A kernel approach to deconvolution of the complex modulus in linear viscoelasticity

The relaxation spectrum of a viscoelastic material holds the key to describing its relaxation mechanisms at a molecular level. It also plays a fundamental role in accessing the molecular weight distribution, and in modelling the dynamics of complex fluids. The relaxation spectrum cannot be measured directly, but it may be locally determined from experimental measurements of viscoelastic response at a macroscopic level. In particular, the relaxation spectrum is a continuous distribution of relaxation times which may be recovered, at least locally, from measurements of the complex modulus of the material. Although mathematical expressions for the continuous spectrum have been known for well over a century, these were inaccessible to numerical implementation for decades. Regularization methods for approximating discrete line spectra were first introduced in the 1980s, but it was not until 2012 that methods for recovering continuous spectra in a mathematical framework were proposed. In this paper, we analyze spectrum recovery within the framework of reproducing kernel Hilbert spaces and identify such spaces as natural homes for the complex modulus and spectrum. Theorems are proved establishing the convergence of inverse operators expressed as series of derivatives of the complex modulus. This enables a detailed characterization of the native spaces of the real and imaginary parts of the complex modulus, and leads to a further theorem which identifies a hierarchy of trial spaces for the spectrum. Homeomorphic trial spaces for data and spectra are then specified in detail, and their efficacy demonstrated by means of a case study.