse problem in the sense of weighted least-squares. An iterative representer-based algorithm is derived to approximate the Euler–Lagrange equations. It is proved that the linearized problem can be solved exactly by an extension of the representer method which inherits the same well-known properties of this technique. Implementation of the numerical algorithm for synthetic numerical experiments shows improved estimates of the poorly known prior guess.

Development of objective‐oriented groundwater models: 1. Robust parameter identification

When the structure of an aquifer is complex and unknown, determining the complexity of model structure and identifying its spatially distributed parameters become very difficult. This two‐paper series describes an approach for objective‐oriented model construction that can identify a model with simplified structure and assure its reliability for predetermined model applications. The theoretical basis of the methodology includes the definition of structure error, the conditions of structure reducibility, the calculation of the worst‐case parameter (WCP), and the solution of a generalized inverse problem. In the first paper of the series we extend the basic theory established in our previous works from zonation parameterization to any kind of parameterization. The extension makes the methodology more general and complete. We present an effective direct search algorithm that can form a nonnested structure series with an optimized pattern for solving the generalized inverse problem. We use numerical examples to show how the WCP depends on the structure complexity, prior information, boundary conditions, and the objectives of model application. Generally speaking, with more prior information, simpler boundary conditions, fewer objectives, and lower accuracy requirement, the WCP will have a simpler shape. As a result, a reliable model can be identified with less observation data. In part 2 of this series we will give the conditions of structure reducibility and the method for robust experimental design. A robust experimental design is independent of the true parameter and can provide sufficient information for identifying an objective‐oriented model.

Acoustoelastic analysis of reflected waves in nearly incompressible, hyper-elastic materials: Forward and inverse problems

Many materials (e.g., rubber or biologic tissues) are "nearly" incompressible and often assumed to be incompressible in their constitutive equations. This assumption hinders realistic analyses of wave motion including acoustoelasticity. In this study, this constraint is relaxed and the reflected waves from nearly incompressible, hyper-elastic materials are examined. Specifically, reflection coefficients are considered from the interface of water and uni-axially prestretched rubber. Both forward and inverse problems are experimentally and analytically studied with the incident wave perpendicular to the interface. In the forward problem, the wave reflection coefficient at the interface is evaluated with strain energy functions for nearly incompressible materials in order to compute applied strain. For the general inverse problem, mathematical relations are derived that identify both uni-axial strains and normalized material constants from reflected wave data. The validity of this method of analysis is demonstrated via an experiment with stretched rubber. Results demonstrate that applied strains and normalized material coefficients can be simultaneously determined from the reflected wave data alone if they are collected at several different (but unknown) levels of strain. This study therefore indicates that acoustoelasticity, with an appropriate constitutive formulation, can determine strain and material properties in hyper-elastic, nearly incompressible materials.

Inversion of gravity data using a binary formulation

SUMMARY We present a binary inversion algorithm for inverting gravity data in salt imaging. The density contrast is restricted to being one of two possibilities: either zero or one, where one represents the value expected at a given depth. The algorithm is designed to easily incorporate known density contrast information, and to overcome difficulties in salt imaging associated with nil zones. The problem of salt imaging may be formulated as a general inverse problem in which a piecewise constant density contrast is constructed as an indirect means of identifying the salt boundary. Difficulty arises when the salt body crosses the nil zone in depth. As a result, part of the salt structure is invisible to the surface data and many inversion algorithms have difficulties in recovering the salt structure correctly. The binary condition places a strong restriction on the admissible models so that the non-uniqueness caused by nil zones might be resolved. In this paper, we will present the binary formulation for inversion of gravity data, develop the solution strategy, illustrate it with numerical examples, and discuss limitations of the technique.

The collage coding method and its application to an inverse problem for the Lorenz system

A Lorenz system inverse problem featuring noisy partial data sets is considered via the method of collage coding. Collage coding is a rigorously-established approach to such problems, with its roots in fractal imaging, and its theoretical basis being Banach’s fixed point theorem and some recent related results. For comparison purposes, results for this inverse problem obtained by using other methods are mentioned. A framework for treating general inverse problems involving partial data sets in a similar way is also outlined.

The role of multilevel data in potential field interpretation

We start from the general inverse problem for potential fields and discuss the validity and suitability of using not only horizontal variations of them, but also their vertical ones. Hence multilevel data sets are considered, being they obtained either from measured data at different levels or by measuring the field just at the lower level and then generating by upward continuation the data at the remaining upper levels. In the latter case the upward continued data must be considered as the true ones plus some errors due to experimental errors and to the fact that they are generated from a discrete data set known on a finite survey area. Several forms of a priori information, such as that assuming the source can be represented by a fault or a sheet sandwich model, may allow the information contained in multilevel data to be effectively used for the interpretation of gravity and magnetic data. Recently established techniques of analysis, such as the continuous wavelet transform, are applied successfully to potential fields using implicitly the information contained in multilevel data. Assuming a block model for the source domain, the upward continuation formula may help to improve the solution of either 1D, 2D or 3D inverse problems enlarging the system with the equations related to the vertical variations of the fields, thereby reducing the algebraic ambiguity. Meaningful and computationally suitable quantities, such as weighted averages of potential fields, are also closely related to the use of multilevel data, providing useful insights for the determination of the depth distribution of the sources.

Structure reduction and robust experimental design for distributed parameter identification

Designing an experiment for identifying a distributed parameter is a very challenging problem when the spatial variability of the parameter is unknown. First, we need to determine an appropriate structure complexity for the unknown parameter. A more complicated structure needs more data for identification. Second, we need to evaluate the sufficiency of an experiment design before it is actually conducted. In this paper, the complexity of parameter structure is determined by the accuracy requirement of model application, and the sufficiency of data is evaluated by solving a generalized inverse problem. The worst-case parameter (WCP) defined in this paper is the one that causes the maximum deviation in model application when its structure is simplified. The WCP of a structure can be found by solving a discrete optimization problem with the genetic algorithm. Quantitative relationships between structure complexity, parameter identifiability, data sufficiency and model reliability are derived. For example, it has been proven that if an experimental design is sufficient for identifying the WCP then it must be sufficient for identifying all other parameters with the same structure or simplified structures in the admissible region, and thus, it is a robust design. Based on the theoretical results and algorithms presented in this paper, we may develop a cost-effective methodology for constructing reliable distributed parameter models.

Deconvolution of Sparse Positive Spikes

Deconvolution is usually regarded as one of the ill-posed problems in applied mathematics if no constraints on the unknowns are assumed. This article discusses the idea of well-defined statistical models being a counterpart of the notion of well-posedness. We show that constraints on the unknowns such as positivity and sparsity can go a long way towards overcoming the ill-posedness in deconvolution. We show how these issues are dealt with in a parametric deconvolution model introduced recently. From the same perspective we take a fresh look at two familiar deconvolvers: the widely used Jansson method and another one that minimizes the Kullback-Leibler divergence between observations and fitted values. In the latter case, we point out that in the context of deconvolution and the general linear inverse problems with positivity contraints, a counterpart of the EM algorithm exists for the problem of minimizing the Kullback-Leibler divergence. We graphically compare the performance of these deconvolvers using data simulated from a spike-convolution model and DNA sequencing data.

Pre-stack migration applied to GPR for landmine detection

Detection of buried landmines by ground penetrating radar (GPR) normally suffers from very strong clutter that will decrease the image quality in GPR data. Problems are also encountered when imaging steeply dipping landmines by GPR. To solve these problems, we have developed a stepped-frequency continuous-wave array antenna ground penetrating radar system, called SAR-GPR, that can acquire common middle point multi-offset data. As an approximate solution to the general wavefield inversion problem, migration algorithms were used to refocus the scattered landmine information to improve signal–clutter ratio and re-construct the landmine image. Also, pre-stack migration was found to efficiently deal with the steeply dipping landmine problem. Before migration processing, subtracting antenna coupling was used. The SAR-GPR system was tested under two conditions. The first condition is designed to simulate inhomogeneous soil under rough ground conditions and the second condition is to simulate steeply dipping buried landmine. Very strong clutter in the GPR data was exhibited in the first condition. After pre-stack migration, strong clutter was efficiently suppressed and a high quality landmine image was re-constructed in both experiments.

A numerical solution of the linear multidimensional unsteady inverse heat conduction problem with the boundary element method and the singular value decomposition

In this paper, we present a new method for solving the general linear multidimensional unsteady inverse heat conduction problem. The direct numerical method is based on the Boundary Element Method formulation. Taking into account future time steps, the ill-conditioned linear system is solved using a procedure based on the Singular Value Decomposition technique which handles both spatial and temporal instabilities. The regularization method is essentially a spectral truncation method with a single hyperparameter. The optimal value of this hyperparameter can be chosen a priori from the knowledge of the data uncertainty. In the second part of this paper, an experiment is described which illustrates an application of the method on a two-dimensional problem. The physical problem consists in identifying the heat flux on a plate exposed to a moving front of hot fluid from temperature measurements collected on the opposite side of the plate. Numerical results obtained are discussed in comparison to direct heat flux measurements. The purpose of this experiment was to cross-check two types of heat flux sensors in real unsteady two-dimensional situations. It is also shown that the method can be applied to real 3D problems.

A general analytical model of electrical permanent magnet machine dedicated to optimal design

What is new in this work is the generic capabilities of the proposed analytical model of permanent magnet machines associated with a novel deterministic global optimization method. That allows to solve some more general inverse problem of designing. The analytical approach is powerful to take into account various kinds of constraints (electromagnetical, thermal, etc.). The inverse problem associated with the optimal design of actuators could then be formulated as a mixed‐constrained optimization problem. In order to solve these problems, interval Branch and Bound algorithms which have already proved their efficiency, have made it possible to determine some optimized rotating machines.

Modal inverse methods: An overview

Geoacoustic inversion using modal inverse methods is based on a very simple experimental geometry. Data are acquired from synthetic aperture measurements of a point source acoustic field using a single hydrophone translating through the water column at constant depth. The modal composition of the propagating field is extracted by exploiting the Hankel transform relationship between the acoustic field, measured as a function of range, and the horizontal wavenumber spectrum for horizontally stratified media. Modal content, estimated as discrete values of the horizontal wavenumber, are used as input data for a general linear inverse problem, where small perturbations to a background sound speed profile are related to changes in individual horizontal wavenumbers. An overview of the method is presented with an emphasis placed on the application to range-dependent shallow water environments. Inversion results are presented from a recent workshop on geoacoustic inversion for range-dependent environments, as well as from recent at-sea measurements. [Work supported by ONR.]

Global‐local optimization for parameter structure identification in three‐dimensional groundwater modeling

This paper develops two global‐local optimization methods for three‐dimensional parameter structure identification in groundwater modeling. Parameter structure identification is formulated in terms of solving a generalized inverse problem, which allows for a determination of an appropriate level of parameter structure complexity, and the identification of its pattern and the associated parameter values. The structure complexity is determined by calculating the parameter structure error measured in the prediction space or management space, while the structure pattern and the associated parameter values are identified simultaneously by minimizing the fitting residual measured in the observation space. The former requires solving a continuous maximum‐minimum (max‐min) optimization problem, and the latter requires solving a combinatorial optimization problem. In this paper, Voronoi zonation is used to parameterize the unknown distributed parameter. For each given level of structure complexity a sequential global‐local optimization method, which consists of a genetic algorithm, a quasi‐Newton method, and local search, is developed to solve the combinatorial problem. In addition, the continuous max‐min problem is transformed to a hierarchical optimization problem and solved by a hybrid global‐local method. Sensitivities of state variables to the unknown parameters are calculated by the sensitivity equations. The validity and applicability of the proposed methodology are demonstrated by numerical experiments. We have shown that the choice of an objective function in model application impacts the determination of the parameter structure complexity.

A General Inverse Problem for a Memory Kernel in One-Dimensional Viscoelasticity

A general inverse problem for the identification of a memory kernel in viscoelasticity in one space dimension is dealt with, where the kernel is represented by a finite sum of products of known spatially dependent functions and unknown time-dependent functions. Using the Laplace transform method an existence and uniqueness theorem for the memory kernel is proved.

Les perceptrons multicouches : de la régression non-linéaire aux problèmes inverses

The problem of nonlinear least square regression using multi-layered perceptrons is addressed in this paper. A general formulation using maximum likelihood is given and a particular attention is paid to algorithm accounting for uncertainties. The generalization to density estimation is given. In the second part of the paper, a general framework allows to modelize general inverse problems.

Ill-Posedness and Accuracy in Connection with the Recovery of a Single Parameter from a Single Measurement

A particular one-parameter problem, that of the reflection and refraction of a plane wave at a planar boundary between two media, which can be solved in closed form in both the direct and inverse contexts, is chosen to illustrate the ill-posed nature of a class of more general inverse problems and to show how the quality of the reconstruction of the parameter varies with the accuracy of the data on the one hand, and the accuracy of the interaction model employed in the inversion scheme on the other hand.

A nonlinear inverse problem inspired by three‐dimensional diffuse tomography: Explicit formulas

AbstractWe use time of flight information to obtain explicit inversion formulas for a simple model of optical tomography. This can be considered as an instance of a more general nonlinear inversion problem for multi‐terminal networks. In this case the network is nonplanar with lots of cycles but the underlying physical model yields a highly structured situation. © 2003 Wiley Periodicals, Inc. Int J Imaging Syst Technol 12, 198–203, 2002; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/ima.10031

Determination of the optical properties of two-layer turbid media by use of a frequency-domain hybrid Monte Carlo diffusion model

The general two-layer inverse problem in biomedical photon migration is to estimate the absorption and scattering coefficients of each layer as well as the top-layer thickness. We attempted to solve this problem, using experimental and simulated spatially resolved frequency-domain (FD) reflectance for optical properties typical of skin overlying muscle or skin overlying fat in the near infrared. Two forward models of light propagation were used: a two-layer diffusion solution [Appl. Opt. 37, 779 (1998)] and a hybrid Monte Carlo (MC) diffusion model [Appl. Opt. 37, 7401 (1998)]. MC-simulated FD reflectance data were fitted as relative measurements to the hybrid and the pure diffusion models. It was found that the hybrid model could determine all the optical properties of the two-layer media studied to ~5%. Also, the same accuracy could be achieved by means of fitting MC-simulated cw reflectance data as absolute measurements, but fitting them as relative ones is an ill-posed problem. In contrast, two-layer diffusion could not retrieve the top-layer optical properties as accurately for FD data and was ill-posed for both relative and absolute cw data. The hybrid and the pure diffusion models were also fitted to experimental FD reflectance measurements from two-layer tissue-simulating phantoms representative of skin-on-fat and skin-on-muscle baseline optical properties. Both the hybrid and the diffusion models could determine the optical properties of the lower layer. The hybrid model demonstrated its potential to retrieve quantitatively the transport scattering coefficient of skin (the upper layer), which was not possible with the pure diffusion model. Systematic discrepancies between model and experiment may compromise the accuracy of the deduced top-layer optical properties. Identifying and eliminating such discrepancies is critical to practical application of the method.

The range of the iterated matrix adjoint operator

The following inverse problem is considered: for a given n × n real matrix B, does there exist a real matrix A such that ▪ where the classical adjoint operation is intended? The rank of B and the number of applications of the adjoint operator determine the character of this general inverse problem for the iterated adjoint operator. Thus, for given B, the question of interest is whether or not B lies in the range of the iterated matrix adjoint operator. Maple V R5 is used as an aid to obtain results indicated here.

A three-dimensional inverse method using navier—stokes equations for turbomachinery blading

A new numerical method for solving fully three-dimensional inverse shape design problem of turbomachinery blading has been developed. The general inverse problem refers to the problem in which the pressure distributions on suction and pressure surfaces of blade are given, but the corresponding blade profile is unknown. In this paper, the calculations are based on the 3D Navier-Stokes equations expressed in terms of nonorthogonal curvilinear coordinates and corresponding nonorthogonal velocity components, and the explicit time marching algorithm and Baldwin-Lomax turbulence model are adopted. A special treatment for boundary conditions on blade surfaces is employed to satisfy the given pressure distribution. In computational process, an initial blade profile is supposed at starting, and then the blade surfaces will move regularly with time steps in the time marching process until the convergence is reached. The movement velocities at every point of blade surfaces are obtained from the solution of the Navier-Stokes equations. After each revision of the blade profile, the grid is reconstructed, and the aerodynamic parameters need to be transferred between the old and new grid points by an accurate interpolation method. Thus the viscous inverse problem is solved in a new process. The computational results for two test cases indicate that the method presented in this paper is very effective.