Systems and methods for generating regularized solutions for inverse problems, where the regularization is performed using the Duncan-Horn formulation of a Kalman filter. The various alternative embodiments of the invention are applicable to many different types of inverse problems. In one electrocardiography implementation, electrical potentials are measured within a chamber of the heart and these measurements are processed to generate a mapping of the electrical activity at the interior surface of the heart chamber (these potentials at the heart s surface resulted in the potentials that were measured in the heart chamber). The generation of the mapping of the potentials at the heart s surface is regularized by using a Duncan-Horn formulation of a Kalman filter. The regularization may be based upon multiple steps in time, and the number of time steps may be selectable.