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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 ,
67 where α1 , . . . , αn+1 ∈ C. With these approximations, we obtain the problem
maximize |ϕ′ (a)|
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
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.
- Fall '13
- The Aeneid