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1996 | Buch

An Introduction to Inverse Problems in Partial Differential Equations for Physicists, Scientists and Engineers Kapitel lesen Erstes Kapitel lesen

Parameter Identification and Inverse Problems in Hydrology, Geology and Ecology

herausgegeben von: Johannes Gottlieb, Paul DuChateau

Verlag: Springer Netherlands

Buchreihe : Water Science and Technology Library

Enthalten in: Professional Book Archive

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Über dieses Buch

The Workshop on Parameter Identification and Inverse Problems in Hydrology, Geology and Ecology, Karlsruhe, April 10-12, 1995, was organized to bring to­ gether an interdisciplinary group drawn from the areas of science, engineering and mathematics for the following purposes: - to promote, encourage and influence more understanding and cooperation in the community of parameter identifiers from various disciplines, - to forge unity in diversity by bringing together a variety of disciplines that attempt to understand the reconstruction of inner model parameters, un­ known nonlinear constitutive relations, heterogeneous structures inside of geological objects, sources or sinks from observational data, - to discuss modern regularization tools for handling improperly posed pro­ blems and strategies of incorporating a priori knowledge from the applied problem into the model and its treatment. These proceedings contain some of the results of the workshop, representing a bal­ anced selection of contributions from the various groups of participants. The reviewed invited and contributed articles are grouped according to the broad headings of hydrology, non-linear diffusion and soil physics, geophysical methods, mathematical analysis of inverse and ill-posed problems and parallel algorithms for inverse problems. Some of the issues adressed by the articles in these proceedings include the rela­ tion between least squares and direct formulations of inverse problems for partial differential equations, nonlinear regularization, identification of nonlinear consti­ tutive relations, fast parallel algorithms for large scale inverse problems, reduction of model structures, geostatistical inversion techniques.

Inhaltsverzeichnis

Frontmatter

Tutorial on Inverse Problems

Frontmatter
An Introduction to Inverse Problems in Partial Differential Equations for Physicists, Scientists and Engineers
A Tutorial
Abstract
In an inverse problem where properties of a physical system are to be found indirectly from measured outputs, one seeks to define and then somehow invert an input/output mapping. When the unknown physical properties are characterized by a small number of constant parameters, the problem is referred to as a parameter identification problem and in such cases the input/output mapping is often simple enough that important properties of the map become transparent. In this first part of the tutorial, examples of such problems are presented to show that the input/output mapping is often a monotone map and to show how monotonicity can be exploited to invert the input/output mapping. Other examples illustrate how a parameterization that fails to exploit monotonicity can lead t incorrect or inferior results. These observations provide a basis for understanding more complex inverse prolems.
Paul Duchateau

Hydrology

Frontmatter
Interpretation of Field Tests in Low Permeability Fractured Media. Recent Experiences
Abstract
We have been treating fractured media as the result of embedding conductive 2D fractures in a 3D continuum medium. Automatic calibration has normally succeeded in producing models capable of predicting independent data sets. However, in recent times, we have faced a set of tests performed in very low conductive, highly heterogeneous granite where using exclusively the standard procedure has not produced good results. Difficulties include the following: (1) complex fracture geometry; (2) spurious effects caused by ill-shaped tetrahedra; (3) weak responses to pumping; (4) superposition of natural head data (controlled by boundary conditions) with drawdowns caused by pumping; (5) coupling these two flow conditions to produce the flow field needed for the tracer test is extremely sensitive to small errors. Finally, these difficulties are aggravated by the large computer times required by a fully 3D medium. Suggestions for overcoming these problems are outlined.
J. Carrera, L. Vives, P. Tume, M. Saaltink, G. Galarza, J. Guimerà, A. Medina
New Front Limitation Algorithm
Fast Finite - Difference Method for the Advection - Dispersion Problem and Parameter Identification
Abstract
Due to complexity of the numerical model of the transport problem and the large number of unknown parameters, existing techniques for parameter estimation cannot be applied successfully without modifications to reduce the computational time by maintaining sufficient accuracy. This paper presents a new effective approach (so-called FRONT LIMITATION algorithm) to solve the advection-dispersion equation without restriction for grid PECLET number. The method prevents numerical dispersion, brings slight grid-orientation effect and it only has to take into consideration a weak COURANT number and source/sink limits.
The technique utilizes the control volume method with the full implicit diffusion-dispersion and source/sink terms and with the special explicit handling of the advection term.
The performance of the algorithm using models for one-, two- and three-dimensional examples is examined.
The Front Limitation algorithm is considered as the basis of a parameter identification method with sensitivity analysis.
Frieder Haefner, Siegrun Boy, Steffen Wagner, Aron Behr, Vladimir Piskarev, Iskander Zakirov, Boris Palatnik
Identification and Reduction of Model Structure for Modeling Distributed Parameter Systems
Abstract
In this paper, the problem of how to determine the model structure for modeling a distributed parameter system is considered. The model structure error of using model M A to replace model M B is defined as a weighted sum of their distances in both observation and prediction spaces. The value of so defined structure error can be obtained by solving a max-min problem. In order to select an appropriate complexity of model structure, the concept of extended identifiabilities is further extended to the case that model structure error is involved. A stepwise regression procedure is then presented for simultaneously determining the model structure and model parameters. Numerical examples are given to explain the presented concepts and methodology.
Ne-Zheng Sun
Results from a Comparison of Geostatistical Inverse Techniques for Groundwater Flow
Transmissivity Fields and Groundwater Travel Time CDFs
Abstract
The Waste isolation Pilot Plant (WIPP) is a U.S. Department of Energy (DOE) facility located in southeastern New Mexico which is currently being evaluated to assess its suitability for isolating transuranic wastes generated by defense programs in the U.S. The proposed repository is located within the bedded salt of the Salado Formation at a depth of about 660 meters. The Culebra Dolomite, located within the Rustler Formation at a depth of about 250 meters, has been characterized as the most transmissive, laterally continuous hydrogeologic unit above the repository, and is considered a potentially important transport pathway for offsite radionuclide migration within the subsurface. This could occur if, for example, in the future, a well drilled for exploration purposes created an artificial connection between the waste and the Culebra aquifer. Such a scenario is part of a Probabilistic Performance Assessment (PA) that the U.S. Environmental Protection Agency (EPA) requires the DOE to perform to demonstate the compliance of the repository system with regulations governing disposal of radioactive wastes. Because the EPA regulation is probabilistic, Pas must accurately reflect the variability and uncertainty within all factors that contribute to the simulation of repository performance for isolating the wastes.
D. A. Zimmerman, C. L. Axness, G. De Marsily, M. G. Marietta, C. A. Gotway

Nonlinear Diffusion and Soil Physics

Frontmatter
Identification of the Hydraulic Diffusivity of a Soil by Inverse Method with Dual-Energy Gamma Ray Attenuation Measurements
Abstract
Hydraulic diffusivity is fundamental to the study of a natural forestry site. Complementary field and laboratory experiments were carried out to identify hydraulic diffusivity. A transient physical process of water transport is required for this, for which a dynamic imbibition experiment was chosen. Time variations of moisture content are measured at various locations by dual-energy gamma ray attenuation. The diffusion transport occurring in this experiment has been described by the classical diffusion equation in terms of the Boltzmann variable with various models of non-linear diffusivity. An inverse method is then applied to determine the parameters of the hydraulic diffusivity model.
F. Barataud, D. Stemmelen, C. Moyne
Identification of Parameters for Heat Conductivity Equations
Abstract
An algorithm for identification of unknown parameters of nonlinear heat conductivity equations is proposed. Solutions of equations observed with an error are input data of the algorithm. Finite dimensional approximations of input signals and their derivatives are used. The algorithm utilizes the idea of the direct minimization of the residual of equations written in an appropriate variational form. The convergence of the algorithm output to the set of all parameters compatible with the exact solution is proved. The paper is illustrated by computer simulations.
N. D. Botkin
Sensitivity Analysis in Parameter Identification, Test Planning and Test Evaluation Procedures for Two-Phase Flow in Porous Media
Abstract
The flow model describing unsaturated one dimensional vertical water flow in porous media is given by
$$ \left( {\frac{\partial }{{\partial z}}k(\theta ) \times \left( {\frac{{\partial {h_p}}}{{\partial z}} + 1} \right)} \right) = \frac{{\partial \theta }}{{\partial t}} - {w_0} $$
(1a)
and
$$ \frac{{\partial \theta }}{{\partial t}} = C({h_0}) \cdot \frac{{\partial {h_p}}}{{\partial t}} $$
(1b)
where the independent variables are time t and spatial coordinate z, taken positive up-wards. The dependent variables of equation (1) are the water pressure head hp = pw/ρw·g (hc=-hp and the water content θ. w0 is the sink/source term. The capillary capacity function C(hc) is the first derivative of the hysteretic soil water retention curve. The unsaturated hydraulic conductivity k(θ) depends on the water content in the soil.
O. Kemmesies, L. Luckner
An Inverse Problem for Porous Medium Equation
Abstract
An inverse problem for a nonlinear degenerate diffusion equation is considered. We look for unknown coefficient a(u) of the equation u t = (a(u(u x ) x with zero initial values and nonmonotone impulse-like Dirichlet boundary condition. Measurements in an interior point x 0 are taken as overposed data. An optimization approach is used. The case of the more general equation c(u)u t = (a(u)u x ) x + b(u)u x + s(u) is also considered.
R. Nabokov
Evaluation of Different Boundary Conditions for Independent Determination of Hydraulic Parameters Using Outflow Methods
Abstract
Measuring pressure induced water outflow in combination with inverse modeling is a suitable method to determine the hydraulic properties of soil columns. To study the dependence of the inverse method on the imposed experimental boundary conditions, cumulative water outflow from a soil column is simulated numerically. The column is assumed to be homogeneous with a height of 15 cm. Boundary conditions chosen for the bottom of the soil column are: one-step (OS), multi-step (MS) and linear (LPD) pressure decrease in time. Hydraulic properties are described by the closed forms proposed by van Genuchten (1980). Sensitivity coefficients for each of the parameters α, n, θ s , K s , and τ together with response surfaces for different parameter combinations are estimated for each of the boundary conditions. Then, the influence of random errors in the outflow data on the performance of the different boundary conditions is examined. It is shown that the MS-method is the only one resulting in unique estimates of all parameters investigated. The LPD- and MS-method are comparable if θ s and K s are not optimized simultaneously. The OS-method is very sensitive to the magnitude of the pressure step. A large step may lead to non-unique solutions of the inverse problem, caused by correlations between most of the parameters. Reducing the step leaves only the parameters n and θ s as not identifiable.
T. Zurmühl

Geophysical Methods

Frontmatter
Gravity Data Inversion Using the Subspace Method
Abstract
The solution of a nonlinear inverse gravity problem is expanded in terms of orthogonal basis functions and expansion coefficients. The basis functions chosen are the normalized eigenvectors of the second derivatives of the objective function (the Hessian matrix) calculated for an initial model. Of the expansion coefficients obtained in this way a limited number (a subspace) will be used as new model parameters. A new objective function is defined in terms of these new model parameters and is minimized in the subspace of the original ones. The matrix inversion in the subspace of the model parameters will be better conditioned due to less dimensionality and the limited number of eigenvectors used in the inversion. Since the most significant eigenvectors corresponding to the largest eigenvalues are taken in the inversion this will eliminate those elements of the model which are likely to have less influence in fitting the data or lead to local minima. Choosing this strategy makes inversion fast and stable against the noise. The efficiency of the method is tested with synthetic and real gravity data. The tests will prove fast convergence and stability of the inversion against the noise.
M. Mirzaei, J. W. Bredewout, R. K. Snieder
A Method to Determine Parameters of a Linear Functional Equations Set and its Application to the Lightning Location Problem
Abstract
Let H be Hilbert space with a scalar product (•,•), and corresponding norm ‖•‖. Let {A ηk } η∈Rm, k = 1,2,… n be parametric families of linear operators acting in H. In this paper a problem of identifiing the parameter η ∈ D ⊂ R m, D is a compactum, in the set of the linear functional equations
$$ {A_{{nk}}}x = uk,k = 1,2,...,n $$
(1)
is considered with respect to preassigned right parts (u 1, u 2,…u n ) ∈ H n and the unknown x ∈ H. The operators A ηk , k = 1,2,…n are supposed to be irreversible. Their kernels may depend on the unknown parameter η, and equations (1) may be incompatible.
A. V. Panyukov, V. A. Strauss

Mathematical Analysis of Inverse and Ill-Posed Problems

Frontmatter
Stability Estimates for Inverse Problems
Abstract
In this paper the stability of inverse problems is discussed. It is taken into account that in inverse problems the structure of the solution space is often completely different from the structure of the data space, so that the definition of stability is not trivial. We solve this problem by assuming that under experimental conditions both the model and the data can be characterized by a finite number of parameters. In the formal definition that we present, we compare distances in data space and distances in model space under variations of these parameters. Moreover, a normalization is introduced to ensure that these distances do not depend on physical units. We note that it is impossible to obtain an objective estimate of stability due to the freedom one has in the choice of the norm in the solution space and the data space. This definition of stability is used to examine the stability of the Marchenko equation. It is shown explicitly that instabilities arise from the non-linearity of the inverse problem considered.
H. J. S. Dorren, R. K. Snieder
Identifiability of Distributed Physical Parameters
Abstract
Considering differential equations of second order which contain a term in the form
$$ {(a{u_x})_x} + bu $$
where a = a(x,u) and b = b(x,u), conditions for the identifiability of the coefficients a and b are given.
S. Handrock-Meyer
Inverse Scattering Problem for the Wave Equations and its Applications
Abstract
One of the most effective methods for studying inverse scattering problems (ISP) is the method of the Ge’fand-Levitan-Marchenko type integral equations.
L. P. Nizhnik
A Descriptive Regularization Approach for a Class of Ill-Posed Nonlinear Integral Equations
Abstract
There are many problems in material sciences, geophysics, optics and meteorology, where a real–valued vertical profile x(t) with 0 ≤ t ≤ 1 of a physical quantity in a layer of thickness 1 is to be determined. If the profile cannot be measured directly, we have an inverse problem. That means, the unknown profile is to be identified from indirect measurements y(s). The really available observations concern in general a variety of linear or nonlinear functionals y(s) = f s (x) depending on a real parameter s, for which we in the sequel assume that 0 ≤ s ≤ 1. An especially difficult subproblem occurs if the functionals are nonlinear of the form \( {f_s}(x) = \int\limits_0^1 {k(s,x(t))dt} \) with a given kernel function k(s, x) (cf. [3]). This kernel function expresses the transmissibility properties of the layer under consideration with respect to the rays or waves passing through the layer and yielding the measurements y(s) depending on the angle of incidence or wave length s. In our specific problem, which can be written as a nonlinear Urysohn integral equation
$$ \int_0^1 {k(s,x(t))dt = y(s)\quad, \quad \quad 0 \leqslant s \leqslant 1} $$
(1)
we have the situation that the transmissibility conditions depend on the layer level t only via x(t). With respect to the data y(s) this leads to an essential loss of information about the unknown function x(t).
Torsten Schröter, Bernd Hofmann
Parameter Estimation in Nonlinear Models by Using Total Least Squares
Abstract
In this paper we introduce a stable and efficient approach to estimate unknown parameters in nonlinear models, where all variables are affected by noise. This technique is also known as the total least squares (TLS) or errors in variables method (EVM). We discuss the possibility of adding nonlinear restrictions to the unknown parameters and error margins for the independent variables. Special attention is paid to the reliability of the parameters. The statistical assumptions with respect to the measurement errors and the consequences for the numerical approach are also highlighted.
Walter J. H. Stortelder
Tikhonov Regularization for Identification Problems in Differential Equations
Abstract
In this paper we investigate the method of Tikhonov regularization for solving nonlinear ill-posed inverse problems
$$ F(x) = y, $$
(1)
where instead of y noisy data y δ ∈ Y with ∥y — y δ ∥ ≤ δ are given, F: D(F) → Y is a nonlinear operator with domain D(F) ⊂ X and X, Y are Hubert spaces with corresponding inner products (•, •) and norms ∥ • ∥, respectively. Nonlinear ill-posed inverse problems arise in a number of applications and can be divided into explicit and implicit ill-posed inverse problems. A large class of explicit ill-posed inverse problems can be described by nonlinear integral equations of the first kind; implicit ill-posed inverse problems arise e.g. in problems connected with the identification of unknown coefficients q (which are in general functions) in distributed systems from certain observations y δ ∈ Y of the noise-free data y. Distributed systems are governed by differential equations, in general, which may be described by an operator equation of the form
$$ T\left( {q,u} \right) = b, $$
(2)
where T maps the couple (q, u) from the product space Q × U into the space of the right hand side of equation (2). This is of course formal and has to be made precise in each particular case.
Ulrich Tautenhahn

Parallel Algorithms for Inverse Problems

Frontmatter
Parameter Estimation in Multispecies Transport Reaction Systems Using Parallel Algorithms
Abstract
The authors present a general approach to estimate parameters in reactive multispecies transport models. The basic idea is to treat the discretized model equations as nonlinear constraints of a large finite-dimensional optimization problem. Multiple shooting in time is a specific feature of the discretization. The resulting constrained least squares problem is solved by a generalized Gauss-Newton method. Parallelism on several levels is exploited in the algorithmn. A practical parameter estimation problem from contaminant hydrology is considered and successfully solved. The reliability of the parameter estimates is confirmed by the results of a sensitivity analysis. A more detailed description of the presented methods and results can be found in [9].
M. W. Zieße, H. G. Bock, J. V. Gallitzendörfer, J. P. Schlöder
On Design and Implementation of Parallel Algorithms for Solving Inverse Problems
Abstract
A new strategy for design and optimization of parallel algorithms is proposed. This strategy is based on an abstract machine model that allows the programmer to focus on the inherent parallelism of the problems at hand rather than on the parallelism of the target machines. It is possible to transform a subclass of these parallel algorithms into programs running efficiently on a great variety of parallel machines, including workstation clusters. From a practical point of view, this subclass is large. It includes e.g. conjugate gradient methods for which we discuss the transformation. The proposed method is also applicable to more complex problems like inverse problems in partial differential equations. We demonstrate this for an identification problem arising from hydrology.
Wolf Zimmermann, Welf Löwe, Johannes Gottlieb
Backmatter
Metadaten
Titel
Parameter Identification and Inverse Problems in Hydrology, Geology and Ecology
herausgegeben von
Johannes Gottlieb
Paul DuChateau
Copyright-Jahr
1996
Verlag
Springer Netherlands
Electronic ISBN
978-94-009-1704-0
Print ISBN
978-94-010-7263-2
DOI
https://doi.org/10.1007/978-94-009-1704-0