This book presents a new “factorized” formulation, for boundary value problems for linear elliptic partial differential equations. Based on the invariant embedding method of Richard Bellman, well-known for the synthesis of closed loop optimal control, and here applied to solving boundary value problems, this formulation is comprised of two decoupled Cauchy problems and a Riccati equation for Dirichlet Neumann type operators. After presenting and justifying the formal calculation of the factorization using a simple model problem, the authors discuss the application of this method to more complex situations. In this context, a link is built, on a discretized version of the problem, between invariant embedding and the Gaussian factorization. Finally, the book examines how factorization can be extended to other classic linear equations of elliptic type and to the QR factorization