Mohammad Motamed

A stochastic collocation method for the second order wave equation with a discontinuous random speed

In this paper we propose and analyze a stochastic collocation method for solving the second order wave equation with a random wave speed and subjected to deterministic boundary and initial conditions. The speed is piecewise smooth in the physical space and depends on a finite number of random variables. The numerical scheme consists of a finite difference or finite element method in the physical space and a collocation in the zeros of suitable tensor product orthogonal polynomials (Gauss points) in the probability space. This approach leads to the solution of uncoupled deterministic problems as in the Monte Carlo method. We consider both full and sparse tensor product spaces of orthogonal polynomials. We provide a rigorous convergence analysis and demonstrate different types of convergence of the probability error with respect to the number of collocation points for full and sparse tensor product spaces and under some regularity assumptions on the data. In particular, we show that, unlike in elliptic and parabolic problems, the solution to hyperbolic problems is not in general analytic with respect to the random variables. Therefore, the rate of convergence may only be algebraic. An exponential/fast rate of convergence is still possible for some quantities of interest and for the wave solution with particular types of data. We present numerical examples, which confirm the analysis and show that the collocation method is a valid alternative to the more traditional Monte Carlo method for this class of problems.

A fast phase space method for computing creeping rays

Creeping rays can give an important contribution to the solution of medium to high frequency scattering problems. They are generated at the shadow lines of the illuminated scatterer by grazing incident rays and propagate along geodesics on the scatterer surface, continuously shedding diffracted rays in their tangential direction.In this paper, we show how the ray propagation problem can be formulated as a partial differential equation (PDE) in a three-dimensional phase space. To solve the PDE we use a fast marching method. The PDE solution contains information about all possible creeping rays. This information includes the phase and amplitude of the field, which are extracted by a fast post-processing. Computationally, the cost of solving the PDE is less than tracing all rays individually by solving a system of ordinary differential equations.We consider an application to mono-static radar cross section problems where creeping rays from all illumination angles must be computed. The numerical results of the fast phase space method and a comparison with the results of ray tracing are presented.

On the Linear Stability of the Fifth-Order WENO Discretization

We study the linear stability of the fifth-order Weighted Essentially Non-Oscillatory spatial discretization (WENO5) combined with explicit time stepping applied to the one-dimensional advection equation. We show that it is not necessary for the stability domain of the time integrator to include a part of the imaginary axis. In particular, we show that the combination of WENO5 with either the forward Euler method or a two-stage, second-order Runge–Kutta method is linearly stable provided very small time step-sizes are taken. We also consider fifth-order multistep time discretizations whose stability domains do not include the imaginary axis. These are found to be linearly stable with moderate time steps when combined with WENO5. In particular, the fifth-order extrapolated BDF scheme gave superior results in practice to high-order Runge–Kutta methods whose stability domain includes the imaginary axis. Numerical tests are presented which confirm the analysis.

Finite difference schemes for second order systems describing black holes

In the harmonic description of general relativity, the principal part of Einsteins equations reduces to 10 curved space wave equations for the components of the space-time metric. We present theorems regarding the stability of several evolution-boundary algorithms for such equations when treated in second order differential form. The theorems apply to a model black hole space-time consisting of a spacelike inner boundary excising the singularity, a timelike outer boundary and a horizon in between. These algorithms are implemented as stable, convergent numerical codes and their performance is compared in a 2-dimensional excision problem.

Analysis and computation of the elastic wave equation with random coefficients

We consider the stochastic initial-boundary value problem for the elastic wave equation with random coefficients and deterministic data. We propose a stochastic collocation method for computing statistical moments of the solution or statistics of some given quantities of interest. We study the convergence rate of the error in the stochastic collocation method. In particular, we show that, the rate of convergence depends on the regularity of the solution or the quantity of interest in the stochastic space, which is in turn related to the regularity of the deterministic data in the physical space and the type of the quantity of interest. We demonstrate that a fast rate of convergence is possible in two cases: for the elastic wave solutions with high regular data; and for some high regular quantities of interest even in the presence of low regular data. We perform numerical examples, including a simplified earthquake, which confirm the analysis and show that the collocation method is a valid alternative to the more traditional Monte Carlo sampling method for approximating quantities with high stochastic regularity.

A wavefront-based Gaussian beam method for computing high frequency wave propagation problems

We present a novel wavefront method based on Gaussian beams for computing high frequency wave propagation problems. Unlike standard geometrical optics, Gaussian beams compute the correct solution of the wave field also at caustics. The method tracks a front of two canonical beams with two particular initial values for width and curvature. In a fast post-processing step, from the canonical solutions we recreate any other Gaussian beam with arbitrary initial data on the initial front. This provides a simple mechanism to include a variety of optimization processes, including error minimization and beam width minimization, for a posteriori selection of optimal beam initial parameters. The performance of the method is illustrated with two numerical examples.

A Sparse Stochastic Collocation Technique for High-Frequency Wave Propagation with Uncertainty

A Sparse Stochastic Collocation Technique for High-Frequency Wave Propagation with Uncertainty

A MultiOrder Discontinuous Galerkin Monte Carlo Method for Hyperbolic Problems with Stochastic Parameters

We present a new multiorder Monte Carlo algorithm for computing the statistics of stochastic quantities of interest described by linear hyperbolic problems with stochastic parameters. The method is...

Analysis and Computation of Hyperbolic PDEs with Random Data

Hyperbolic partial differential equations (PDEs) are mathematical models of wave phenomena, with applications in a wide range of scientific and engineering fields such as electromagnetic radiation, geosciences, fluid and solid mechanics, aeroacoustics, and general relativity. The theory of hyperbolic problems, including Friedrichs and Kreiss theories, has been well developed based on energy estimates and the method of Fourier and Laplace transforms [8, 16]. Moreover, stable numerical methods, such as the finite difference method [14], the finite volume method [17], the finite element method [6], the spectral method [4], and the boundary element method [11], have been proposed to compute approximate solutions of hyperbolic problems. However, the development of the theory and numerics for hyperbolic PDEs has been based on the assumption that all input data, such as coefficients, initial data, boundary and force terms, and computational domain, are exactly known.

A Fast Method for the Creeping Ray Contribution to Scattering Problems

Creeping rays can give an important contribution to the solution of medium to high frequency scattering problems. They are generated at the shadow lines of the illuminated scatterer by grazing incident rays and propagate along geodesics on the scatterer surface, continuously shedding diffracted rays in their tangential direction. In this paper we show how the ray propagation problem can be formulated as a partial differential equation (PDE) in a three‐dimensional phase space. To solve the PDE we use a fast marching method. The PDE solution contains information about all possible creeping rays. This information includes the phase and amplitude of the field, which are extracted by a fast postprocessing. Computationally the cost of solving the PDE is less than tracing all rays individually by solving a system of ordinary differential equations(ODE). We consider an application to monostatic radar cross section problems where creeping rays from all illumination angles must be computed.