Abstract
We construct high-order derivative operators for smooth functions represented via discontinuous multiwavelet bases. The need for such operators arises in order to avoid artifacts when computing functionals involving high-order derivatives of solutions of integral equations. Previously high-order derivatives had to be formed by repeated application of a first-derivative operator that, while uniquely defined, has a spectral norm that grows quadratically with polynomial order and, hence, greatly amplifies numerical noise (truncation error) in the multiwavelet computation. The new constructions proceed via least-squares projection onto smooth bases and provide substantially improved numerical properties as well as permitting direct construction of high-order derivatives. We employ either b-splines or bandlimited exponentials as the intermediate smooth basis, with the former maintaining the concept of approximation order while the latter preserves the pure imaginary spectrum of the first-derivative operator and provides more direct control over the bandlimit and accuracy of computation. We demonstrate the properties of these new operators via several numerical tests as well as application to a problem in nuclear physics.
Highlights
In this paper we revisit the issue of computing high-order derivatives of smooth functions represented in multiwavelet bases
The amplification of numerical noise inherent in computing higher derivatives with the original scheme, the sensitivity of some non-linear functionals to point-wise rather than norm-wise errors, and the sensitivity of some functionals to regions where the density is small, combine to make computations with functionals that incorporate derivatives more expensive compared to those that do not include derivatives. This is largely due to the need to compute much more accurately in order to obtain accurate derivatives and function tails. Another issue arising from the norm of the original derivative operator and the need to compute higher derivatives by its repeated application is that differential operators and their pseudo-inverse do not numerically commute within the precision of computation — i.e., the order of operations matters and there is a lack of consistency between the numerical and formal calculus
We have described a systematic approach for the construction of derivative operators by projecting to and from an intermediate basis with the desired properties and forming the derivative in that basis
Summary
In this paper we revisit the issue of computing high-order derivatives of smooth functions represented in multiwavelet bases. Another issue arising from the norm of the original derivative operator and the need to compute higher derivatives by its repeated application is that differential operators and their pseudo-inverse (i.e., convolution with the corresponding Green’s function) do not numerically commute within the precision of computation — i.e., the order of operations matters and there is a lack of consistency between the numerical and formal calculus This has emerged as a significant issue for scientists making first use of MADNESS, which typically requires reformulating partial differential equations as integral equations with free-space or periodic boundary conditions.
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