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In many engineering scenarios, the output of a given model is tied to design guidelines that must be fulfilled. In thermoacoustics, for instance, a specific flame response — characterized by its gain and phase over certain frequencies of interest — may be desired to ensure system stability once the flame is coupled with its acoustic environment. Which parameters (e.g., fuel and air mass flow rates, global equivalence ratio, degree of premixedness, wall temperature at the burner mouth, etc.) should be selected to achieve this target flame response? This constitutes a classical inverse problem and, as such, requires a well-defined optimization framework for its solution. With the advent of numerical solvers featuring automatic differentiation capabilities, solving such optimization problems is becoming increasingly accessible -- though many challenges still remain.