Two types of problems are studied in this thesis.One part of the thesis is devoted to high frequency computation. Motivated by fast multiscale Gaussian wavepacket transforms and multiscale Gaussian beam methods which were originally designed for initial value problems of wave equations in the high frequency regime, we develop fast multiscale Gaussian beam methods for wave equations in bounded convex domains in the high frequency regime. To compute the wave propagation in bounded convex domains, we have to take into account reflecting multiscale Gaussian beams, which are accomplished by enforcing reflecting boundary conditions during beam propagation and carrying out suitable reflecting beam summation. To propagate multiscale beams efficiently, we prove that the ratio of the squared magnitude of beam amplitude and the beam width is roughly conserved, and accordingly we propose an effective indicator to identify significant beams. We also prove that the resulting multiscale Gaussian beam methods converge asymptotically. Numerical examples demonstrate the accuracy and efficiency of the method.The second part of the thesis studies the reduction of backscatter radar cross section (RCS) for a cavity embedded in the ground plane. One approach for RCS reduction is through the coating material. Assume the bottom of the cavity is coated by a thin, multilayered radar absorbing material (RAM) with possibly different permittivities. The objective is to minimize the backscatter RCS by the incidence of a plane wave over a single or a set of incident angles and frequencies. By formulating the scattering problem as a Helmholtz equation with artificial boundary condition, the gradient with respect to the material permittivities is determined efficiently by the adjoint state method, which is integrated into a nonlinear optimization scheme. Numerical example shows the RCS may be significantly reduced.Another approach is through shape optimization. By introducing a transparent boundary condition, the unbounded scattering problem from a cavity is reduced to a bounded domain problem. RCS reduction for the cavity is formulated as a shape optimization problem involving the Helmholtz equation. The existence of the minimizer is proved under an appropriate constraint. Descent directions of the objective function with respect to the boundary may be found via the domain derivative. It is used in a gradient-based optimization scheme to find the optimal shape of the cavity. Numerical examples show that the RCS is effectively reduced at different incident frequencies.
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