In the master’s thesis, we explore the use of genetic algorithms to solve the inverse
problem of isospectral two-dimensional membranes. The idea originates from Mark
Kac’s article 'Can One Hear the Shape of a Drum?', questioning whether the shape
of a two-dimensional membrane can be determined from its vibration spectrum.
Later, it was shown that the answer is negative, as demonstrated by the authors of
'One Cannot Hear the Shape of a Drum', who showed that multiple membranes can
share the same spectrum. The introduction of the thesis presents the theoretical
background related to the inverse problem and the basics of genetic algorithms,
with a particular focus on the theory of genetic algorithms as a foundation for
understanding their application in solving inverse problems.
We then systematically test genetic algorithms on various cases, which become
increasingly complex. The aim is to explore and demonstrate their potential and
effectiveness in solving inverse problems. The results of our simulations often show
that genetic algorithms produce good solutions, with the advantage being that instead
of a single solution, we obtain an entire family of solutions. This leads to
the possibility of combining genetic algorithms with classical optimization methods,
which can lead to even better results in less computational time.
Our main focus is to demonstrate the usefulness of genetic algorithms and their
potential. The goal of the thesis is not to find the best way to solve this specific
problem, nor is it to precisely determine the parameters of the genetic algorithm for
this problem. One of the objectives of the thesis is also to serve as an insight into
the use of genetic algorithms and as inspiration for further work and improvement
of the results obtained in this thesis. The intention is for the reader to become acquainted
with the possibilities offered by genetic algorithms as a different approach
to problem-solving.
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