Numerical differentiation of approximated functions with limited order-of-accuracy deterioration

摘要

We consider the problem of numerical differentiation of a function f from approximate or noisy values of f on a discrete set of points; such discrete approximate data may result from a numerical calculation (such as a finite element or finite difference solution of a partial differential equation), from experimental measurements, or, generally, from an estimate of some sort. In some such cases it is useful to guarantee that orders of accuracy are not degraded: assuming the approximating values of the function are known with an accuracy of order O(hr), where h is the mesh size, an accuracy of O(hr) is desired in the value of the derivatives of f. Differentiation of interpolating polynomials does not achieve this goal since, as shown in this text, n-fold differentiation of an interpolating polynomial of any degree ≥ (r - 1) obtained from function values containing errors of order O(hr) generally gives rise to derivative errors of order O(hr-n); other existing differentiation algorithms suffer from similar degradations in the order of accuracy. In this paper we present a new algorithm, the LDC method (low degree Chebyshev), which, using noisy function values of a function f on a (possibly irregular) grid, produces approximate values of derivatives f(n) (n = 1, 2...) with limited loss in the order of accuracy. For example, for (possibly nonsmooth) O(hr) errors in the values of an underlying infinitely differentiable function, the LDC loss in the order of accuracy is "vanishingly small": derivatives of smooth functions are approximated by the LDC algorithm with an accuracy of order O(hr -) for all r- < r. The algorithm is very fast and simple; a variety of numerical results we present illustrate the theory and demonstrate the efficiency of the proposed methodology.