Biomedical Engineering Reference
In-Depth Information
4.6 Reduced-order modelling by proper orthogonal
decomposition
The model that has been proposed in the previous sections is based on a coupling
between two expensive problems, the Poisson equation in the torso and the bidomain
equations in the heart. Even with the uncoupled schemes that have been presented,
the computational cost may be prohibitive in some situations. This is for example
the case if an inverse problem has to be solved or if several heart beats have to be
simulated. In those cases, a reduced-order model may be interesting. We propose
some preliminary results obtained with a reduced order model based on the Proper
Orthogonal Decomposition (POD) method. After briefly recalling the POD method,
we present three examples of ECGs obtained with POD. All the simulations of this
section assume a weak coupling with the torso.
4.6.1 POD in a nutshell
For convenience, we briefly recall some notions about POD in this paragraph. For
more details about this well-known method, we refer the reader to [36, 56] for in-
stance.
The basic idea is to replace the finite-element basis by a new basis that contains
the main features of the expected solution. To generate this new basis a numerical
simulation, or set of simulations, is run and some solutions u
(
t k ) ,
1
k
p (called
N , p where
“snapshots” in this context) are stored in a matrix B
=(
u
(
t 1 ) ,...,
u
(
t p )) R
N
p is the dimension of the finite-element basis. Then, the singular value decom-
position (SVD) of this matrix is computed:
USV ,
B
=
N
,
N and V
p
,
p
N
,
p
where U
R
R
are orthogonal matrices, S
R
is the diagonal
matrix of the singular values ordered by decreasing value.
The first N modes POD basis functions
{ Ψ
}
N modes are then given by the first
N modes columns of U , and the POD Galerkin problem is solved by looking for a
solution of the type
i
1
i
N modes
i = 1 α i ( t ) Ψ i .
u
=
The N
N sparse system of the finite-element method is thus replaced by a full
system of size N modes ×
×
N modes with the POD method. To give a rough idea, it is
generally possible to achieve a good accuracy for the problems at hand with N modes
100. With the time scheme used in this work and described by the procedure (4.15)-
(4.16), the matrix is constant in time, since all the nonlinearities are treated explicitly.
The matrix is therefore projected on the POD basis and factorized only once at the
beginning of the computation. As a consequence, for the simulations presented in
this section, the reduced-order model resolution is about one order of magnitude
faster than the full order one.
Search WWH ::




Custom Search