A spectral measure estimation problem in rheology

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Highlights

  • The determination of a spectral measure is a standard inverse problem with convex constraints.

  • The convex constraints come in from the experimental measurement errors.

  • The method of maximum entropy in the mean handles this class of problems in a natural way.

Abstract

In this paper we consider an inverse problem appearing in rheology, consisting of determining a spectral measure over the set of relaxation times, that yields an observed collection of loss and storage moduli. Mathematically speaking, the problem consists of solving a system of Fredholm equations. To solve it, we propose an extended version of the maximum entropy method in the mean which is flexible enough to incorporate potential measurement errors.

Section snippets

Introduction and preliminaries

To describe the tensile strength of synthetic fibers, or to be more precise, of polymer melts, some standard experimental tests are carried on. The rheological measurements include linear viscoelastic shear oscillations, as well as linear elongations. The idea behind the measurements is to determine the relationship between the force (stress) and strain (deformation) in the material. When a material is deformed, part of the work exerted is stored as in the deformation, and part is spent as

Discretization and augmentation of (5)(6)

Before discretizing (5)–(6) we integrate them into one single system of Fredholm equations. For that we consider a 2M-dimensional (column) vector G=(G(ω1),,G(ωM),G(ω1),,G(ωM))t, where as usual we denote by xt the transpose of the vector x. Here {ω1,ω2,,ωM} is the collection of frequencies at which both G(ω) and G(ω) were measured.

Even though the output of the discretization process is similar to (1)–(2), the inner logic is different. Here we choose a uniform partition of [τmin,τmax]

The basics of MEM

One possible interpretation of the method of maximum entropy in the mean (MEM), is that it consists of transforming (9) into a problem consisting in determining a probability measure that satisfies some constraints. For that we think of the unknown vector as the expected value of a random variable defined on an appropriate space. The expected value is to be taken with respect to a measure that has to be such that the expected value satisfies the constraint, which happens to be the equation to

Implementation of the procedures

In this section we use the data obtained by Berger  [1] which is the same used by Wolpert et al.  [3], to compare the result of our maxentropic procedure to theirs. We provide below the pseudo-code with which the estimation steps carried out:

  • Normalization of the data to avoid numerical stability and overflow issues: Since the range of the data is of several orders of magnitude, we renormalized the problem by multiplying both sides of (9) by a diagonal matrix D in such a way that the order of

Acknowledgments

We thank Dr. L. Berger for making a copy of his work available to us. We thank the referees for their comments. They definitely helped us to clarify our presentation.

References (7)

  • L. Berger

    Untersuchung zum rheologishen Verhalten von Polybutadienen mit Bimodalen Molmassenvertailung

    (1988)
  • L. Berger et al.

    Linear viscoelasticity, simple and planar melt extension of linear polybutadienes with bimodal mass distributions

    Rheol. Acta

    (1992)
  • R.L. Wolpert et al.

    A non-parametric Bayesian approach to inverse problems (with discussion)

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