This paper presents ecient computational techniques for solving an optimizationproblem in cardiac debrillation governed by the monodomain equations. Timedependentelectrical currents injected at dierent spatial positions act as the control.Inexact Newton-CG methods are used, with reduced gradient computation by adjointsolves. In order to reduce the computational complexity, adaptive mesh renement forstate and adjoint equations is performed. To reduce the high storage and bandwidthdemand imposed by adjoint gradient and Hessian-vector evaluations, a lossy compressiontechnique for storing trajectory data is applied. An adaptive choice of quantizationtolerance based on error estimates is developed in order to ensure convergence.The eciency of the proposed approach is demonstrated on numerical examples.