Hi Simon,
To this end, I am solving n linear systems, where n is the number of
parameters:
K * sensitivity[i] = rhs[i] i=1,...,n
K is the (symmetric) tangent matrix which I already use to solve for the
solution of my pde, that is,
I just assemble the rhs vectors anew.
The result is the Jacobian matrix
J = [ sensitivity[1] ;...; sensitivity[n] )
which has as many rows as my pde has degrees of freedom.
J has dimensions 320x8, that is,
my pde has 320 dofs and I want to optimise 8 parameters.
The solution of my pde (the dofs) are in the interval [0 to 1] and the
parameters are in the interval [0 to 1e5] , i.e., the entries of J are
rather small per nature (1e-6 to 1e-10).
I evaluated the condition number of J in matlab: cond(J) = 1e13, which
is too large given that the optimization algorithm works on J^t*J.
I might be missing something, but how do you define the condition number
of a non-square matrix?
Separately, though: what is the relationship between the condition
number of J and solving linear systems with K? If I understand you right...
Currently, I solve the above n linear systems using the direct solver
UMFPACK. I know that there is no such thing like a preconditioner for
direct solvers.
Thus, I solved the linear systems using the iterative SolverCG with a
preconditioner:
const int solver_its = 1000
const double tol_sol = 1e-15
SolverControl solver_control(solver_its, tol_sol);
GrowingVectorMemory<Vector<double> > GVM;
SolverCG<Vector<double> > solver_CG(solver_control, GVM);
PreconditionSSOR<> preconditioner;
preconditioner.initialize(tangent_matrix, 1.2);
//solve the linear systems...
However, the condition number of J is still around 1e13.
...then this is the code you use to solve with K. K being whatever it
is, the preconditioner you use only affects *how* you solve the linear
systems with K, but it has no effect on the *solution* of the linear
systems, and so this piece of code has no influence on J: J is what it
is, and the condition number of J is what it is independent of the code
you use to *solve for J*.
My idea was to push the condition number of J at least some order of
magnitudes when applying a preconditioner to my tangent matrix.
Is this consideration reasonable?
The condition number of J is determined by your choice of parameters. If
I understand things right, your J^T J is 8x8, and if it is poorly
conditioned, you need to choose the 8 parameters differently, for
example (i) scaled in a different way, and (ii) as linear combinations
of what you use right now.
Best
W.
--
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Wolfgang Bangerth email: bange...@colostate.edu
www: http://www.math.colostate.edu/~bangerth/
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