A spectral measure estimation problem in rheology
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 -dimensional (column) vector , where as usual we denote by the transpose of the vector . Here is the collection of frequencies at which both and 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
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:
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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 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.
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2019, Physica A: Statistical Mechanics and its Applications