===affil2:
Sandia National Laboratories
===firstname:
Eric
===firstname4:
Mike
===firstname3:
Mike
===lastname2:
Cyr
===lastname:
Phipps
===firstname5:

===affil6:

===lastname3:
Eldred
===email:
etphipp@sandia.gov
===lastname6:

===affil5:

===otherauths:

===lastname4:
Parks
===affil4:
Sandia National Laboratories
===lastname7:

===affil7:

===firstname7:

===postal:
Sandia National Laboratories
PO Box 5800 MS-1318
Albuquerque, NM  87185
===firstname6:

===ABSTRACT:
Quantifying uncertainties in computational simulations is a key
challenge for predictive computational science.  An important task
within uncertainty quantification is the propagation of input data
uncertainty to the corresponding simulation output quantities of
interest.  While many uncertainty propagation approaches have been
investigated throughout the literature, many of these approaches
involve sampling the simulation at a prescribed set of values for the
input data.  For example, Monte Carlo methods require evaluation of
the simulation at random realizations of the input data, whereas
stochastic collocation and non-intrusive polynomial chaos methods
require evaluations on either structured or unstructured grids.  In
each of these cases, a large number of samples can be required when
high accuracy is required, the space of uncertain input data has high
dimension, or the simulation quantities of interest lack regularity
with respect to the uncertain input data.  If the simulations
themselves are computationally expensive, the large number of samples
can lead to intractable uncertainty quantification problems.  Thus
significant improvements can be made by reducing the number of samples
required and reducing the cost of evaluating the simulation on an
ensemble of input data realizations.

To this end, we pursue two approaches to reducing the cost of
sampling-based uncertainty propagation methods.  The first is adjoint
enhancement whereby gradients of quantities of interest are computed
alongside response values.  Often the gradient with respect to an
arbitrarily large number of input parameters can be computed in a
small multiple of the time required for a single simulation, if the
simulation code has an intrusive adjoint propagation capability.  The
challenge is designing uncertainty propagation approaches that
incorporate gradient information effectively to reduce the overall
number of samples.  We present several techniques based on gradient
enhancement of global orthogonal polynomial expansions and local and
global interpolants, where the gradients are
computed by automatic differentiation techniques.  These
adjoint-enhanced approaches can then be embedded within adaptive
collocation procedures that estimate anisotropy or sparsity.

Second, we investigate reducing the cost of computing multiple
simulation samples by propagating batches of them together.  
For large-scale simulation codes involving Newton-type nonlinear
solvers for both steady-state and implicit time integration, 
this involves computing a sequence of residual vectors and
Jacobian matrices for each sample, and then solving each
linear system. For these methods, we leverage
template-based generic programming techniques to intrusively evaluate
multiple residuals/Jacobians together.  This can significantly improve
vectorization and data locality making the cost of propagating $N$
samples significantly less than $N$ times the cost of one sample.
Furthermore,  because the Jacobian matrices are relatively small perturbations from one another, we explore reusing a single preconditioner over 
a range of samples. This amortizes the cost of preconditioner
construction which can be a significant contribution for large-scale
computations.
Finally, we apply Krylov basis recycling to accelerate convergence for families of linear systems.  Recycling solvers target the solution of slowly changing sequences of linear systems, and reuse information generated from solving previous systems in the sequence to accelerate convergence for the next system in the sequence.

To demonstrate the effectiveness of these techniques, they are applied
to large-scale uncertainty propagation problems in fluid dynamics that are representative of a class of challenging problems
in multi-physics simulation. For solving the linear systems arising from
discretizing the fluid dynamics problem, we pursue application of 
block-preconditioners which have good parallel scalability properties but are expensive to construct.
===affil3:
Sandia National Laboratories
===lastname5:

===affilother:

===title:
Scalable Uncertainty Quantification Through Simultaneous Propagation, Krylov Subspace Recycling and Adjoint Enhancement
===firstname2:
Eric