quantification of uncertainty in extreme scale ... · uncertainty is represented using probability...
Post on 05-Jul-2020
5 Views
Preview:
TRANSCRIPT
Introduction SW Alg Closure
Quantification of Uncertaintyin Extreme Scale Computations
www.quest-scidac.org
Habib N. Najm
Sandia National Laboratories,Livermore, CA
2014 SciDAC-3 PI MeetingJuly 30 – August 1, 2014
Washington, DC
SNL Najm QUEST 1 / 24
Introduction SW Alg Closure
Acknowledgement
QUEST Team:
SNL M. Eldred, B. Debusschere, J. Jakeman,K. Chowdhary, C. Safta, K. Sargsyan
USC R. Ghanem
Duke O. Knio, O. Le Maı̂tre, J. Winokur
UT O. Ghattas, R. Moser, C. Simmons, A. AlexanderianT. Bui-Thanh, N. Petra, G. Stadler
LANL D. Higdon, J. Gattiker
MIT Y. Marzouk, P. Conrad, T. Cui, A. Gorodetsky
This work was supported by:
US Department of Energy (DOE), Office of Advanced Scientific ComputingResearch (ASCR), Scientific Discovery through Advanced Computing (SciDAC)
Sandia National Laboratories is a multiprogram laboratory operated by Sandia Corporation, a Lockheed MartinCompany, for the United States Department of Energy under contract DE-AC04-94-AL85000.
SNL Najm QUEST 2 / 24
Introduction SW Alg Closure
Outline
1 Introduction
2 Progress Highlights – SoftwareDAKOTAQUESO
3 Progress Highlights – AlgorithmsSparsityRandom FieldsAdaptive Sparse QuadratureAsympotoically Exact MCMC
4 Closure
SNL Najm QUEST 3 / 24
Introduction SW Alg Closure
Why UQ? Why in SciDAC?
Why UQ?Assessment of confidence in computational predictionsValidation and comparison of scientific/engineering modelsDesign optimizationUse of computational predictions for decision-supportAssimilation of observational data and model construction
Why UQ in SciDAC?Explore model response over range of parameter variationEnhanced understanding extracted from computationsParticularly important given cost of SciDAC computations
SNL Najm QUEST 4 / 24
Introduction SW Alg Closure
Uncertainty Quantification and Computational Science
!"#$%&'(")'*+,"-.*+
y = f(x)
/)$%&0+
1%&$%&+
23.-45(")0+
2'3'#.&.30+
!"#$%&'(")'*+,"-.*+
y = f(x) x+ y+
Forward problem
SNL Najm QUEST 5 / 24
Introduction SW Alg Closure
Uncertainty Quantification and Computational Science
!"#$%&'(")'*+,"-.*+
y = f(x)
/)$%&0+
1%&$%&+
23.-45(")0+
2'3'#.&.30+
,.'0%3.#.)&+,"-.*+
z = g(x)
+
,.'0%3.#.)&+,"-.*+
z = g(x)
!"#$%&'(")'*+,"-.*+
y = f(x) x+ y+
6'&'+
Inverse & Forward problems
SNL Najm QUEST 5 / 24
Introduction SW Alg Closure
Uncertainty Quantification and Computational Science
!"#$%&'(")'*+,"-.*+
y = f(x)
/)$%&0+
1%&$%&+
23.-45(")0+
2'3'#.&.30+
,.'0%3.#.)&+,"-.*+
z = g(x)
+
,.'0%3.#.)&+,"-.*+
z = g(x)
!"#$%&'(")'*+,"-.*+
y = f(x) x+ y+
6'&'+ zd+
Inverse & Forward UQ
SNL Najm QUEST 5 / 24
Introduction SW Alg Closure
Uncertainty Quantification and Computational Science
!"#$%&'(")'*+,"-.*+
y = f(x)
/)$%&0+
1%&$%&+
23.-45(")0+
2'3'#.&.30+
,.'0%3.#.)&+,"-.*+
z = g(x)
+
,.'0%3.#.)&+,"-.*+
z = g(x)
!"#$%&'(")'*+,"-.*+
y = f(x) x+ y+
6'&'+ zd+
yd+
y ={f1(x), f2(x)7+87+fM(x)}+
Inverse & Forward UQModel validation & comparison, Hypothesis testing
SNL Najm QUEST 5 / 24
Introduction SW Alg Closure
Key Elements of our UQ strategy
Probabilistic frameworkUncertainty is represented using probability theory
Parameter Estimation, Model CalibrationExperimental measurementsRegression, Bayesian Inference
– Markov Chain Monte Carlo (MCMC) methods
Forward propagation of uncertaintyPolynomial Chaos (PC) Stochastic Galerkin methods
– Intrusive/non-intrusiveStochastic Collocation methods
Model comparison, selection, and validationExperimental design and uncertainty management
SNL Najm QUEST 6 / 24
Introduction SW Alg Closure
QUEST UQ Software tools
DAKOTA: Optimization and calibration; non-intrusive UQ;global sensitivity analysis; ∼10K registered downloads.
QUESO: Bayesian inference; multichain MCMC; modelcalibration and validation; decision under uncertainty.
GPMSA: Bayesian inference; Gaussian processemulation; model calibration; model discrepancy analysis.
UQTk: Intrusive and non-intrusive forward PC UQ; customsparse PCE; random fields.
MUQ: Adaptive forward PC UQ; advanced MCMC andvariational methods for inference; efficient surrogates.
SNL Najm QUEST 7 / 24
Introduction SW Alg Closure DAKOTA QUESO
DAKOTA – Recent Developments – dakota.sandia.gov
High-dimensionality:Sparse representations
Memory conserving approaches to high-dimensionalcompressed sensing and variance-based decompositionMultifidelity compressed sensing
PCE regression with high dimensional basis adaptation.Model Complexity:
Orthogonal least interpolationTail probability estimation – adaptive importance samplingImproved response QoI scalability
Software Integration:Bayesian calibration with QUESO/GPMSA/DREAM
Architecture:Dynamic multi-level job schedulers (MPI & hybrid)
SNL Najm QUEST 8 / 24
Introduction SW Alg Closure DAKOTA QUESO
QUESO – Recent Developments UT Austin
Migration to Github; expanded user base significantlyhttps://github.com/libqueso/queso
Software quality and usability improvementsFull user documentation and a large number of examplesDeveloper documentation in developmentQUESO-Dakota interface
Ongoing effort to add Gaussian process (GP) basedemulation capabilities to QUESOUsing GPMSA as a referenceEnabling Dakota to access such new capabilities in QUESO
Inference of random fieldsInitial support for fault tolerant samplingInitial support for heterogenous architectures
SNL Najm QUEST 9 / 24
Introduction SW Alg Closure Sparsity RF ASQ MCMC
Sparsity and Compressive Sensing
Many physical models have a large # of uncertain inputsUQ in this high-dimensional setting is a majorcomputational challenge
too many samples and/or large # PC modesYet physical models typically exhibit sparsity
A small number of inputs are importantSeek sparse PC representation on input space
Small number of dominant terms
Compressed sensing (CS) is useful for discovering sparsityin high dimensional modelsIdentify terms that contribute most to model outputvariationIdeal for when data is limited
SNL Najm QUEST 10 / 24
Introduction SW Alg Closure Sparsity RF ASQ MCMC
Sparse Representations – developments SNL
CS algorithms have been developed for under-determinedsolutions of the coefficients of PC expansions (PCEs)
basis pursuit, basis pursuit denoisingorthogonal matching pursuitleast angle regressionleast absolute shrinkage and selection operator (LASSO).
Orthogonal least interpolation (OLI)determines the lowest order PCE that can interpolate agiven (unstructured) data set.
New capabilities include:support for gradient (adjoint) enhancementfault tolerancecross-validation of algorithm parameterseither structured (sub-sampled tensor product) orunstructured (Latin hypercube) grids
SNL Najm QUEST 11 / 24
Introduction SW Alg Closure Sparsity RF ASQ MCMC
Adaptive Basis Selection
Set 1 Set 2 Set 3
Cardinality of total degree basis grows factorially with thenumber of uncertain inputs.Even for lower dimensional problems redundant basisterms can degrade accuracyTo reduce redundancy and improve accuracy the PCEtruncation can be chosen adaptively.
SNL Najm QUEST 12 / 24
Introduction SW Alg Closure Sparsity RF ASQ MCMC
Random Fields – Relevance
Many applications involve uncertain inputs/outputs thathave spatial or time dependenceSuch an uncertain function, represented probabilistically, isa random field/process.
It is a random variable at each space/time locationGenerally with some correlation structure in space/timeAn infinite-dimensional object
The Karhunen Loeve expansion (KLE) provides an optimalrepresentation of random fields, employing a (small)number of eigenmodes of its covariance function
SNL Najm QUEST 13 / 24
Introduction SW Alg Closure Sparsity RF ASQ MCMC
Random Fields – sparse data SNL
Developed a Bayesian procedure for KLE constructiongiven sparse data
Bayesian Principal Component Analysis (BPCA)Address challenges arising due to
approximate knowledge of the covariance matrixlack of positive definiteness of sample covariance matrix
BPCA framework explores the space of orthonormalvectors, seeking those that best explain the data
Likelihood density p(Φ) is peaked at
Φ∗ = arg minΦ∈Vk(Rd)
n∑i=1
‖xi − PΦxi‖2
where Vk(Rd) is the space of k orthonormal d-dimensional vectors
Resulting KLE incorporates uncertainty due to smallnumber of samples
SNL Najm QUEST 14 / 24
Introduction SW Alg Closure Sparsity RF ASQ MCMC
BPCA Example – Data from a 3D MVN
15 10 5 0 5 10 15
105
05
10
15
10
5
0
5
10
15
Samples of random variables from a3D Multivariate Normal (MVN) distribution
1.00.5
0.00.5
1.0
0.50.00.5
0.5
0.0
0.5
Samples from p(Φ) using 100 samples, xi
1.00.5
0.00.5
1.00.5
0.00.5
0.5
0.0
0.5
First two principal components.Black is the vector with maximum variance
1.00.5
0.00.5
1.0
0.50.00.5
0.5
0.0
0.5
Samples from p(Φ) using 300 samples, xi
SNL Najm QUEST 15 / 24
Introduction SW Alg Closure Sparsity RF ASQ MCMC
BPCA Example – Brownian motion – 25 samples
500-dimensional Brownianmotion stochastic process.Using only 25 samples, wecompute samples from p(Φ) andplot the first three principalcomponents.The dark solid lines represent theprincipal components and theshaded region represents errorbars based on samples using theBayesian PCA approach.
SNL Najm QUEST 16 / 24
Introduction SW Alg Closure Sparsity RF ASQ MCMC
BPCA Example – Brownian motion – 250 samples
Using 250 samplesModes are evaluated withimproved accuracyLower uncertainty
SNL Najm QUEST 17 / 24
Introduction SW Alg Closure Sparsity RF ASQ MCMC
Random Fields – large scale NOAA data – SVD
KLE for uncertain Sea Surface Temperature (SST)1/4-degree spatial resolution data
106-dimensional random field encompassing spatial andtemporal uncertainty in SST dataSVD using Trilinos / parallelized block Krylov Schur solverHopper / NERSC implementation
Mean SST 1st KL modeSNL Najm QUEST 18 / 24
Introduction SW Alg Closure Sparsity RF ASQ MCMC
Adaptive Sparse Quadrature (ASQ) for UQ Duke/MIT
Non-Intrusive Pseudospectral projection using sparsetensorization of 1-D quadrature formulas:
prevent internalaliasingimprove accuracyreduce number ofsimulations
Accuracy Requirement Comparison
Final projection exactness requirements are significantly reduced!
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 200123456789
1011121314151617181920
j1 orderj 2 o
rder
Polynomial IndexRequired Quadrature Exactness
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 200123456789
1011121314151617181920
j1 order
j 2 ord
er
Polynomial IndexRequired Quadrature Exactness
In this schematic example, we set the polynomial tensor K and prescribedL. In practice we set L which prescribes K
• A PSP with L has the same realization stencil as a quadrature builtwith L but admits more polynomials than a direct projection
• Especially high-order monomial termsJustin Winokur Hierarchical Sparse Adaptive Sampling 8 / 34
Adaptivity:progressive construction by introducing new tensorizationswith cost controlrobust error indicator to guide the adaptation processnested hierarchical approximation (local dimension-wiseerror control)
SNL Najm QUEST 19 / 24
Introduction SW Alg Closure Sparsity RF ASQ MCMC
UQ with ASQ – Ocean Dynamics Simulation
Example of application: uncertainty in subgrid mixing and windcoupling parameterization (4-dimensions) in hurricane Ivansimulations (. 400 realizations)Variance(analysis(
! Based(on:(" mean(SST(in(“skill”(region;(and(" heat(flux((energy(uptake)(beneath(the(hurricane(
−95 −90 −85 −80
20
22
24
26
28
30
Longitude
Latit
ude
At t = 150, ⟨SST⟩ = 28.04
25
26
27
28
29
30
Longitude
Latit
ude
At t = 150, ⟨Q⟩ = -362.70
150150
250
250
250 350
350
350
450
450450
450
550
550
550
550
550
650
650
650650
650
650 750
750
750750
750
750
850
850
850
850
850
850
950
950
950
950
950
950
−95 −90 −85 −80
20
22
24
26
28
30
−800
−600
−400
−200
0
SNL Najm QUEST 20 / 24
Introduction SW Alg Closure Sparsity RF ASQ MCMC
Asymptotically Exact MCMC – I MIT
Forward UQ yields useful surrogates for BayesianinferenceYet surrogates should be most accurate in regions of highposterior probabilityWe have developed a new approach for incrementallyconstructing local approximations during MCMC
earlier times later timesSNL Najm QUEST 21 / 24
Introduction SW Alg Closure Sparsity RF ASQ MCMC
Asymptotically Exact MCMC – II
Algorithm applies approximate MCMC transition kernels,but is provably ergodic with respect to the exact posteriorProbability of evaluating the full forward model during agiven MCMC iteration approaches zeroSpeedups of several orders of magnitude over directMCMC sampling
Applied to large-scaleinference problem with ablack-box forward model :MITgcm for ice-oceandynamics in the WestAntarctic Ice Sheet (with P.Heimbach, MIT)
Code available in the latestrelease of MUQ
102oW 30’ 101oW 30’ 100oW 30’
24’
12’
75oS
48’
36’
0
100
200
300
400
500
600
700
800
900
1000
Satellite image and sample locations
SNL Najm QUEST 22 / 24
Introduction SW Alg Closure Sparsity RF ASQ MCMC
Asymptotically Exact MCMC – III
Elliptic PDE inverse problem: ∇ · (κ(x)∇u(x)) = −fInfer permeability field κ(x) from limited/noisy observationsof pressure u
104 105
MCMC step
10-3
10-2
10-1
100
101
Rela
tive c
ovari
ance
err
or
True model
Linear
Quadratic
GP
Accuracy of chains
104 105
MCMC step
102
103
104
105
Tota
l num
ber
of
evalu
ati
ons True model
Linear
Quadratic
GP
Cost of chains
Only 300 model evaluations needed for 105 MCMC samples!
SNL Najm QUEST 23 / 24
Introduction SW Alg Closure
Closure
Highlights of recent progressSoftwareAlgorithms
Refining and robustifying QUEST algorithms and softwareto address UQ challenges in large-scale problems
high dimensionalitylarge range of scalescomplex models and high computational cost
Addressing UQ needs of SciDAC application partnershipsEight funded active partnerships
Read more at: quest-scidac.org
SNL Najm QUEST 24 / 24
top related