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Houman Owhadi
Principles and Methods of UQA minitutorial, Part I
IPAM Sep 13-14, 2017
DARPA EQUiPS / AFOSR award no FA9550-16-1-0054(Computational Information Games)
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Syllabus
• Principles and method of UQ
• Applications to computation with partial information• Complex energy landscapes (can be observed only at a finite
number of points)• Coarse graining, multiresolution analysis/modeling• Fast eigensubspace projections• Wannier functions• Scalable numerical methods
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3 main approaches to UQ
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The worst case approach ``When in doubt, assume the worst!’’ Au†: Unknown element of A
Robust Optimization/ Min and Max approach
Φ : A −→ Ru −→ Φ(u) Quantity of Interest
What is Φ(u†)?
infu∈AΦ(u) ≤ Φ(u†) ≤ supu∈A Φ(u)
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I. Elishakoff and M. Ohsaki. Optimization and Anti-Optimization of Structures Under Uncertainty. World Scientific, London, 2010.
Saltelli, A.; Ratto, M.; Andres, T.; Campolongo, F.; Cariboni, J.; Gatelli, D.; Saisana, M.; Tarantola, S. (2008). Global Sensitivity Analysis: The Primer. John Wiley & Sons.
Global Sensitivity Analysis
A. Ben-Tal, L. El Ghaoui, and A.Nemirovski. Robust optimization. PrincetonSeries in Applied Mathematics. Princeton University Press, Princeton, NJ, 2009
D. Bertsimas, D. B. Brown, and C. Caramanis. Theory and applications of robustoptimization. SIAM Rev., 53(3):464–501, 2011
Robust Optimization
A. Ben-Tal and A. Nemirovski. Robust convex optimization. Math. Oper. Res.,23(4):769–805, 1998
Optimal Uncertainty QuantificationH. Owhadi, C. Scovel, T. J. Sullivan, M. McKerns, and M. Ortiz. Optimal Uncertainty Quantification. SIAM Review, 55(2):271–345, 2013.
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Stochastic linear programming and Stochastic Optimization
P. Kall. Stochastric programming with recourse: upper bounds and moment problems:a review. Mathematical research, 45:86–103, 1988
A. Madansky. Bounds on the expectation of a convex function of a multivariaterandom variable. The Annals of Mathematical Statistics, pages 743–746, 1959
G. B. Dantzig. Linear programming under uncertainty. Management Sci., 1:197–206, 1955.
Y. Ermoliev, A. Gaivoronski, and C. Nedeva. Stochastic optimization problemswith incomplete information on distribution functions. SIAM Journal on Controland Optimization, 23(5):697–716, 1985
C. C. Huang, W. T. Ziemba, and A. Ben-Tal. Bounds on the expectation of a convexfunction of a random variable: With applications to stochastic programming.Operations Research, 25(2):315–325, 1977.
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Chance constrained/distributionally robust optimization
J. Goh and M. Sim. Distributionally robust optimization and its tractable approximations. Oper. Res., 58(4, part 1):902–917, 2010
R. I. Bot N. Lorenz, and G. Wanka. Duality for linear chance-constrained optimizationproblems. J. Korean Math. Soc., 47(1):17–28, 2010.
L. Xu, B. Yu, and W. Liu. The distributionally robust optimization reformulationfor stochastic complementarity problems. Abstr. Appl. Anal., pages 7, 2014.
S. Zymler, D. Kuhn, and B. Rustem. Distributionally robust joint chance constraintswith second-order moment information. Math. Program., 137(1-2, Ser.A):167–198, 2013.
A. A. Gaivoronski. A numerical method for solving stochastic programming problemswith moment constraints on a distribution function. Annals of OperationsResearch, 31(1):347–369, 1991.
G. A. Hanasusanto, V. Roitch, D. Kuhn, and W. Wiesemann. A distributionallyrobust perspective on uncertainty quantification and chance constrained programming.Mathematical Programming, 151(1):35–62, 2015
Value at Risk
Artzner, P.; Delbaen, F.; Eber, J. M.; Heath, D. (1999). Coherent Measures of Risk. Mathematical Finance 9 (3): 203.
W. Chen, M. Sim, J. Sun, and C.-P. Teo. From CVaR to uncertainty set: implicationsin joint chance-constrained optimization. Oper. Res., 58(2):470–485, 2010.
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Set based design in the aerospace industry
Bernstein JI (1998) Design methods in the aerospace industry: looking for evidence of set-based practices. Master’s thesis. Massachusetts Institute of Technology, Cambridge, MA.
Set based design/analysis
B. Rustem and Howe M. Algorithms for Worst-Case Design and Applications toRisk Management. Princeton University Press, Princeton, 2002.
David J. Singer, PhD., Captain Norbert Doerry, PhD., and Michael E. Buckley,” What is Set-Based Design? ,” Presented at ASNE DAY 2009, National Harbor, MD., April 8-9, 2009. Also published in ASNE Naval Engineers Journal, 2009 Vol 121 No 4, pp. 31-43.
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F min( Yield Strain - Axial Strain )
Ground Acceleration
We want to certify that
Seismic Safety Assessment of a Truss Structure
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Power Spectrum
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
0.1 0.6 1.1 1.6 2.1 2.6 3.1 3.6 4.1 4.6 5.1 5.6 6.1 6.6 7.1 7.6 8.1 8.6 9.1 9.6
Frequency
Am
plitu
de
Mean PSObserved PS
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Power Spectrum
0
0.2
0.4
0.6
0.8
1
1.2
1.4
0.1 0.6 1.1 1.6 2.1 2.6 3.1 3.6 4.1 4.6 5.1 5.6 6.1 6.6 7.1 7.6 8.1 8.6 9.1 9.6
Frequency
Am
plitu
de
Mean PSObserved PS
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Power Spectrum
0
0.2
0.4
0.6
0.8
1
1.2
1.4
0.1 0.6 1.1 1.6 2.1 2.6 3.1 3.6 4.1 4.6 5.1 5.6 6.1 6.6 7.1 7.6 8.1 8.6 9.1 9.6
Frequency
Am
plitu
de
Mean PSObserved PS
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Filtered White Noise Model
White noise
Ground acceleration
Filter0
0.2
0.4
0.6
0.8
1
1.2
1.4
0.1 0.6 1.1 1.6 2.1 2.6 3.1 3.6 4.1 4.6 5.1 5.6 6.1 6.6 7.1 7.6 8.1 8.6 9.1 9.6
Frequency
Am
plitu
de
Mean PSObserved PS
N. Lama, J. Wilsona, and G. Hutchinsona. Generation of synthetic earthquake accelogramsusing seismological modeling: a review. Journal of Earthquake Engineering, 4(3):321–354, 2000.
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Vulnerability Curves (vs earthquake magnitude)
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Identification of the weakest elements
H. Owhadi, C. Scovel, T. J. Sullivan, M. McKerns, and M. Ortiz. Optimal Uncertainty Quantification. SIAM Review, 55(2):271–345, 2013.
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P. L. Chebyshev1821-1894
M. G. Krein1907-1989
A. A. Markov1856-1922
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Answer
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Answer
Markov’s inequality
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Bayesian Approach
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Compute
Numerical Analysis ApproachP. Diaconis
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Compute
Bayesian Approach
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E.g.
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E.g.
E.g.
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− div(a∇u) = g, x ∈ Ω,u = 0, x ∈ ∂Ω,
(1)
ai,j ∈ L∞(Ω)Ω ⊂ Rd ∂Ω is piec. Lip.
a unif. ell.
Approximate the solution space of (1)with a finite dimensional space
Q
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Numerical Homogenization Approach
HMM
Harmonic Coordinates Babuska, Caloz, Osborn, 1994Allaire Brizzi 2005; Owhadi, Zhang 2005
Engquist, E, Abdulle, Runborg, Schwab, et Al. 2003-...
MsFEM [Hou, Wu: 1997]; [Efendiev, Hou, Wu: 1999]
Nolen, Papanicolaou, Pironneau, 2008
Flux norm Berlyand, Owhadi 2010; Symes 2012
Kozlov, 1979
[Fish - Wagiman, 1993]
Projection based method
Variational Multiscale Method, Orthogonal decomposition
Work hard to find good basis functions
Harmonic continuation
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Bayesian Approach
Put a prior on g
Compute E u(x) finite no of observations
Proposition
− div(a∇u) = g, x ∈ Ω,u = 0, x ∈ ∂Ω,
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Replace g by ξ
ξ: White noiseGaussian field with covariance function Λ(x, y) = δ(x− y)
⇔ ∀f ∈ L2(Ω),RΩf(x)ξ(x) dx is N
¡0, kfk2L2(Ω)
¢
Bayesian approach
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Theorem
Letx1, . . . , xN ∈ Ω Ω
xNxi
x1
a = Id
ai,j ∈ L∞(Ω)
[Harder-Desmarais, 1972]
[Duchon 1976, 1977,1978]
[Owhadi-Zhang-Berlyand 2013]
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Theorem
Standard deviation of the statistical error bounds/controls the worst case error
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Pioneering work
“ These concepts and techniques have attracted little attention among numerical analysts” (Larkin, 1972)
Bayesian/probabilistic approach to numerical approximation not new but appears to have remained overlooked
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Bayesian Numerical Analysis
P. Diaconis A. O’ Hagan J. E. H. Shaw
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Information based complexity
H. Wozniakowski G. W. Wasilkowski J. F. Traub E. Novak
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Probabilistic Numerical Methods
Statistical Inference approaches tonumerical approximation and algorithm design
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The Game theoretic approach
John Von Neumann John Nash
J. Von Neumann. Zur Theorie der Gesellschaftsspiele. Math. Ann., 100(1):295–320,1928
J. Von Neumann and O. Morgenstern. Theory of Games and Economic Behavior.Princeton University Press, Princeton, New Jersey, 1944.
N. Nash. Non-cooperative games. Ann. of Math., 54(2), 1951.
Abraham Wald
A. Wald. Statistical decision functions which minimize the maximum risk. Ann.of Math. (2), 46:265–280, 1945.
A. Wald. An essentially complete class of admissible decision functions. Ann.Math. Statistics, 18:549–555, 1947.
A. Wald. Statistical decision functions. Ann. Math. Statistics, 20:165–205, 1949.
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Find the best climate model and incorporate data
Problem
Find a 95% interval of confidence on average global temperatures in 50 years
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E(candidate,model(data))
Player I Player IIChooses candidate Sees data
Chooses model
Framework: Game/Decision Theory
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Player I
Player II
3
1
-2
-2
Deterministic zero sum game
Player I’s payoff
How should I & II play the (repeated) game?
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Player I
Player II
3
1
-2
-2
II should play blue and lose 1 in the worst case
Worst case approach
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Player I
Player II
3
1
-2
-2
Worst case approach
I should play red and lose 2 in the worst case
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Player I
Player II
3
1
-2
-2
No saddle point
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1/2 -1/2
Player II
Average case (Bayesian) approach
3
1
-2
-2
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Mixed strategy (repeated game) solution
3
1
-2
-2II should play red with probability 3/8 and win 1/8 on average
Player II
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Mixed strategy (repeated game) solution
3
1
-2
-2I should play red with probability 3/8 and lose 1/8 on average
Player I
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Game theory
John Von Neumann
Player I
Player II
3
1
-2
-2
q 1− q
p
1− p
Optimal strategies are mixed strategies
Optimal way toplay is at random
Saddle point
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The optimal mixed strategy is determined by the loss matrix
Player I5
1
-2
-2
p
1− p
Player II
II should play red with probability 3/10 and win 1/8 on average
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E(candidate,model(data))
Player I Player IIChooses candidate Sees data
Chooses model
Framework: Game/Decision Theory
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statistical decision theory
Abraham WaldThe best strategy is to play at randomObtained by finding the worst prior inthe Bayesian class of estimators
Incorporation of data naturally done via conditioning
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Non Bayesian
Bayesian
Complete class theorem
Risk
Prior
EstimatorNon cooperative Minmax loss/error
cooperative Bayesian loss/error
Over-estimate risk
Under-estimate risk
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Application to energy landscapes
Phase space
Energy
You can only compute energyat a finite number of points
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Phase space
Energy
Application to energy landscapes
How to quantify the remaining uncertainty on that energy landscape and how to use that quantification?
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Phase space
Energy
Application to energy landscapes
How to quantify the remaining uncertainty on that energy landscape and how to use that quantification?
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Application to energy landscapes
Phase space
Energy
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Φx = yBased on the information that
Φ: Known m× nrank m matrix (m < n)
y: Known element of Rm
A simple approximation problem
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3 approaches to Numerical Approximation
3 approaches to inference andto dealing with uncertainty
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Worst case approach (Optimal Recovery)
Problem
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Solution
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Average case approach (IBC)
Problem
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Solution
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Player I Player II
Max Min
Adversarial game approach
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Loss function
Player I
Player II
No saddle point of pure strategies
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Player I Player II
Max Min
Randomized strategy for player I
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Loss function
Saddle point
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Canonical Gaussian field
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Equilibrium saddle point
Player I
Player II
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Statistical decision theory
Abraham WaldA. Wald. Statistical decision functions which minimize the maximum risk. Ann.of Math. (2), 46:265–280, 1945.
A. Wald. An essentially complete class of admissible decision functions. Ann.Math. Statistics, 18:549–555, 1947.
A. Wald. Statistical decision functions. Ann. Math. Statistics, 20:165–205, 1949.
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The game theoretic solution is equal to the worst case solution
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Generalization
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Restriction Interpolation
Problem
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Application to energy landscapes
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Application to energy landscapes
L
Choose the norm in which you want optimality
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Player I Player II
Max Min
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Examples
Player I Player II
Player I Player II
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Application to energy landscapes
Player I Player II
MaxMin
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Loss function
Player I
Player II
No saddle point of pure strategies
Need to lift the minimax to mixed strategies
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Player I Player II
Max Min
Randomized strategy for player I
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Loss function
Theorem
What is the saddle point?
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Theorem
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Generalization
Canonical Gaussian field
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Examples
L
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Application to energy landscapes
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Theorem
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Optimal bet of player II
Theorem
The bet is both repeated game and worst case optimal
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Variational formulation of the optimal bet
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Kernel formulation of the optimal bet
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Application to energy landscapes
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Best guess
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Variational and Kernel interpretation
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Optimal bet of player II
Theorem
Error of the recovery
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Application to energy landscapes
LTheorem
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Uncertainty Quantification
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The standard deviations controls the deterministic error
Theorem
Proof
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Optimal bet of player II
Gamblets
Optimal recovery splines
ψi: Unique minimizer of
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Example
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Example
Ω
xi
x1
xm
φi(x) = δ(x− xi)
ψi: Polyharmonic splines[Harder-Desmarais, 1972][Duchon 1976, 1977,1978]
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Example
Ω
xi
x1
xm
φi(x) = δ(x− xi)
ψi: Rough Polyharmonic splines
[Owhadi-Zhang-Berlyand 2013]
ai,j ∈ L∞(Ω)
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Thank you
DARPA EQUiPS / AFOSR award no FA9550-16-1-0054(Computational Information Games)