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NASA NTRS1990

Using data with models - Ill-posed problems

An attempt is made to show that the combination of data with models leads to a sequence of conventionally ill-posed problems which can be treated in a systematic and practical way by control-theory techniques. The illustrative approach is to exploit the very great mathematical simplifications which result when the methods of finite dimensional vector spaces are used, thus reducing a lot of difficult mathematics to classical least squares. The immediate motivation for this study comes from ocean acoustic tomography.

Wunsch, C.↗

NASA NTRS1985

The Construction of Curves and Surfaces Using Numerical Optimization Techniques

Numerical optimization techniques are playing an increasing role in curve and surface construction. Often difficult problems in curve and surface construction, especially when some aspect of shape control is involved, can be phrased as a constrained optimization problem. Four such classes of problems are explored: parametric curve fitting with non-linear shape constraints; explicit surface fitting with linear shape constraints; surface fitting to scattered data giving rise to ill-posed problems; finally, variable knot problems. In each of these problems there is a nonlinear aspect: either the shape of the curve or surface is important for manufacturing or engineering reasons or the shape affects the convergence of numerical algorithms which use the curve or surface or the placement of knots affects the accuracy of the fits. In all cases the class of functions used is that of parametric spline curves and tensor or direct product spline surfaces. The reason for choosing this class is that splines provide flexible models that are easily evaluated and stored. Furthermore, the B-spline representation of splines leads to convenient expressions for shape control over regions.

Ferguson, D. R.↗

NASA NTRS1977

Inversion methods in temperature and aerosol remote sounding: Their commonality and differences, and some unexplored approaches

The two remote sensing problems of temperature profiling and aerosol characterization (complex refractive index, size distribution) are considered. These problems differ only in the explicit form of the source function which, for aerosols, includes contributions from both single and multiple scattering processes. When the observables are the spectral extinction or the single scattering of the source radiation, the associated problem is completely analogous to the linearized temperature inversion problem. Methods for obtaining the solution of the linear problem are classified following three main categories: (1) derivation of properties that all solutions satisfy, which must then be properties of the actual solution; (2) regularization of the ill-posed problem; and (3) data changes within their domain of uncertainty in order to avoid the basic instability. A number of unexplored methods are indicated.

Fymat, A. L.↗

NASA NTRS1981

Deconvolution of wide-field-of-view satellite radiometer measurements of reflected solar radiation

An approach is presented by which the deconvolution of wide-field-of-view (WFOV) measurements of solar radiation reflected from earth is reduced from the solution of a two-dimensional integral equation to the solution of a set of one-dimensional integral equations. This reduces the storage needed for the required matrices by an order of magnitude and the computations by two orders of magnitude. Also, the theoretical and operational difficulties associated with the solution of ill-posed problems are greatly simplified by working with one-dimensional integral equations rather than with a two-dimensional integral equation.

Smith, G. L.↗

NASA NTRS1989

Some new results in the investigation of the middle atmosphere from space using the daytime horizon scanning method

The results of the restoration of the optical parameters of the El Chichon aerosol cloud at Lambda 2.2 micron are presented. The eight channel teleradiometer FAZA for visible and near infrared spectral regions is described in some detail. Mathematical aspects of data processing for the ill-posed problems are discussed. The results of the restoration of the volume emission rate of the molecular oxygen on Lambda 1.27 micron in the middle atmosphere and of the aerosol on Lambda 0.84 micron are presented.

Room, R.↗

NASA NTRS1990

Hypersonic waverider design from given shock waves

An attempt is made to generate waverider flows from given shock wave geometries using two approaches. In the first approach, axisymmetric flows are used to construct more general flows based on the concept of osculating cones. Conical waverider design can thus be extended to yield results also for shocks forming a slope surface. The second approach involves solving the ill-posed problem of prescribing a shock wave and finding the flow field behind it. A new numerical marching technique with some features of characteristic cross marching is used to solve the Euler equation. The code selection and some test cases are discussed.

Sobieczky, H.↗

NASA NTRS1972

Rapid optimization of multiple-burn rocket flights.

Different formulations of the fuel optimization problem for multiple burn trajectories are considered. It is shown that certain customary idealizing assumptions lead to an ill-posed optimization problem for which no solution exists. Several ways are discussed for avoiding such difficulties by more realistic problem statements. An iterative solution of the boundary value problem is presented together with efficient coast arc computations, the right end conditions for various orbital missions, and some test results.

Brown, K. R.↗

NASA NTRS1986

The ill-posed nature of the satellite temperature retrieval problem and the limits of retrievability

The inverse problem of satellite temperature profile retrieval is well known to be ill-posed. This means that not only is a vertical temperature profile solution not unique, but that two solutions can be very different from each other. A set of atmosphere-like, and true atmospheric examples of significantly dissimilar inverse solutions, were sought and found, using an 11-channel simulated HIRS sounding radiometer. Using the Riemann-Lebesgue Lemma for guidance, it is shown that simultaneous, numerical solutions of an atmospheric character may differ by as much as 10 K between 10-1000 mb. However, an empirical search for dissimilar solutions in the natural atmosphere reveals an extremely low probability of finding two significantly different RAOBs which produce radiance measurements whose differences cannot be resolved by the satellite radiometer. The empirical results are used to derive a first estimate of the limits of retrievability, analogous to the limits of predictability derivable from the ill-posed nature of the numerical weather prediction problem.

Thompson, Owen E.↗

NASA NTRS1991

Sinc-Galerkin estimation of diffusivity in parabolic problems

A fully Sinc-Galerkin method for the numerical recovery of spatially varying diffusion coefficients in linear partial differential equations is presented. Because the parameter recovery problems are inherently ill-posed, an output error criterion in conjunction with Tikhonov regularization is used to formulate them as infinite-dimensional minimization problems. The forward problems are discretized with a sinc basis in both the spatial and temporal domains thus yielding an approximate solution which displays an exponential convergence rate and is valid on the infinite time interval. The minimization problems are then solved via a quasi-Newton/trust region algorithm. The L-curve technique for determining an approximate value of the regularization parameter is briefly discussed, and numerical examples are given which show the applicability of the method both for problems with noise-free data as well as for those whose data contains white noise.

Smith, Ralph C.↗

NASA NTRS1986

History matching by spline approximation and regularization in single-phase areal reservoirs

An automatic history matching algorithm is developed based on bi-cubic spline approximations of permeability and porosity distributions and on the theory of regularization to estimate permeability or porosity in a single-phase, two-dimensional real reservoir from well pressure data. The regularization feature of the algorithm is used to convert the ill-posed history matching problem into a well-posed problem. The algorithm employs the conjugate gradient method as its core minimization method. A number of numerical experiments are carried out to evaluate the performance of the algorithm. Comparisons with conventional (non-regularized) automatic history matching algorithms indicate the superiority of the new algorithm with respect to the parameter estimates obtained. A quasioptimal regularization parameter is determined without requiring a priori information on the statistical properties of the observations.

Lee, T. Y.↗

NASA NTRS1988

Computing motion using resistive networks

Recent developments in the theory of early vision are described which lead from the formulation of the motion problem as an ill-posed one to its solution by minimizing certain 'cost' functions. These cost or energy functions can be mapped onto simple analog and digital resistive networks. It is shown how the optical flow can be computed by injecting currents into resistive networks and recording the resulting stationary voltage distribution at each node. These networks can be implemented in cMOS VLSI circuits and represent plausible candidates for biological vision systems.

Koch, Christof↗

NASA NTRS1991

Inverse problems and optimal experiment design in unsteady heat transfer processes identification

Experimental-computational methods for estimating characteristics of unsteady heat transfer processes are analyzed. The methods are based on the principles of distributed parameter system identification. The theoretical basis of such methods is the numerical solution of nonlinear ill-posed inverse heat transfer problems and optimal experiment design problems. Numerical techniques for solving problems are briefly reviewed. The results of the practical application of identification methods are demonstrated when estimating effective thermophysical characteristics of composite materials and thermal contact resistance in two-layer systems.

Artyukhin, Eugene A.↗

NASA NTRS1983

PAN AIR modeling studies

PAN AIR is a computer program that predicts subsonic or supersonic linear potential flow about arbitrary configurations. The code's versatility and generality afford numerous possibilities for modeling flow problems. Although this generality provides great flexibility, it also means that studies are required to establish the dos and don'ts of modeling. The purpose of this paper is to describe and evaluate a variety of methods for modeling flows with PAN AIR. The areas discussed are effects of panel density, internal flow modeling, forebody modeling in subsonic flow, propeller slipstream modeling, effect of wake length, wing-tail-wake interaction, effect of trailing-edge paneling on the Kutta condition, well- and ill-posed boundary-value problems, and induced-drag calculations. These nine topics address problems that are of practical interest to the users of PAN AIR.

Towne, M. C.↗

NASA NTRS1979

Calculated global conditions for the unsteady Euler's flow equations

This paper considers the numerical solution of a fluid dynamical system with time-dependent oscillating boundaries. The problem detailed, a compressible and inviscid flow in a one-dimensional pipe with vibrating ends, is examined to develop techniques needed to solve the more complex unsteady flows in turbomachine cascades. The problem is shown to be mathematically ill-posed but by complementing the governing equations with energy conservation, a unique solution is obtained. An artificial damping term is introduced into the governing equations to improve the performance of the time-marching integration.

Kuzo, D. M.↗

NASA NTRS1988

Vortex breakdown in an axisymmetric flow

The breakdown for both an isolated vortex and a vortex flow in a tube is studied by means of a numerical solution of the axisymmetric Navier-Stokes equations for an unsteady, incompressible, and laminar flow. Results are compared to those of other authors and to experiments. Various numerical solution methods and various physical boundary conditions are used in order to explain the existence of different solutions. The computational results reveal that the existence of different solutions is due to slightly different inflow conditions. The numerical problem for the isolated vortex becomes ill-posed at a certain time for the inflow condition of zero axial gradient of the radial velocity component. Different inflow conditions yield breakdown flowfields containing different features of the experimentally observed breakdown structure.

Menne, Stefan↗

NASA NTRS1991

Single Doppler radar and lidar studies of the dynamics of mixed layers and overlying stable regions

Some examples of profiles of PBL quantities retrieved from Doppler radar and lidar observations are shown, and the extent to which the deduced quantities agree with the traditional eddy viscosity concept is examined. It is found in one case that the concept of 'top down bottom up' diffusion introduced by Wyngaard (1983) is more pertinent than that of the eddy viscosity. In another case it is found that, in stable layers with sufficiently large Richardson number, the momentum fluxes could be counter-gradient. In this latter case the use of a negative turbulence diffusion coefficient would result in an ill-posed differential equation, thus confusing the 'diffusion problem'.

Xu, Mei↗

NASA NTRS1984

Neutrino tomography - Tevatron mapping versus the neutrino sky

The feasibility of neutrino tomography of the earth's interior is discussed, taking the 80-GeV W-boson mass determined by Arnison (1983) and Banner (1983) into account. The opacity of earth zones is calculated on the basis of the preliminary reference earth model of Dziewonski and Anderson (1981), and the results are presented in tables and graphs. Proposed tomography schemes are evaluated in terms of the well-posedness of the inverse-Radon-transform problems involved, the neutrino generators and detectors required, and practical and economic factors. The ill-posed schemes are shown to be infeasible; the well-posed schemes (using Tevatrons or the neutrino sky as sources) are considered feasible but impractical.

Wilson, T. L.↗

NASA NTRS1984

CREKID: A computer code for transient, gas-phase combustion of kinetics

A new algorithm was developed for fast, automatic integration of chemical kinetic rate equations describing homogeneous, gas-phase combustion at constant pressure. Particular attention is paid to the distinguishing physical and computational characteristics of the induction, heat-release and equilibration regimes. The two-part predictor-corrector algorithm, based on an exponentially-fitted trapezoidal rule, includes filtering of ill-posed initial conditions, automatic selection of Newton-Jacobi or Newton iteration for convergence to achieve maximum computational efficiency while observing a prescribed error tolerance. The new algorithm was found to compare favorably with LSODE on two representative test problems drawn from combustion kinetics.

Pratt, D. T.↗