The conductivity of cardiac tissue determines excitation propagation and is important for quantifying ischemia and scar tissue and for building personalized models. Estimating conductivity distributions from endocardial mapping data is computationally challenging due to the computational complexity of the monodomain equations describing the cardiac excitation. For computing a maximum posterior estimate, we investigate different algorithmic optimization approaches based on adjoint gradient computation: steepest descent, limited memory BFGS, and recursive multilevel trust region methods using mesh hierarchies or heterogeneous model hierarchies. We compare overall performance, asymptotic convergence rate, and pre-asymptotic progress on some examples in order to assess the benefit of multifidelity acceleration.