Nonnegatively Constrained Confidence Interval Estimation for Ill-Posed Problems

  • Bert W. Rust
Conference paper


Let the m x n linear model
be obtained by discretizing a system of first-kind integral equations
with known functions K i(ξ), unknown function x(ξ) and an m-vector ŷ of measurements corrupted by random errors є drawn from a distribution with zero mean vector and known variance matrix. Consider the problem of estimating linear functions of the form
$$\phi=w^T\bar{x}\approx \int_{a}^{b}w(\xi)x(\xi)d\xi,$$
where w(ξ) is an averaging function designed to elicit some desired information about x(ξ). For such problems, the least squares solution is a highly unstable function of the measurements, and the classical confidence intervals are too wide to be useful. The solution can often be stabilized by imposing physically motivated, a priori non-negativity constraints on x. This paper will show how to extend the classical confidence interval estimation technique to accommodate these nonnegativity constraints and how to compute the resulting much-improved confidence intervals.


Assure Convolution Lution Estima Lawson 


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  1. [1]
    W. R. Burrus, Utilization of A Priori Information by Means of Mathematical Programming in the Statistical Interpretation of Measured Distributions, Ph.D. Thesis, Ohio State University, ORNL-3743, Oak Ridge National Laboratory, June 1965.Google Scholar
  2. [2]
    W. R. Burrus, B. W. Rust, and J. E. Cope, Constrained interval estimation for linear models with ill-conditioned equations, in Information Linkage between Applied Mathematics and Industry II, A. L. Schoenstadt, et al., eds., Academic Press, New York, 1980, pp. 1–38.Google Scholar
  3. [3]
    C. L. Lawson and R. J. Hanson, Solving Least Squares Problems, Prentice-Hall, Englewood Cliffs, 1974.MATHGoogle Scholar
  4. [4]
    D. P. O’Leary and B. W. Rust, Confidence intervals for inequality-constrained least squares problems, with applications to ill-posed problems, SIAM Jour. Sci. Stat. Comput., 7 (1986), pp. 473–489.MathSciNetMATHCrossRefGoogle Scholar
  5. [5]
    D.L. Phillips, A technique for the numerical solution of certain integral equations of the first kind, J. Assoc. Comput. Mach., 9 (1962), pp. 84–97.MathSciNetMATHGoogle Scholar
  6. [6]
    J. E. Pierce and B. W. Rust, Constrained least squares interval estimation, SIAM Jour. Sci. Stat. Comput., 6 (1985), pp. 670–683.MathSciNetMATHCrossRefGoogle Scholar
  7. [7]
    B. W. Rust and W. R. Burrus, Mathematical Programming and the Numerical Solution of Linear Equations, American Elsevier, New York, 1972.MATHGoogle Scholar
  8. [8]
    B. W. Rust and D. P. O’Leary, Confidence intervals for discrete approximations to ill-posed problems, submitted to Jour. Comp. Graph. Statistics, (in review).Google Scholar
  9. [9]
    Phillip Wolfe, A duality theorem for non-linear programming, Quarterly of Appl. Math., 19 (1961), pp. 239–244.MathSciNetMATHGoogle Scholar

Copyright information

© Springer-Verlag New York, Inc. 1992

Authors and Affiliations

  • Bert W. Rust
    • 1
  1. 1.Center for Computing and Applied MathematicsNational Institute of Standards and TechnologyGaithersburgUSA

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