Page 101 - Fundamentals of Magnetic Thermonuclear Reactor Design

Simulation of Electromagnetic Fields Chapter | 4 85

may lead to considerable (exceeding tolerable limits) changes in the output
because of discontinuance of the system solution in the neighbourhood of a
near-degenerate matrix.
Systems of equations and matrices with high conditionality numbers are said
to be ill conditioned, or unstable. In practice, such systems may be indiscernible
within a computer finite precision range. If, however, a system is discernible,
the number of solutions is infinite as in the case of zero matrix determinant that,
in fact, takes place in such calculus.
In many applied problems, the task is to obtain solutions (4.11) insensitive to
small changes in the right-hand side. This follows from the requirement that the
problem is physically determined and that its solution can be interpreted. Thus,
it is necessary to identify and develop solution methods, allowing the output to
be low-sensitive to small errors in the input. In the general case, an approach to
solving a SLAE with a degenerate or ill-conditioned matrix must be the same
as in the case of an ill-posed problem. This would probably require a unique
algorithm to find a solution without identifying the system as degenerate or
non-degenerate. In our context, a poorly conditioned SLAE is the result of the
ill-posed initial problem.
The solution of an ill-posed problem generally requires additional a priori
information, which may be variable by nature. In this case, it seems logical
that the computational algorithm must be parametrised in such a way that a set
of parameters and formulated selection criteria guarantee the possibility of an
approximate stable solution. The information may include a priori qualitative
and quantitative data, such as the input error, smoothness and convexity of a
solution, or whether the solution belongs to a finitely parametrised family. With
a parametrised algorithm one can employ Tikhonov’s regularisation [26,27] as
a method for obtaining approximate solutions to ill-posed problems that are
insensitive to small changes in the input.
We shall now discuss a variational regularisation method that seems to be
most natural for solving the synthesis problems for different applications, in-
cluding nonlinear ones.
Eq. (4.11) can be treated as a matrix and an operator equation, as there are
no formal differences between them due to an isomorphism of matrices and lin-
ear operators. We assume that z and u in Eq. (4.11) to be the elements of linear
metric spaces, F and U, defined by the norm ρ (z z ) = ||z − z ||,
1
2
F
1, 2
z , z ∈ F, ρ (u u ) = ||u − u ||, u u ∈ U, and that A is a linear operator
1,
1,
2
1
2
1
2
U
2
(matrix), acting from F to U.
The search for a solution of (4.11) is said to be a well-posed problem for a
pair of metric spaces, F, U, if
l Eq. (4.11) has a solution z ∈ F for any u ∈ U,
l the solution can be determined unambiguously in F,
l the problem is stable in the spaces (F, U), that is, for each ε > 0 there exists
δ(ε) such that ρ (z , z ) ≤ ε, if ρ (u , u ) ≤ δ.
U
1
2
1
2
F