It describes SIMD, shared memory, and distributed memory machine models; decomposition as a fundamental activity in parallel algorithmic design; key programming models; key concepts of performance analysis and optimization; and three case studies applying these concepts to the Single Shortest Path Problem, the

Eikonal equation, and computation of the two-dimensional convex hull.

The phase of the wave packet can be obtained from the corresponding eikonal equation, which can be solved numerically on the characteristics (rays).

We shall find W given as: W (z, x, y) = A (x, y)W (z, x, y) , whereW 0(z, x, y) is the solution of the vertical spectralproblem (17) normalized as followsFinally, we can obtain a following equationThis equation will be solved in characteristics of the eikonal equation (18).

Expressing the solution U(r) of wave equation in terms of well known Luneberg-Kline series yields

Eikonal equation for phase s(r) and transport equation for amplitude.

It consists of iteratively solving the

Eikonal equation (Eq.

1982) (29) proved that the mild-slope equation includes the

eikonal equation with diffraction effect and the energy conservation equation between two wave rays.

Furthermore, the

eikonal equation and the approximation of the Liouville equation based on GO have been employed for simulating the high frequency wave propagation.

In [7], helix submanifolds in Euclidean space were studied by solving the

Eikonal equation.

There are several ways of doing this, basically grouped into two methods: straight or curved ray theory (Cerveny, 2001) or numerically resolving the wave equation with some constraints, such us the

eikonal equation (Vidale, 1988).

The most general treatment for point source in 3D is given by the

eikonal equation [6, p.

The

eikonal equation in zero-space, expressed in his observable world-coordinates, is [K.

He covers some variational problems in Hilbert spaces, iterative methods in Hilbert spaces, operator-splitting and alternating directions methods, augmented Lagrangians and alternating direct methods of multipliers, the least-squares solution of linear and nonlinear problems, obstacle problems and Bingham flow with applications to control, nonlinear eigenvalue problems,

Eikonal equations, and fully nonlinear elliptic equations.