Later discussion will be facilitated by some preliminary mention of a couple of mathematical points. Suppose that u is an “unknown” vector that we plan to determine by some minimization principle. Let A[u] 0 and B[u] 0 be two positive functionals of ul | 804 Chapter 18. Integral Equations and Inverse Theory Inverse Problems and the Use ofA Priori Information Later discussion will be facilitated by some preliminary mention of a couple of mathematical points. Suppose that u is an unknown vector that we plan to determine by some minimization principle. Let A u 0 and B u 0 be two positive functionals of u so that we can try to determine u by either minimize A u or minimize B u Of course these will generally give different answers for u. As another possibility now suppose that we want to minimize A u subject to the constraint that B u have some particular value say b. The method of Lagrange multipliers gives the variation A u Ax B u - b A u AiB u 0 Ou Ou where Ai is a Lagrange multiplier. Notice that b is absent in the second equality since it doesn t depend on u. Next suppose that we change our minds and decide to minimize B u subject to the constraint that A u have a particular value a. Instead of equation we have c c B u A2 A u - a B u AgA u 0 ou ou with this time A2 the Lagrange multiplier. Multiplying equation by the constant 1 A2 and identifying 1 A2 with Ax we see that the actual variations are exactly the same in the two cases. Both cases will yield the same one-parameter family of solutions say u Ax . As Ax varies from 0 to 1 the solution u Ax varies along a so-called trade-off curve between the problem of minimizing A and the problem of minimizing B. Any solution along this curve can equally well be thought of as either i a minimization of A for some constrained value of B or ii a minimization of B for some constrained value of A or iii a weighted minimization of the sum A AxB. The second preliminary point has to do with degenerate minimization principles. In the example above now suppose that A u has the particular form A u A u - c 2 for some matrix A and vector c. If A has fewer rows than columns or if A is square but degenerate has a nontrivial nullspace see .
