Schur complement

If the matrix Q is positive semidefinite (positive definite) when projected into the null space of the active constraints, then (3-98) is (strictly) convex and the QP is a global (and unique) minimum. Otherwise, local solutions exist for (3-98), and more extensive global optimization methods are needed to obtain the global solution. Like LPs, convex QPs can be solved in a finite number of steps. However, as seen in Fig. 3-57, these optimal solutions can lie on a vertex, on a constraint boundary, or in the interior. A number of active set strategies have been created that solve the KKT conditions of the QP and incorporate efficient updates of active constraints. Popular methods include null space algorithms, range space methods, and Schur complement methods. As with LPs, QP problems can also be solved with interior point methods [see Wright (1996)]. 

The material balance condition must apply at the nodes of the network such that no adsorption or reaction occurs there, that is, no accumulation at the nodes. For each pore a boundary value problem has to be solved. Adopting the methods of Rieckmann and Keil, it was necessary to discretize each pore using a finite difference scheme. This led to a very large system of nonlinear equations, which were solved using the Schur complement method, described by Rieckmann and Keil [44]. 

From these equations Keller [7] suggests an exact algorithm for the solution of (4.4). It consists of successive steps for the solution of the Schur complement equation and finally a backward solution to determine Ax. Chan [5] further develops this algorithm using specific approximations of the vector p and the matrix C. To get an impression of his method we want to present his suggestion on the computation of p. Since... 

Solve the discrete Pressure Schur Complement equation for the pressure update q... 

For explaining the Schur complement, consider the stmcture with two subdomains in Fig. la. Accordingly, the stiffness matrix, displacement vector, and the force vector are partitioned, and the static equilibrium condition is written as... 

L21. Lloyd, W., and Schur, P. H., Immune complexes, complement, and anti-DNA in exacerbations of systemic lupus erythematosus (SLE). Medicine (Baltimore) 60, 208-217 (1981). 

K2. Kammer, G., and Schur, P. H., Binding of circulating immune complexes to human peripheral blood lymphocytes Effect of complement. Clin. Immunol. Immunopathol. 10, 202-213 (1978). 

Z6. Zvaifler, N. J., and Schur, P., Reactions of aggregated mercaptoethanol treated gamma globulin with rheumatoid factor—precipitin and complement fixation studies. Arthritis Rheum. 11, 523-536 (1968),... 

Tannenbaum, H., Ruddy, S. and Schur, P. H. (1975) Acute anaphylaxis associated with serum complement depletion. J. Allergy din. Immunol, 56, 226. 

---