in this thesis we study the approximate inverse based multigrid algorithm fapin for the solution of large sparse linear systems of equations. this algorithm, which is closely related to the well known multigrid v-cycle, has proven successful in the numerical solution of several second order boundary value problems. here we are mainly concerned with its application to fourth order problems. in particular, we demonstrate good multigrid performance with discrete problems arising from the beam equation and the biharmonic (plate) equation. the work presented also represents new experience with fapin using cubic b-spline, bicubic b-spline and piecewise bicubic hermite basis functions. we recast a convergence proof in matrix notation for the nonsingular case. central to our development are the concepts of an approximate inverse and an approximate pseudo-inverse of a matrix. in particular, we use least squares approximate inverses (and related approximate pseudo-inverses) found by solving a frobenius matrix norm minimization problem. these approximate inverses are used in the multigrid smoothers of our fapin algorithms.