MIT6_079F09_hw08 - 6.079/6.975, Fall 2009-10 S. Boyd &...

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Unformatted text preview: 6.079/6.975, Fall 2009-10 S. Boyd & P. Parrilo Homework 8 1. Conformal mapping via convex optimization. Suppose that Ω is a closed bounded region in C with no holes ( i.e. , it is simply connected). The Riemann mapping theorem states that there exists a conformal mapping ϕ from Ω onto D = { z ∈ C | | z | ≤ 1 } , the unit disk in the complex plane. (This means that ϕ is an analytic function, and maps Ω one-to-one onto D .) One proof of the Riemann mapping theorem is based on an infinite dimensional opti- mization problem. We choose a point a ∈ int Ω (the interior of Ω). Among all analytic functions that map ∂ Ω (the boundary of Ω) into D , we choose one that maximizes the magnitude of the derivative at a . Amazingly, it can be shown that this function is a conformal mapping of Ω onto D . We can use this theorem to construct an approximate conformal mapping, by sampling the boundary of Ω, and by restricting the optimization to a finite-dimensional subspace of analytic functions. Let b 1 ,...,b N be a set of points in ∂ Ω (meant to be a sampling of the boundary). We will search only over polynomials of degree up to n , n ϕ ˆ( z ) = α 1 z + α 2 z n − 1 + + α n z + α n +1 , ··· where α 1 ,...,α n +1 ∈ C . With these approximations, we obtain the problem maximize | ϕ ˆ ( a ) | subject to | ϕ ˆ( b i ) | ≤ 1 , i = 1 ,...,N, with variables α 1 ,...,α n +1 ∈ C . The problem data are b 1 ,...,b N ∈ ∂ Ω and a ∈ int Ω. (a) Explain how to solve the problem above via convex or quasiconvex optimization. (b) Carry out your method on the problem instance given in conf_map_data.m . This file defines the boundary points b i and plots them. It also contains code that will plot ˆ ϕ ( b i ), the boundary of the mapped region, once you provide the values of α j ; these points should be very close to the boundary of the unit disk. (Please turn in this plot, and give us the values of α j that you find.) The function polyval may be helpful. Remarks. • We’ve been a little informal in our mathematics here, but it won’t matter. • You do not need to know any complex analysis to solve this problem; we’ve told you everything you need to know. • A basic result from complex analysis tells us that ˆ ϕ is one-to-one if and only if the image of the boundary does not ‘loop over’ itself. (We mention this just for fun; we’re not asking you to verify that the ˆ ϕ you find is one-to-one.) 1 2. Optimal amplifier gains. We consider a system of n amplifiers connected (for simplicity) in a chain, as shown below. The variables that we will optimize over are the gains a 1 ,...,a n > of the amplifiers. The first specification is that the overall gain of the system, i.e. , the product a 1 ··· a n , is equal to A , which is given....
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This note was uploaded on 01/31/2012 for the course ECE 202 taught by Professor Boyde during the Fall '06 term at Goldsmiths.

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MIT6_079F09_hw08 - 6.079/6.975, Fall 2009-10 S. Boyd &...

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