STABLE AGGREGATES OF FLEXIBLE ELEMENTS : MAKING A SILK PURSE FROM SOW ’ S EARS
We discuss the application of a ray-based method for wave propagation to problems involving refraction and diffraction. The method assigns a Gaussian contribution to each ray such that their sum is insensitive to the Gaussian’s width. Diffraction is modelled by including contributions from rays scattered by the edges of obstacles. Although the Gaussian components are individually only poor approximations to physical waves, they collectively give accurate solutions even for problems where the wavelength approaches the dimensions of system components. INTRODUCTION A framework for building field estimates based on ray information was introduced in a recent series of papers [1-6]. In this framework, each ray is assigned a field contribution of Gaussian profile. These contributions, however, are not parabasal Gaussian beamlets, as each of them is not required to satisfy the wave equation. They are poor approximations to fields (the “sow’s ears” of the title), but their individual errors are forced to cancel when they are summed, leading to a field estimate of extraordinary accuracy. This estimate is also asymptotically (for small wavelength) insensitive to the width of the individual contributions. The sums are therefore referred to as a stable aggregates of flexible elements (SAFE). Here, we discuss the use of SAFE for modelling refraction and reflection at interfaces, as well as diffraction by opaque obstacles. WAVES IN INHOMOGENEOUS MEDIA Consider a scalar monochromatic field U in a two-dimensional inhomogeneous medium. This field satisfies [k∇ 2 + n(x,z)]U(x,z) = 0, (1) where k = 2π λ is the wave number, and n is the refractive index. We assume that this field propagates essentially towards larger z, so it is sufficient to specify U(x,0) in order to find U at any z > 0. Finding a closed form solution is generally not possible, so numerical schemes are necessary. Some of these schemes, based on rays, are particularly suited for problems where the wavelength is much smaller than any other characteristic length. It is natural to assume that the rays associated to the field described above also travel towards larger z. Each of these rays can then be described by an equation of the form x = X(z) . We assume that a solution to Eq. (1) is associated with a one-parameter family of rays, described collectively by a function X(ξ;z) , where ξ identifies the rays. The propagation of the rays in the inhomogeneous medium is ruled by the equations ∂ X ∂z (ξ;z) = P(ξ;z) H(ξ;z) , ∂P ∂z (ξ;z) = 1 2H(ξ;z) ∂n ∂x [X(ξ;z),z], (2) where P and H are respectively the xand z-components of a vector of length n(X,z) locally tangent to the ray. The optical path length of the rays, measured from a common phase front, is denoted by L(ξ;z). In order to find a ray-based estimate for the field at z > 0, one must first determine from U(x,0) the location and direction of the rays at z = 0, as well as their weight. (This process is discussed in [2].) The rays are then propagated by using Eqs. (2). Finally, the field estimate is constructed from the propagated rays, i.e. from X, P, H, L, and the weight function. For SAFE, this estimate is a sum of ray contributions of the form Uγ (x,z) = w(ξ ,γ;z) g(ξ ,γ; x,z) dξ ∫ , (3) where each contribution has a Gaussian profile in x and is centred at its corresponding ray: g(ξ ,γ; x,z) = k 2π exp − kγ 2 x − X(ξ;z) [ ]2 + ik L(ξ;z) + x − X(ξ;z) [ ] P(ξ;z) { } ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ . (4)
Stable aggregates of flexible elements give a stronger link between rays and waves.
A recently proposed ray-based method for wave propagation is used to provide a meaningful criterion for the validity of rays in wave theory. This method assigns a Gaussian contribution to each ray in order to estimate the field. Such contributions are inherently flexible. By means of a simple example, it is shown that superior field estimates can result when the contributions are no longer forced to evolve like parabasal beamlets.
Distribution-free monitoring of univariate processes
We consider statistical process control (SPC) of univariate processes when observed data are not normally distributed. Most existing SPC procedures are based on the normality assumption. In the literature, it has been demonstrated that their performance is unreliable in cases when they are used for monitoring non-normal processes. To overcome this limitation, we propose two SPC control charts for applications when the process data are not normal, and compare them with the traditional CUSUM chart and two recent distribution-free control charts. Some empirical guidelines are provided for practitioners to choose a proper control chart for a specific application with non-normal data.