Backward Error Estimation

Estimation of unknowns in the presence of noise and uncertainty is an active area of study, because no method handles all cases well, or even satisfactorily. The basic problem considered is the linear system, Ax ≈ b, where A and b are given matrices with noise and uncertainty from measurements or modeling. The goal is to get the ”best ” estimate of x. The problem is useful in a wide variety of situations since it covers how to invert a matrix that contains uncertainty. Mainstream methods like least squares and total least squares fail dramatically when A is ill-conditioned. Other methods like Tikhonov regression, ridge regression, and min max (bounded data uncertainty) provide robustness at the cost of fine details (by reducing �x�). This paper presents a new family of regression methods based off the backward error criteria which can add robustness and when possible enhances fine details (by increasing �x�). This paper covers the motivation, proof, and application of backward error estimation. The resulting algorithms are compared against existing methods in numerical examples from image processing. This paper is concerned with the estimation of unknowns that are related to some measurements by a linear model that is subject to uncertainty. Consider the set of linear equations, Ax = b, where A ∈ R m×n and b ∈ R m are given. The goal is to calculate the value of x ∈ R n. If the equation is exact and A is not singular, the solution can be readily found by a variety of techniques, such as taking the QR factorizatio