Use and optimization of the Proper Generalized
Decomposition (PGD) to solve complex problems
Name of the supervisor: L. Chamoin ([email protected])
Context : Nowadays, numerous scientific problems remain intractable using classical
numerical techniques as they involve a huge number of unknowns of parameters. This is for
instance the case for quantum mechanics problems, or for structures mechanics problems with
uncertain data. This phenomenon is currently denoted “curse of dimensionality”. However, a
very promising approach, called Proper Generalized Decomposition (PGD), is intensely
studied to circumvent the issue; it consists in a separated variables decomposition of the
solution, with modes, of the form (for a space-time problem):
In many applications, few modes are necessary to obtain a very accurate (and with a
reasonable computational cost) numerical solution (cf. Fig1).
space function s
space function s
space function s
space function s
3
2
3
2
1.5
2
2.5
1
1
1
0.5
2
0
0
0
ï0.5
1.5
ï1
ï1
ï1
ï2
ï1.5
1
ï2
ï2
0.5
ï3
ï2.5
ï4
ï3
ï3
0
ï3
0.14
0
0.015
0.12
ï0.005
0.01
0.1
ï0.01
0.005
0.08
ï0.015
0
8
x 10
6
4
h
h
h
h
2
ï0.02
0.06
0
ï0.005
ï2
0.04
ï0.025
ï0.01
0.02
ï0.03
ï0.015
0
ï0.035
0
1
2
3
4
5
t
6
7
8
9
10
0
1
2
3
4
5
t
6
7
8
9
10
ï0.02
ï4
ï6
ï8
0
1
2
3
4
5
6
7
8
9
10
t
0
1
2
3
4
5
6
7
8
9
10
t
Figure 1 : representation of the first four modes for the solution of a transient thermics
problem : space functions
(upper row), and time functions
(lower row).
Objectives : In this work, we aim at optimizing the performances of PGD as regards two
criteria. First, we want to set up error indicators that enable to control and adapt the PGD
representation (do we need more modes? do we need a better FE mesh to compute theses
modes?....). In a second stage, we want to optimize the PGD computation when it is dedicated
to the prediction of some quantities of interest instead rather than the global solution itself.
Schedule of the training:
1) Bibliography on PGD and application on simple test cases;
2) Calculation of error indicators to drive the adaptation process;
3) Validation of adaptation performances on numerical examples;
4) Optimization of PGD to predict quantities of interest.