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
1. Presentation of the Formal Computation of Factorization.
2. Justification of the Factorization Computation.
3. Complements to the Model Problem.
4. Interpretation of the Factorization through a Control Problem.
5. Factorization of the Discretized Problem.
6. Other Problems.
7. Other Shapes of Domain.
8. Factorization by the QR Method.
9. Representation Formulas for Solutions of Riccati Equations.
Jacques Henry is Emeritus Director of Research at INRIA Bordeaux Sud-ouest, France. Having studied under J.-L. Lions, his work focuses on optimal control of partial differential equations, inverse problems and their applications in biology.
Angel M Ramos is Associate Professor at the Complutense University of Madrid, Spain and has a doctorate in applied mathematics. He is the Director of the MOMAT research group, whose works are devoted to modeling, simulation and mathematical optimization.
Table of Contents
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