0 to pose this as a nite dimensional optimization

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Unformatted text preview: al 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 b1 , . . . , bN 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, ϕ(z ) = α1 z n + α2 z n−1 + · · · + αn z + αn+1 , ˆ 67 where α1 , . . . , αn+1 ∈ C. With these approximations, we obtain the problem maximize |ϕ′ (a)| ˆ subject to |ϕ(bi )| ≤ 1, ˆ i = 1, . . . , N, with variables α1 , . . . , αn+1 ∈ C. The problem data are b1 , . . . , bN ∈ ∂ 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 bi and plots them. It also contains code that will plot ϕ(bi ), the boundary ˆ of the mapped region, once you provide the values of αj ; these points should be very cl...
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This note was uploaded on 09/10/2013 for the course C 231 taught by Professor F.borrelli during the Fall '13 term at University of California, Berkeley.

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