Thursday, Feb. 26, 10-12. The weak formulation of this system leads to a saddle point problem which is subject to the inf-sup condition. The inf-sup condition poses restrictions on finite element spaces when finite element method is employed and leads to indefinite algebraic systems. We introduce three possible least-squares functionals for the system of first order. Subject to some regularity conditions their minimization recovers the solution.
All formulations lead to finite element systems with symmetric and positive definite matrices.
We discuss briefly the merits and the deficiencies of the three least squares formulations. The main objective of the talk is to describe and to analyze the formulation based on a discrete minus one inner product. The algorithm is based on lattice basis reduction, and first finds short vectors satisfying the diophantine equations. The next step is to branch on linear combinations of these vectors, which either yields a vector that satisfies the bound constraints or provides a proof that no such vector exists. The research was motivated by the need for solving constrained diophantine equations as subproblems when designing integrated circuits for video signal processing.