Much  of  the  uncertainty  quantification  (UQ)  research  over  the  last  fifteen  year  has  given  little  attention  to  critical  problems   necessary   for   predictive   modelling   of   stochastic   multiscale   problems.     They   include   modelling   of  correlations   in   space   and   time   necessary   to   alleviate   issues   related   to   high   stochastic   dimensionality,   ability   to  perform   UQ   tasks   with   limited   data,   accounting   for   the   phenomenology   of   coarse   graining   and   selection   of  effective  variables,  and  many  more.  In  this  presentation,  we  will  advocate  the  exploration  of  synergies  between  the  machine   learning   and   uncertainty   quantification   research   communities   towards   addressing   the   aforementioned  problems.   In   particular,   we   will   present   a   data-­‐driven   probabilistic   graphical   model   based   methodology   to  efficiently  perform  uncertainty  quantification  in  multiscale  systems.  Both  the  stochastic  input  and  model  responses  are  treated  as  random  variables  in  this  framework.  Their  relationships  are  modeled  by  graphical  models  which  give  explicit  factorization  of  the  high-­‐dimensional  joint  probability  distribution.  The  hyperparameters  in  the  probabilistic  model   can   be   learned   locally   in   the   graph   using   various   techniques   including   sequential   Monte   Carlo   (SMC)  method,  EM  or  variational  methods.  The  effective  coarse  grained  variables  arise  naturally   in  the  graphical  model  and   their   marginal   distributions   can   be   computed   non-­‐parametrically   in   a   data-­‐driven   manner.   We   make  predictions  from  the  probabilistic  graphical  model  using  loopy  belief  propagation  algorithms.  Numerical  examples  will  be  presented  to  show  the  accuracy  and  efficiency  of  the  predictive  capability  of  the  developed  graphical  model  in  multiscale   fluid   flow  and  materials   simulations.  We  will   conclude  with  a  discussion  of   the  many  exciting  open  problems  and  unexplored  research  directions  

Graph  Theoretic  Models  for  the  Solution  of  Stochastic  Multiscale  Problems  

             Monday,  September  19,  2016                                            4:15  PM  –  5:15  PM    

127  Hayes-­‐Healy  Center   Colloquium Tea 3:45 PM to 4:15 PM 154 Hurley Hall

Nicholas  Zabaras  Department  of  Aerospace  &  Mechanical  

Engineering  University  of  Notre  Dame  

 

Department  of  Applied  and  Computational    Mathematics  and  Statistics  Colloquium