Representation of Green's Function Integral Operators Using Wavelet Transforms

In this paper, we analyze the representation of integral operators whose kernels are Green's functions for a class of linear differential equations using wavelets with a finite number of vanishing moments. In particular, we show how wavelets can be used to generate a sparse representation of these operators. We show that the matrix associated with the discretized integral operator represented in the wavelet basis is sparse and, in particular, contains multiple bands of various widths. The particular banded structure of the wavelet representation of the operator follows from the fact that the associated Green's function is smooth away from the source point and is singular at some order; that is, for some T, its Tth derivative is discontinuous at the source point. We derive bounds on the magnitude of the coefficients of the integral operator in the wavelet basis as a function of scale and position and, in particular, in terms of whether or not the coefficient lies within a band. Based on these bounds, we can approximate the operator by ignoring coefficients not lying within these bands, thus producing a sparse representation. This sparse representation is extremely beneficial for numerical applications in which one would like to apply the Green's function operator efficiently: normally if such an operator mapped N points into N points, it would require O(N ²) operations; however, with the wavelet transform, the mapping would require only O((4M + 2γLM + 2γ(1 - γ)M - 3)N) operations, where 2M is the length associated with the support of the wavelet function, L = log₂ N - 1, and γ = 1/ln2. An application example in which this is important is the control of smart structures in which a large number of embedded sensors and actuators must be coordinated to achieve disturbance rejection on the surface of the structure.