Edge based finite elements are finite elements whose degrees of freedom are assigned to edges of finite elements rather than nodes. Compared with conventional node based counterparts, they offer many useful properties. For example, they enforce tangential continuity only on inter-element boundaries but no normal continuity; they allow a vector field separated as the sum of the gradient of a scalar function and the remaining part. This dissertation presents a magnetic vector potential formulation implemented with edge elements to simulate eddy current phenomenon. The additional degree of freedom associated with the magnetic vector potential is fixed with the help of tree and co-tree separation from graph theory. The validity of the method is verified using well-known benchmark problems.;A phenomenological signal inversion scheme is proposed to characterize defect profiles from eddy current probe signals. The method relies on the edge element based forward model to predict probe responses and a minimization algorithm to minimize an objective function representing the squared error between the modal prediction and the observed signal. A gradient-based minimization algorithm is first investigated. The long computation time associated with the gradient calculation is reduced using the adjoint equation based method. However, gradient-based methods tend to converge to a poorer local minimum. A genetic algorithm and a simulated annealing algorithm are employed to improve performance. The performance of these stochastic methods in the context of the defect characterization problem is studied. The preliminary results show the effectiveness of the stochastic methods.