Songting Luo

Fast sweeping method for the factored eikonal equation

We develop a fast sweeping method for the factored eikonal equation. By decomposing the solution of a general eikonal equation as the product of two factors: the first factor is the solution to a simple eikonal equation (such as distance) or a previously computed solution to an approximate eikonal equation. The second factor is a necessary modification/correction. Appropriate discretization and a fast sweeping strategy are designed for the equation of the correction part. The key idea is to enforce the causality of the original eikonal equation during the Gauss-Seidel iterations. Using extensive numerical examples we demonstrate that (1) the convergence behavior of the fast sweeping method for the factored eikonal equation is the same as for the original eikonal equation, i.e., the number of iterations for the Gauss-Seidel iterations is independent of the mesh size, (2) the numerical solution from the factored eikonal equation is more accurate than the numerical solution directly computed from the original eikonal equation, especially for point sources.

Factored singularities and high-order Lax-Friedrichs sweeping schemes for point-source traveltimes and amplitudes

In the high frequency regime, the geometrical-optics approximation for the Helmholtz equation with a point source results in an Eikonal equation for traveltime and a transport equation for amplitude. Because the point-source traveltime field has an upwind singularity at the source point, all formally high-order finite-difference Eikonal solvers exhibit first-order convergence and relatively large errors. In this paper, we propose to first factor out the singularities of traveltimes, takeoff angles, and amplitudes, and then we design high-order Lax-Friedrichs sweeping schemes for point-source traveltimes, takeoff angles, and amplitudes. Numerical examples are presented to demonstrate the performance of our new method.

Fast Sweeping Methods for Factored Anisotropic Eikonal Equations: Multiplicative and Additive Factors

The viscosity solution of static Hamilton-Jacobi equations with a point-source condition has an upwind singularity at the source, which makes all formally high-order finite-difference scheme exhibit first-order convergence and relatively large errors. To obtain designed high-order accuracy, one needs to treat this source singularity during computation. In this paper, we apply the factorization idea to numerically compute viscosity solutions of anisotropic eikonal equations with a point-source condition. The idea is to factor the unknown traveltime function into two functions, either additively or multiplicatively. One of these two functions is specified to capture the source singularity so that the other function is differentiable in a neighborhood of the source. Then we design monotone fast sweeping schemes to solve the resulting factored anisotropic eikonal equation. Numerical examples show that the resulting monotone schemes indeed yield clean first-order convergence rather than polluted first-order convergence and both factorizations are able to treat the source singularity successfully.

Fast Huygens sweeping methods for Helmholtz equations in inhomogeneous media in the high frequency regime

In some applications, it is reasonable to assume that geodesics (rays) have a consistent orientation so that the Helmholtz equation may be viewed as an evolution equation in one of the spatial directions. With such applications in mind, we propose a new Eulerian computational geometrical-optics method, dubbed the fast Huygens sweeping method, for computing Green functions of Helmholtz equations in inhomogeneous media in the high-frequency regime and in the presence of caustics. The first novelty of the new method is that the Huygens–Kirchhoff secondary source principle is used to integrate many locally valid asymptotic solutions to yield a globally valid asymptotic solution so that caustics associated with the usual geometrical-optics ansatz can be treated automatically. The second novelty is that a butterfly algorithm is adapted to carry out the matrix–vector products induced by the Huygens–Kirchhoff integration in O(NlogN) operations, where N is the total number of mesh points, and the proportionality constant depends on the desired accuracy and is independent of the frequency parameter. To reduce the storage of the resulting traveltime and amplitude tables, we compress each table into a linear combination of tensor-product based multivariate Chebyshev polynomials so that the information of each table is encoded into a small number of Chebyshev coefficients. The new method enjoys the following desired features: (1) it precomputes a set of local traveltime and amplitude tables; (2) it automatically takes care of caustics; (3) it constructs Green functions of the Helmholtz equation for arbitrary frequencies and for many point sources; (4) for a specified number of points per wavelength it constructs each Green function in nearly optimal complexity in terms of the total number of mesh points, where the prefactor of the complexity only depends on the specified accuracy and is independent of the frequency parameter. Both two-dimensional (2-D) and three-dimensional (3-D) numerical experiments are presented to demonstrate the performance and accuracy of the new method.

High-Order Factorization Based High-Order Hybrid Fast Sweeping Methods for Point-Source Eikonal Equations

The solution for the eikonal equation with a point-source condition has an upwind singularity at the source point as the eikonal solution behaves like a distance function at and near the source. As...

A uniformly second order fast sweeping method for eikonal equations

A uniformly second order method with a local solver based on the piecewise linear discontinuous Galerkin formulation is introduced to solve the eikonal equation with Dirichlet boundary conditions. The method utilizes an interesting phenomenon, referred as the superconvergence phenomenon, that the numerical solution of monotone upwind schemes for the eikonal equation is first order accurate on both its value and gradient when the solution is smooth. This phenomenon greatly simplifies the local solver based on the discontinuous Galerkin formulation by reducing its local degrees of freedom from two (1-D) (or three (2-D), or four (3-D)) to one with the information of the gradient frozen. When considering the eikonal equation with point-source conditions, we further utilize a factorization approach to resolve the source singularities of the eikonal by decomposing it into two parts, either multiplicatively or additively. One part is known and captures the source singularities; the other part serves as a correction term that is differentiable at the sources and satisfies the factored eikonal equations. We extend the second order method to solve the factored eikonal equations to compute the correction term with second order accuracy, then recover the eikonal with second order accuracy. Numerical examples are presented to demonstrate the performance of the method.

A New Approximation for Effective Hamiltonians for Homogenization of a class of Hamilton–Jacobi Equations

We propose a new formulation to compute effective Hamiltonians for homogenization of a class of Hamilton–Jacobi equations. Our formulation utilizes an observation made by Barron and Jensen [Comm. Partial Differential Equations, 15 (1990), pp. 1713–1742] about viscosity supersolutions of Hamilton–Jacobi equations. The key idea is to link the effective Hamiltonian to a suitable effective equation. The main advantage of our formulation is that only one auxiliary equation needs to be solved in order to compute the effective Hamiltonian H¯(p) for all p. Error estimates and stability are proved and numerical examples are presented to demonstrate the performance.

Some Inverse Problems in Periodic Homogenization of Hamilton-Jacobi Equations

We look at the effective Hamiltonian

Adjoint State Method for the Identification Problem in SPECT: Recovery of Both the Source and the Attenuation in the Attenuated X-Ray Transform

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