Inversion using simulated annealing

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Abstract
The problem here is finding the profile m̄ that minimizes E(m̄) = ‖d̄ − ḡ(m̄)‖ − ε‖m̄ − m̄0‖ in which d̄ is a data set, m̄0 is the bias, and ḡ is a Green's function. If ḡ is nonlinear and the dimension of m̄ is large, then finding Emin is difficult. Simulated annealing (SA) seeks Emin by sampling from the probability distribution p(m̄) = exp[− E(m̄)/T]/Z in which T is so small that p spikes at Emin. This would be silly if one had to generate Z numerically, but SA does not need to know Z. The physical analogy for SA is that each component of m̄ is an atom in a melt having temperature T and free energy E. Rapid cooling of the melt gives a glass (high E) but slow cooling gives a single crystal (low E). Problems with SA are that slow cooling consumes the computer budget faster than it finds Emin and that the freezing temperature Tc is difficult to determine. Accordingly, much research on SA concerns cooling schedules that do not require Tc. In wave field inversion problems, it was found that known cooling schedules gave poor results and that knowledge of Tc was essential. A rapid method to find Tc was discovered. Also, SA was no longer thought of as a process that terminates; rather Tc is determined accurately keeping the melt at Tc and letting SA find many good m̄. A measure of resolution in m̄ space is the “order parameter” η = ‖m̄ − m̂‖, where m̂ is the unknown true profile. Expected values of η(E) can be obtained from synthetic data.

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