Solving the Inverse Problem for Measures Using Iterated Function Systems: A New Approach

  • Forte B
  • Vrscay E
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We present a systematic method of approximating, to an arbitrary accuracy, a probability measure μ on x=[0,1]q, q≥ 1, with invariant measures for iterated function systems by matching its moments. There are two novel features in our treatment. 1. An infinite set of fixed affine contraction maps on X, W={w1,w2,⋯ }, subject to an 'ε-contractivity' condition, is employed. Thus, only an optimization over the associated probabilities pi is required. 2. We prove a collage theorem for moments which reduces the moment matching problem to that of minimizing the collage distance between moment vectors. The minimization procedure is a standard quadratic programming problem in the pi which can be solved in a finite number of steps. Some numerical calculations for the approximation of measures on [0, 1] are presented.

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  • B. Forte

  • E. R. Vrscay

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