PCE with many inputs

UQ with UQLab Community Q&A and How To
1

Hello all,
I have a model with 44 input parameters and I have about 2700 FEM simulations. However, for some of my outputs (I have different outputs based for different frequencies), my PCE leave-one-out error is about 20%. I am creating PCE up to degree of 4 and I cannot go beyond that because of computational cost.

  • Do you think increasing the number of simulations will help? If yes, how much should I increase it?
  • Do you think reducing the coefficient of variation in the input parameters can be helpful?
  • Do you think there is any benefit in using PCK rather than PCE?
  • Do you know any methods to accurately determine unimportant parameters to remove them and reduce the dimension before creating PCE?
  • Any other suggestions?
    Unfortunately, I cannot publicly provide the data.

Your help is greatly appreciated.

2

@Aep93

Your number of simulations (N=2700) for d=44 dimensions is not so low, so you should be able to build a good polynomial chaos expansion surrogate model. I’m not sure you used the right options though, especially in terms of sparse PCE:

  • you should use ‘LARS’ as solver
  • you can specify a range of maximal degrees, say 3:10 instead of a single one: LARS will test all of them and keep the best.
  • for this high dimension, you should use a q-norm truncation scheme, which allows you to select a much smaller candidate basis before LARS is run.
    I suggest something like this:
MetaOpts.Method = 'LARS';
MetaOpts.Degree = 3:10;            
Metaopts.TruncOptions.qNorm = 0.3:0.1:0.8;
MetaOpts.ExpDesign.X =    ;      % your experimental design (X)
MetaOpts.ExpDesign.Y =    ;      % your experimental design (Y)
myPCE_LARS = uq_createModel(MetaOpts);

You can also limit the interactions terms in your polynomial basis (somehow similar effect as a low q=0.3 or so, depending on the dimension) wit MetaOpts.TruncOptions.MaxInteraction = 2;. In most cases there are only low interaction, but you need much higher univariate degrees to get a good accuracy.

From the obtained PCE, you can easily get the Sobol indices:

SobolOpts.Type = 'Sensitivity';
SobolOpts.Method = 'Sobol';
SobolFromPCE = uq_createAnalysis(SobolOpts);    
% if you put this right after creating the PCE, the latter is used 
% to compute Sobol' indices analytically.

I hope it will solve your problem. Regarding the data: the nice thing is that you could post anonymized data easily: what is needed a N \times d array of input parameters and the corresponding outputs. We don’t need to know what the parameters stand for (call them X_1, \dots , X_{44}). If the inputs are uniform distributions, they could be easily normalized to [0,1]. This way you can get targeted help from the community !

Best regards
Bruno

3 Likes
3

Thank you very much @bsudret for your detailed explanations. I want to say that I am still working on this and I will update you when I come to a conclusion.
Thanks and regards