Stochastic generalized Fourier integral operators

The goal of the proposed project is to develop a novel method of analyzing wave propagation in media with spatially distributed random perturbations. The program is to establish a stochastic theory for systems of linear hyperbolic partial differential equations (PDEs) or pseudodifferential equations with random field coefficients, based on the modern technique of Fourier integral operators. The project has far reaching implications in the fields of partial differential equations, generalized functions and stochastic analysis. The motivation comes from system identification problems in continuum mechanics and from wave propagation in seismology. The results of the project will aid the mentioned fields in solving the problem of the interplay of irregular stochastic perturbations, wave propagation described by partial differential equations, and the description of the (measurable) stochastic properties of the solutions. The novelty in the project lies in the fact that with the aid of its results the degree of irregularity of the perturbations (and solutions) can be increased to a level which has not been reachable by the existing mathematical theories, but which is dictated by the models used in practical applications. For example, measurement data indicate that random material properties in continuum mechanics as well as medium scale density fluctuations of the earth crust should be modeled by continuous, but non-differentiable random fields, which are too irregular for the classical theory of hyperbolic partial differential equations. The project sets out to solve these problems by placing the equations in algebras of generalized functions on the one hand, and by shifting the stochastic model from the coefficients of the equation to the phase function and amplitude of the Fourier integral operators determining the solution on the other hand. The approach has become possible due to recent progress in Fourier integral operators in algebras of generalized functions, combined with new results in stochastic partial differential equations in the same setting. The project will deliver a novel method for describing the stochastic properties of the solutions in a way that makes them comparable with measurements and will ultimately lay the basis for a new approach to parameter fitting and system identification in the mentioned areas of application. In addition, the project will contribute new methods in generalized functions, in particular, the novel concept of stochastic generalized Fourier integral operators.

The goal of the project was to develop a completely new approach to modelling wave propagation in random media. An acoustic wave propagating in a medium with spatially varying properties encounters disturbances which are of a stochastic nature. In the traditional approach, wave propagation is described through systems of partial differential equations. The randomness enters in variations of the coefficients of the equations. The direct problem is to solve the equations, knowing the coefficients. The inverse problem consists in determining the coefficients, and thereby properties of the material or medium, from knowing the solution. Generally, solutions to the equations describing wave propagation can be represented by so-called Fourier integral operators. The structure of these operators allows one to encode the propagation geometry as well as the wave amplitudes at a given frequency. For example, when the material properties are constant, a point excitation propagates along the light cone. The new idea of the project was to model the randomness of the material not through the coefficients of the equations, but through random variations in the terms of the Fourier integral operators. This approach replaces the computationally expensive task of solving the random partial differential equations by the much cheaper task of evaluating the stochastic Fourier integral operators. The main issues that had to be addressed were: (1) probabilistic properties of the newly introduced stochastic operators, mainly relying on continuous dependence of the values of the solutions on the terms of the operators; (2) the requirement of studying the equations in spaces of generalized functions, due to the high irregularity of the random fluctuations; (3) the choice of suitable types of random field models describing spatial random variations; (4) validation of the concept in certain explicitly solvable cases; (5) establishing how the parameters of the stochastic models relate to empirically measurable solutions in order to be able to solve the inverse problem. One such explicitly solvable model that could be completely described by the newly introduced methods was transport in a so-called Goupillaud medium, a randomly layered medium with layers of ever decreasing thickness. The main bulk of work, however, addressed acoustic waves in linearly elastic materials with spatially randomly changing properties. The stochastic Fourier integral operator solutions could be calibrated by means of Monte Carlo samples of the true responses of the material to the excitation by an acoustic wave, generated by computer simulations. This procedure would allow one to detect changes in the material parameters through monitoring changes in the parameters of the Fourier integral operators. Thereby, the foundations of future applications in material sciences were laid down.