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Houman OwhadiThe worst case approach to UQ

“The gods to-day stand friendly, that we may,Lovers of peace, lead on our days to age!But, since the affairs of men rests still uncertain,Let’s reason with the worst that may befall.”

Julius Caesar, Act 5, Scene 1William Shakespeare (1564 –1616)

You want to certify that

Problem

and

You want to certify that

Problem

and

You only know

Worst and best caseoptimal bounds P[G(X) ≥ a]given available information.

Compute

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.

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.

P. L. Chebyshev1821-1894

M. G. Krein1907-1989

A. A. Markov1856-1922

Answer

Answer

Markov’s inequality

History of classical inequalitiesS. Karlin and W. J. Studden. Tchebycheff Systems: With Applications in Analysisand Statistics. Pure and Applied Mathematics, Vol. XV. Interscience PublishersJohn Wiley & Sons, New York-London-Sydney, 1966.

Theory of majorization

A. W. Marshall and I. Olkin. Inequalities: Theory of Majorization and its Appli-cations, volume 143 of Mathematics in Science and Engineering. Academic PressInc. [Harcourt Brace Jovanovich Publishers], New York, 1979

M. G. Krein. The ideas of P. L. Cebysev and A. A. Markov in the theory of limitingvalues of integrals and their further development. In E. B. Dynkin, editor, Elevenpapers on Analysis, Probability, and Topology, American Mathematical SocietyTranslations, Series 2, Volume 12, pages 1–122. American Mathematical Society,New York, 1959.

Classical Markov-Krein theorem and classical works of Krein, Markov and Chebyshev

Connections between Chebyshev inequalities and optimization theoryH. J. Godwin. On generalizations of Tchebychef’s inequality. J. Amer. Statist.Assoc., 50(271):923–945, 1955.

K. Isii. On a method for generalizations of Tchebycheff’s inequality. Ann. Inst.Statist. Math. Tokyo, 10(2):65–88, 1959.

K. Isii. On sharpness of Tchebycheff-type inequalities. Ann. Inst. Statist. Math.,14(1):185–197, 1962/1963.

A. W. Marshall and I. Olkin. Multivariate Chebyshev inequalities. Ann. Math.Statist., 31(4):1001–1014, 1960.

A. W. Marshall and I. Olkin. Inequalities: Theory of Majorization and its Appli-cations, volume 143 of Mathematics in Science and Engineering. Academic PressInc. [Harcourt Brace Jovanovich Publishers], New York, 1979

I. Olkin and J. W. Pratt. A multivariate Tchebycheff inequality. Ann. Math.Statist, 29(1):226–234, 1958.

K. Isii. The extrema of probability determined by generalized moments. I. Boundedrandom variables. Ann. Inst. Statist. Math., 12(2):119–134; errata, 280, 1960.

L. Vandenberghe, S. Boyd, and K. Comanor. Generalized Chebyshev bounds viasemidefinite programming. SIAM Rev., 49(1):52–64 (electronic), 2007

Connection between Chebyshev inequalities and optimization theory

H. Joe. Majorization, randomness and dependence for multivariate distributions.Ann. Probab., 15(3):1217–1225, 1987.

J. E. Smith. Generalized Chebychev inequalities: theory and applications in decisionanalysis. Oper. Res., 43(5):807–825, 1995.

D. Bertsimas and I. Popescu. Optimal inequalities in probability theory: a convexoptimization approach. SIAM J. Optim., 15(3):780–804 (electronic), 2005.

E. B. Dynkin. Sufficient statistics and extreme points. Ann. Prob., 6(5):705–730,1978.

A. F. Karr. Extreme points of certain sets of probability measures, with applications.Math. Oper. Res., 8(1):74–85, 1983.

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

A. Madansky. Inequalities for stochastic linear programming problems. Manage-ment science, 6(2):197–204, 1960.

G. B. Dantzig. Linear programming under uncertainty. Management Sci., 1:197–206, 1955.

J. R. Birge and R. J.-B. Wets. Designing approximation schemes for stochasticoptimization problems, in particular for stochastic programs with recourse. Math.Prog. Stud., 27:54–102, 1986

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.

J. Zjackovja. On minimax solutions of stochastic linear programming problems.Casopis Pest. Mat., 91:423–430, 1966.

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.

W. Wiesemann, D. Kuhn, and M. Sim. Distributionally robust convex optimization.Oper. Res., 62(6):1358–1376, 2014.

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.

Optimal Uncertainty Quantification

S. Han, M. Tao, U. Topcu, H. Owhadi, and R. M. Murray. Convex optimal uncertaintyquantification. SIAM Journal on Optimization, 25(23):1368–1387, 2015.

S. Han, U. Topcu, M. Tao, H. Owhadi, and R. Murray. Convex optimal uncertaintyquantification: Algorithms and a case study in energy storage placement for powergrids. In American Control Conference (ACC), 2013, pages 1130–1137. IEEE,2013

H. Owhadi, C. Scovel, T. J. Sullivan, M. McKerns, and M. Ortiz. Optimal Uncertainty Quantification. SIAM Review, 55(2):271–345, 2013.

T. J. Sullivan, M. McKerns, D. Meyer, F. Theil, H. Owhadi, and M. Ortiz. Optimaluncertainty quantification for legacy data observations of Lipschitz functions.ESAIM Math. Model. Numer. Anal., 47(6):1657–1689, 2013.

J Chen, MD Flood, R Sowers. Measuring the Unmeasurable: An Application of Uncertainty Quantification to Financial Portfolios, OFR WP, 2015

L Ming, W Chenglin. An improved algorithm for convex optimal uncertainty quantification with polytopic canonical form. Control Conference (CCC), 2015

H. Owhadi and Clint Scovel. Extreme points of a ball about a measure with finite support (2015). arXiv:1504.06745

H. Owhadi, C. Scovel and T. Sullivan. Brittleness of Bayesian Inference under Finite Information in a Continuous World. Electronic Journal of Statistics, vol 9, pp 1-79, 2015. arXiv:1304.6772

Our proof relies on• Winkler (1988, Extreme points of moment sets)• Follows from an extension of Choquet theory (Phelps 2001, lectures on Choquet’s

theorem) by Von Weizsacker & Winkler (1979, Integral representation in the set ofsolutions of a generalized moment problem)

• Kendall (1962, Simplexes & Vector lattices)

G. Winkler. On the integral representation in convex noncompact sets of tightmeasures. Mathematische Zeitschrift, 158(1):71–77, 1978

G. Winkler. Extreme points of moment sets. Math. Oper. Res., 13(4):581–587,1988.

H. von Weizsacker and G. Winkler. Integral representation in the set of solutionsof a generalized moment problem. Math. Ann., 246(1):23–32, 1979/80.

D. G. Kendall. Simplexes and vector lattices. J. London Math. Soc., 37(1):365–371,1962.

Theorem


Further Reduction of optimization variables

McDiarmid inequality’s

Another example: Optimal concentration inequality H. Owhadi, C. Scovel, T. J. Sullivan, M. McKerns, and M. Ortiz. Optimal Uncertainty Quantification. SIAM Review, 55(2):271–345, 2013.

Reduction of optimization variables

Theorem

Theorem m = 2

C = (1, 1)hC(s) = a− (1− s1)D1 − (1− s2)D2

Explicit Solution m=2

Theorem m = 2

Corollary

Explicit Solution m=2

μ

f

A

Each piece of information is a constrainton an optimization problem.

Optimization concepts (binding, active) transfer to UQ concepts

Non bindingconstraint

Binding but nonactive constraint

Active constraint

Extremizer/Worst case scenario

Optimal Hoeffding= Optimal McDiarmid for m=2

Theorem m = 3Explicit Solution m=3

F min( Yield Strain - Axial Strain )

Ground Acceleration

We want to certify that

Seismic Safety Assessment of a Truss Structure

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

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

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.

Vulnerability Curves (vs earthquake magnitude)

Identification of the weakest elements


Caltech Small Particle Hypervelocity Impact Range

G

Projectile velocity

Plate thickness

Plate Obliquity

Perforation area

We want to certify that

Problem

What do we know?

Projectile velocity

Plate thickness

Plate Obliquity

Thickness, obliquity, velocity: independent random variables

Mean perforation area: in between 5.5 and 7.5 mm^2

Bounds on the sensitivity of the response function w.r. to each variable

We only know

Worst case bound

Reduction calculus

What if we know the response function?

Deterministic surrogate model for the perforation area (in mm^2)

Optimal bound on the probability of non perforation

The measure of probability can be reduced to the tensorization of2 Dirac masses on thickness, obliquity and velocity

Application of the reduction calculus

The optimization variables can be reduced to the tensorizationof 2 Dirac masses on thickness, obliquity and velocity

Support Points at iteration 0

Numerical optimization


Velocity and obliquity marginals each collapse to a single Dirac mass. The plate thickness marginal collapses to have support on the extremes of its range.

Iteration1000

Probability non-perforation maximized by distribution supported on minimal, not maximal, impact obliquity. Dirac on velocity at a non extreme value.

Important observations

Extremizers are singular

They identify key playersi.e. vulnerabilities of the physical system

Extremizers are attractors

Initialization with 3 support points per marginal


Unknown response function G + Legacy data

Constraint on the mean perf. area

Modified Lipschitz continuity constraints on response function

Objective

Constraints on input variables

Legacy Data

32 data points(steel-on-aluminium shots A48–A81) from summer 2010 at Caltech’s SPHIR facility:

These constrain the value of G at 32 points

T. J. Sullivan, M. McKerns, D. Meyer, F. Theil, H. Owhadi, and M. Ortiz. Optimal uncertainty quantification for legacy data observations of Lipschitz functions. ESAIM Math. Model. Numer. Anal., 47(6):1657–1689, 2013.

The numerical results demonstrate agreement with the Markov bound

Only 2 data points out of 32 carry information about the optimal bound

Legacy Data

32 data points(steel-on-aluminium shots A48–A81) from summer 2010 at Caltech’s SPHIR facility:

Only A54 and A67 carry information

The other 30 data points carry noinformation about least upper boundand could have be ignored.


What if we have model uncertainty?

What do we want?

What do we know?

PSAAP numerical model

F

Plate thickness Perforation areaProjectile velocity

Plate Obliquity=0

v

F

4.5 km/s 7 km/s0mm2

30mm2

59 data/experimental points

G

Plate thickness Perforation areaProjectile velocity

Plate Obliquity=0

v

F

4.5 km/s 7 km/s0mm2

30mm2

Confidence sausage around the model

(h, v)

PerforationArea

F

Cy

Admissible set

What we compute

Confidence sausage

v

F

4.5 km/s 7 km/s0mm2

30mm2

The extremizers led to the identification of a bug in an old model

Caltech PSAAP Center UQ analysis

P.-H. T. Kamga, B. Li, M. McKerns, L. H. Nguyen, M. Ortiz, H. Owhadi, andT. J. Sullivan. Optimal uncertainty quantification with model uncertainty andlegacy data. Journal of the Mechanics and Physics of Solids, 72:1–19, 2014

A. A. Kidane, A. Lashgari, B. Li, M. McKerns, M. Ortiz, H. Owhadi, G. Ravichan-dran, M. Stalzer, and T. J. Sullivan. Rigorous model-based uncertainty quantificationwith application to terminal ballistics. Part I: Systems with controllable inputs andsmall scatter. Journal of the Mechanics and Physics of Solids, 60(5):983–1001,2012.

M. M. McKerns, L. Strand, T. J. Sullivan, A. Fang, and M. A. G. Aivazis. Buildinga framework for predictive science. In Proceedings of the 10th Python in ScienceConference (SciPy 2011), 2011.

L. J. Lucas, H. Owhadi, and M. Ortiz. Rigorous verification, validation, uncertaintyquantification and certification through concentration-of-measure inequalities.Comput. Methods Appl. Mech. Engrg., 197(51-52):4591–4609, 2008.



Reduced numerical optimization problems solved using

• mystic: http://trac.mystic.cacr.caltech.edu/project/mystic– a highly-configurable optimization framework

• pathos: http://trac.mystic.cacr.caltech.edu/project/pathos– a distributed parallel graph execution framework providing a high-

level programmatic interface to heterogeneous computing

Mike McKerns

Important observations

In presence of incomplete information on the distribution of input variables the dependence of the least upper bound on the accuracy of the model is very weak

We need to extract as much information as possible from the sample/experimental data on the underlying distributions

How do we reason with the worst in presence of data sampled from an unknown distribution?

Quantity of Interest

You observe

You know μ† ∈ A

Problem:

θ(d)

Player I Player II

Chooses θ

Mean squared error

Confidence error

Max Min

Game theory and statistical decision theory

John Von Neumann Abraham Wald

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.

A. Wald. Contributions to the theory of statistical estimation and testing hypotheses.Ann. Math. Statist., 10(4):299–326, 1939.

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.

Player I

Player II

3

1

-2

-2

Deterministic zero sum game

Player I’s payoff

Player I & II both have a blue and a red marbleAt the same time, they show each other a marble

How should I & II play the game?

Pure strategy solution

3

1

-2

-2II should play blue and loose 1 in the worst case

I should play red and loose 2 in the worst case

Mixed strategy (repeated game) solution

3

1

-2

-2II should play red with probability 3/8 and win 1/8 on average

I should play red with probability 3/8 and loose 1/8 on average

3

1

-2

-2 Player I’s payoff

J. Von Neumann

Max Min

Optimal bound on the statistical errormaxμ∈A

E(μ, θ)

Optimal statistical estimatorsminθ

maxμ∈A

E(μ, θ)

Pure strategy solution for Player II

Max Min

Mixed strategy (repeated game) solution for Player II

Mixed strategy (repeated game) solution for Player I

Theorem

Can we have equality?

Theorem

The best mixed strategy for I and II = worst prior for II

A. Dvoretzky, A. Wald, and J. Wolfowitz. Elimination of randomization in certainstatistical decision procedures and zero-sum two-person games. Ann. Math.Statist., 22(1):1–21, 1951.

The best estimator is not random if the loss function is strictly convex

Non Bayesian

Bayesian

Complete class theorem

Risk

Prior

EstimatorNon cooperative Minmax loss/error

cooperative Bayesian loss/error

Over-estimate risk

Under-estimate risk

L. Le Cam. An extension of Wald’s theory of statistical decision functions. Ann.Math. Statist., 26:69–81, 1955

L. J. Savage. The theory of statistical decision. Journal of the American StatisticalAssociation, 46:55–67, 1951.

Further generalization of Statistical decision theory

A. Shapiro and A. Kleywegt. Minimax analysis of stochastic problems. Optim.Methods Softw., 17(3):523–542, 2002.

M. Sniedovich. The art and science of modeling decision-making under severeuncertainty. Decis. Mak. Manuf. Serv., 1(1-2):111–136, 2007

M. Sniedovich. A classical decision theoretic perspective on worst-case analysis.Appl. Math., 56(5):499–509, 2011.

L. D. Brown. Minimaxity, more or less. In Statistical Decision Theory and RelatedTopics V, pages 1–18. Springer, 1994.

L. D. Brown. An essay on statistical decision theory. Journal of the AmericanStatistical Association, 95(452):1277–1281, 2000.

I. Gilboa and D. Schmeidler. Maxmin expected utility with non-unique prior.Journal of Mathematical Economics, 18(2):141–153, 1989

H. Owhadi and C. Scovel. Towards Machine Wald. Handbook for UncertaintyQuantication, 2016. arXiv:1508.02449.

If we want to make decision theory practical for UQ we need to introduce computational complexity constraints

Impact in econometrics and social sciences

R. Leonard. Von Neumann, Morgenstern, and the Creation of Game Theory: FromChess to Social Science, 1900–1960. Cambridge University Press, 2010.

O. Morgenstern. Abraham Wald, 1902-1950. Econometrica: Journal of the Econo-metric Society, pages 361–367, 1951.

G. Tintner. Abraham Wald’s contributions to econometrics. Ann. Math. Statistics,23:21–28, 1952.

How do we do that?

Is there a natural relation between game theory, computational complexity and numerical approximations?

Ax = b

Φx = y

A: Known n× n symmetricpositive definite matrix

b: Unknown element of Rn

Approximate solution x of

Based on the information that

Φ: Known m× nrank m matrix (m < n)

y: Known element of Rm

A simple approximation problem

AMax Min

A


3

1

-2

-2Player I’s payoff

How should I and II play the game?

Pure strategy (classical numerical analysis) solution

3

1

-2

-2II should play blue and loose 1 in the worst case

I should play red and loose 2 in the worst case

Mixed strategy (repeated game) solution

3

1

-2

-2II should play red with probability 3/8 and win 1/8 on average

I should play red with probability 3/8 and loose 1/8 on average

3

1

-2

-2 Player I’s payoff

J. Von Neumann

Game theoretic formulationAx = b

Max Min

Abraham Wald

Continuous game but as in decision theory under compactness it can be approximated by a finite game

Best strategy: lift minimax to measuresAx = b

Max Min

The best strategy for I is to play at randomPlayer II’s best strategy live

in the Bayesian class of estimators

Player II’s mixed strategy

Ax = b AX = ξξ ∼ N (0, Q)

Player II’s bet

Player II’s mixed strategy

Ax = b AX = ξξ ∼ N (0, Q)

Theorem

Owhadi 2015, Multi-grid with rough coefficients and Multiresolution PDE decomposition from Hierarchical Information Games, arXiv:1503.03467, SIAM Review (to appear)

Main Question

Can we turn the process of discovery of a scalable numerical method into a UQ problem and, to some degree, solve it as such in an automated fashion?

Can we use a computer, not only to implement a numerical method but also to find the method itself?

− div(a∇u) = g, x ∈ Ω,u = 0, x ∈ ∂Ω,

(1)

Ω ⊂ Rd ∂Ω is piec. Lip.

a unif. ell.ai,j ∈ L∞(Ω)

Example: Find a method for solving (1) as fast as possible to a given accuracy

log10(a)

Multigrid Methods

Multiresolution/Wavelet based methods[Brewster and Beylkin, 1995, Beylkin and Coult, 1998, Averbuch et al., 1998]

Multigrid: [Fedorenko, 1961, Brandt, 1973, Hackbusch, 1978]

• Linear complexity with smooth coefficients

Severely affected by lack of smoothnessProblem

[Mandel et al., 1999,Wan-Chan-Smith, 1999,Xu and Zikatanov, 2004, Xu and Zhu, 2008], [Ruge-Stuben, 1987]

Robust/Algebraic multigrid

• Some degree of robustness but problem remains open with rough coefficients

Why?Don’t know how to bridge scales with rough coefficients!

Interpolation operators are unknown

[Vassilevski - Wang, 1997, 1998]

Stabilized Hierarchical bases, Multilevel preconditioners

[Panayot - Vassilevski, 1997]

[Chow - Vassilevski, 2003]

[Panayot - 2010]

[Aksoylu- Holst, 2010]

Low Rank Matrix Decomposition methodsFast Multipole Method: [Greengard and Rokhlin, 1987]

Hierarchical Matrix Method: [Hackbusch et al., 2002]

[Bebendorf, 2008]:

N ln2d+8N complexityTo achieve grid-size accuracy in L2-norm

Their process of discovery is based on intuition, brilliant insight, and guesswork

Common theme between these methods

Can we turn this process of discovery into an algorithm?

YESAnswer:

Identify gameFind optimal strategy

N ln3dN complexityResulting method:

Compute fast

This is a theorem

Compute with partial information

Play adversarial Information game

To achieve grid-size accuracy in H1-normSubsequent solves: N lnd+1N complexity

Owhadi 2015, Multi-grid with rough coefficients and Multiresolution PDE decomposition from Hierarchical Information Games, arXiv:1503.03467, SIAM Review (to appear)

Resulting method:

H10 (Ω) = W(1) ⊕aW(2) ⊕a · · ·⊕aW(k) ⊕a · · ·

(− div(a∇u) = g in Ω,

u = 0 on ∂Ω,

For v ∈W(k)

C12k≤ kvka

k div(a∇v)kL2(Ω)≤ C2

2k

Theorem

< ψ,χ >a:=RΩ(∇ψ)Ta∇χ = 0 for (ψ,χ) ∈W(i) ×W(j), i 6= j

kvk2a :=< v, v >a=RΩ(∇v)T a∇v

Looks like an eigenspace decomposition

w(k) = F.E. sol. of PDE in W(k)

Can be computed independently

u = w(1) + w(2) + · · ·+ w(k) + · · ·

u=

w(1) w(2) w(3)

w(4) w(5) w(6)

8× 10−3

1.5× 10−3 4× 10−4 4× 10−5

0.030.14

+

+

+

+

Multiresolution decomposition of solution space

Quacks like an eigenspace decomposition

w(k) = F.E. sol. of PDE in W(k)

Can be computed independently

B(k): Stiffness matrix of PDE in W(k)

Theorem λmax(B(k))

λmin(B(k))≤ C

Just relax in W(k) to find w(k)

u = w(1) + w(2) + · · ·+ w(k) + · · ·

Swims like an eigenspace decomposition

μ(x)∂2t u− div(a∇u) = g(x, t)

Application to time dependent problems

μ(x)∂tu− div(a∇u) = g(x, t)

[Owhadi-Zhang 2016, From gamblets to near FFT-complexitysolvers for wave and parabolic PDEs with rough coefficients]

Hyperbolic and parabolic PDEs with rough coefficientscan be solved in O(N ln3dN) (near FFT) complexity

u=

w(1) w(2) w(3)

w(4) w(5) w(6)

8× 10−3

1.5× 10−3 4× 10−4 4× 10−5

0.030.14

+

+

+

+

Doesn’t have the complexity of an eigenspace decomposition

Theorem

Can be performed and stored in

V: F.E. space of H10 (Ω) of dim. N

V = W(1) ⊕aW(2) ⊕a · · ·⊕aW(k)

The decomposition

O(N ln3dN) operations

ψ(1)i χ

(2)i χ

(3)i

χ(4)i χ

(5)i

χ(6)i

Basis functions look like and behave like wavelets:Localized and can be used to compress the operator

and locally analyze the solution space

u

H−1(Ω)H10 (Ω)

um gm

div(a∇·)

g

Inverse Problem

Reduced operator

∈ RmRmNumerical implementation requirescomputation with partial information.

um ∈ Rm u ∈ H10 (Ω)Missing information

φ1, . . . ,φm ∈ L2(Ω)

um = (Ωφ1u, . . . , Ω φmu)

Discovery process (− div(a∇u) = g in Ω,

u = 0 on ∂Ω,

φ1, . . . ,φm ∈ L2(Ω)

u− u∗a

Player I Player IIChoosesg ∈ L2(Ω) Sees

Ωuφ1, . . . , Ω uφm

Chooses u∗ ∈ L2(Ω)kgkL2(Ω) ≤ 1

Max Min

Identify underlying information game

Measurement functions:

kfk2a :=RΩ(∇f)T a∇f

Player I

Player II

3

1

-2

-2


Player I’s payoff

Player I & II both have a blue and a red marbleAt the same time, they show each other a marble

How should I & II play the (repeated) game?

Game theory

John Von Neumann

John Nash

Player I

Player II

3

1

-2

-2

= 3pq + (1− p)(1− q)− 2p(1− q)− 2q(1− p)=1− 3q + p(8q − 3) =− 1

8 for q = 38

q 1− q

p

1− p

Optimal strategies are mixed strategies

Optimal way toplay is at random

Abraham Wald

The best strategy for A is to play at randomPlayer B’s best strategy live

in the Bayesian class of estimators

Player A Player BChoosesg ∈ L2(Ω) Sees

RΩuφ1, . . . ,

RΩuφm

Chooses u∗ ∈ L2(Ω)kgkL2(Ω) ≤ 1

Continuous game but as in decision theoryunder compactness it can be approximatedby a finite game

°°u− u∗°°a

Player II’s class of mixed strategies


u = 0 on ∂Ω,

(− div(a∇v) = ξ in Ω,

v = 0 on ∂Ω,

ξ: Random field

u∗(x) := E£v(x)

¯ RΩv(y)φi(y) dy =

RΩu(y)φi(y) dy,∀i

¤Player II’s bet

g ∈ L2(Ω)

Pretend that player I is choosing g at random

min max problemover distribution of ξ

Player II’s optimal strategy?

Theorem

ψi(x) := Eξ∼N (0,Γ)hv(x)

¯ RΩv(y)φj(y) dy = δi,j , j ∈ 1, . . . ,m

iψi: Elementary gambles/bets

ψiGamblets

Elementary gambles form deterministic basis functions for player B’s bet

u∗(x) =Pm

i=1 ψi(x)RΩu(y)φi(y) dy

ξ ∼ N (0,Γ)Computational efficiency

Depend onWhat are these gamblets?

Example

Γ(x, y) = δ(x− y)φi(x) = δ(x− xi)

Ω

xi

x1

xm

a = Id ψi: Polyharmonic splines[Harder-Desmarais, 1972][Duchon 1976, 1977,1978]

ai,j ∈ L∞(Ω) ψi: Rough Polyharmonic splines[Owhadi-Zhang-Berlyand 2013]

• Γ: Covariance function of ξ (Player B’s decision)

• (φi)mi=1: Measurements functions (rules of the game)

What is Player II’s best choice for

Γ(x, y) = E£ξ(x)ξ(y)

¤What is Player II’s best strategy?

Γ = LL = − div(a∇·)

RΩξ(x)f(x) dx ∼ N (0, kfk2a)

kfk2a :=RΩ(∇f)Ta∇f

Why? See algebraic generalization

?

Theorem

u∗(x) is the F.E. solution of (1) in spanL−1φi|i = 1, . . . ,mku− u∗ka = infψ∈spanL−1φi:i∈1,...,m ku− ψka

If Γ = L then

The recovery is optimal (Galerkin projection)

L = − div(a∇·)(− div(a∇u) = g, x ∈ Ω,

u = 0, x ∈ ∂Ω,(1)

Theorem ψi: Unique minimizer of(Minimize kψkaSubject to ψ ∈ H1

0 (Ω) andRΩφjψ = δi,j , j = 1, . . . ,m

Pmi=1 wiψi minimizes kψka

over all ψ such thatRΩφjψ = wj for j = 1, . . . ,m

TheoremOptimal variational properties

Variational characterization

Selection of measurement functions

Theorem ku− u∗ka ≤ Hλmin(a)

kgkL2(Ω)

φi = 1τi τi

Ω

diam(τi) ≤ H

τj

Indicator functions of aExample

Partition of Ω of resolution H

Elementary gamble

ψi

(− div(a∇u) = g, x ∈ Ω,

u = 0, x ∈ ∂Ω,(1)

Your best bet on the value of u

τi

Ωτj

10000

0000

0000

0

00

given the information thatRτiu = 1 and

Rτju = 0 for j 6= i

Exponential decay of gamblets

TheoremRΩ∩(B(τi,r))c(∇ψi)

Ta∇ψi ≤ e−rlH kψik2a

x-axis slice

ψi

ψi

log10¡10−10 + |ψi||

¢x-axis slice

4

−10

r

Ω

τi

r

Ω

τiSr

Theorem

ku− u∗,locka ≤ 1√λmin(a)

HkgkL2(Ω)

u∗,loc(x) =Pm

i=1 ψloc,ri (x)

RΩu(y)φi(y) dy

If r ≥ CH ln 1H

No loss of accuracy iflocalization ∼ H ln 1

H

ψloc,ri : Minimizer of(Minimize kψkaSubject to ψ ∈ H1

0 (Sr) andRSr

φjψ = δi,j

for τj ∈ Sr

Localization of the computation of gamblets

Formulation of the hierarchical game

Hierarchy of nested Measurement functions

φ(1)2 = 1

τ(1)2

φ(2)2,3 = 1

τ(2)2,3

φ(k)i1,...,ik

with k ∈ 1, . . . , qφ(k)i =

Pj ci,jφ

(k+1)i,j

τ(1)2 τ

(2)2,3

τ(3)2,3,1

φ(3)2,3,1 = 1

τ(3)2,3,1

Example

φ(1)i1

φ(2)i1,j1

φ(2)i1,j2

φ(2)i1,j3

φ(2)i1,j4

φ(3)i1,j2,k1

φ(3)i1,j2,k2

φ(3)i1,j2,k3

φ(3)i1,j2,k4

φ(k)i : Indicator functions of a

hierarchical nested partition of Ω of resolution Hk = 2−k

iΠ1,2i

I1 I2 I3

j ∈ Π1,2i ⊂ Π2I Π2,3j

Π1,3i

τ(1)i τ

(2)j

φ(1)i φ

(2)i φ

(3)i

φ(4)i φ

(5)i φ

(6)i

In the discrete setting simply aggregate elements(as in algebraic multigrid)

Player I Player IIChoosesg ∈ L2(Ω)

Formulation of the hierarchy of games

kgkL2(Ω) ≤ 1


u = 0 on ∂Ω,

Must predict

Sees Ωuφ

(k)i , i ∈ Ik

u and Ωuφ

(k+1)j , j ∈ Ik+1

Player II’s best strategy(− div(a∇u) = g in Ω,

u = 0 on ∂Ω,


v = 0 on ∂Ω,

ξ ∼ N (0,L)

u(k)(x) := E£v(x)

¯ RΩv(y)φ

(k)i (y) dy =

RΩu(y)φ

(k)i (y) dy, i ∈ Ik

¤Player II’s bets

Fk = σ(RΩvφ

(k)i , i ∈ Ik) v(k)(x) := E

£v(x)

¯Fk¤

Theorem Fk ⊂ Fk+1v(k)(x) := E

£v(k+1)(x)

¯Fk¤

The sequence of approximations forms a martingale under the mixed strategy emerging from the game

Player II’s best strategy(− div(a∇u) = g in Ω,

u = 0 on ∂Ω,


v = 0 on ∂Ω,

ξ ∼ N (0,L)

u(k)(x) := E£v(x)

¯ RΩv(y)φ

(k)i (y) dy =

RΩu(y)φ

(k)i (y) dy, i ∈ Ik

¤Player II’s bets

u(1) u(2) u(3)

u(4) u(5) u(6)

Gamblets Elementary gambles form a hierarchy of deterministic basis functions for player II’s hierarchy of bets

Theorem u(k)(x) =P

i ψ(k)i (x)

RΩu(y)φ

(k)i (y) dy

ψ(k)i : Elementary gambles/bets at resolution Hk = 2−k

ψ(k)i (x) := E

hv(x)

¯ RΩv(y)φ

(k)j (y) dy = δi,j , j ∈ Ik

iψ(1)i ψ

(2)i ψ

(3)i

ψ(4)i ψ

(5)i ψ

(6)i

Theorem

V(k) ⊂ V(k+1)

V(k) := spanψ(k)i , i ∈ Ik ψ(1)i1

ψ(2)i1,j1

ψ(2)i1,j2

ψ(2)i1,j3

ψ(2)i1,j4

ψ(3)i1,j2,k1

ψ(3)i1,j2,k2

ψ(3)i1,j2,k3

ψ(3)i1,j2,k4

Gamblets are nested

ψ(k)i (x) =

Pj∈Ik+1 R

(k)i,j ψ

(k+1)j (x)

R(k)i,j = E

£ RΩv(y)φ

(k+1)j (y) dy

¯ RΩv(y)φ

(k)l (y) dy = δi,l, l ∈ Ik

¤Interpolation/Prolongation operator

10 0

0

R(k)i,j

Your best bet on the value ofRτ(k+1)j

u

given the information thatRτ(k)iu = 1 and

Rτlu = 0 for l 6= i

τ(k)i R

(k)i,j

τ(k+1)j

At this stage you can finish withclassical multigrid

But we want multiresolution decomposition

Elementary gamble

Ω

00

0

χ(k)i

000

00τ

(k)i τ

(k)j

00

00

Your best bet on the value of u

given the information thatRτ(k)i

u = 1,Rτ(k)

i−u = −1 and

Rτ(k)j

u = 0 for j 6= i

100

-1

τ(k)i−

+1−1

χ(k)i = ψ

(k)i − ψ

(k)i−

+1

−1

+1−1

i = (i1, . . . , ik−1, ik)

i− = (i1, . . . , ik−1, ik − 1)

ψ(1)i1

ψ(2)i1,j1

ψ(2)i1,j2

ψ(2)i1,j3

ψ(2)i1,j4

+1−1

ψ(1)i χ

(2)i χ

(3)i

χ(4)i χ

(5)i

χ(6)i

χ(k)i = ψ

(k)i − ψ

(k)i−

Theorem

W(k+1): Orthogonal complement of V(k) in V(k+1)

with respect to < ψ,χ >a:=RΩ(∇ψ)T a∇χ

H10 (Ω) = V(1) ⊕aW(2) ⊕a · · ·⊕aW(k) ⊕a · · ·

Multiresolution decomposition of the solution space

V(k) := spanψ(k)i , i ∈ IkW(k) := spanχ(k)i , i

Theorem

u(k+1) − u(k) = F.E. sol. of PDE in W(k+1)

u=

u(1) u(2) − u(1) u(3) − u(2)

u(4) − u(3) u(5) − u(4) u(6) − u(5)

8× 10−3

1.5× 10−3 4× 10−4 4× 10−5

0.030.14

+

+

+

+

Multiresolution decomposition of the solution

Subband solutions u(k+1) − u(k)can be computed independently

Uniformly bounded condition numbers

A(k)i,j :=

ψ(k)i ,ψ

(k)j

®a

B(k)i,j :=

χ(k)i ,χ

(k)j

®a

4.5log10(

λmax(A(k))

λmin(A(k)))

log10(λmax(B

(k))λmin(B(k))

)

Theoremλmax(B

(k))

λmin(B(k))≤ C

Just relax!In v ∈W(k)

to getu(k) − u(k−1)

V = W(1) ⊕aW(2) ⊕a · · ·⊕aW(k)

Ranges of eigenvalues in V andW(k) (k = 1, . . . , 5) in log scale

£infψ∈V

kψk2akψk2

L2, supψ∈V

kψk2akψk2

L2

¤

£infψ∈W(k)

kψk2akψk2

L2, supψ∈W(k)

kψk2akψk2

L2

¤

c(1)i

c(2)j

c(3)j

c(4)j

0 200 400 600 800-6

-4

-2

0

2

4x 10

-4

c(5)j c

(6)j

u =P

i c(1)i

ψ(1)i

kψ(1)i ka+Pq

k=2

Pj c(k)j

χ(k)j

kχ(k)j ka

0 1000 2000 3000 4000-4

-2

0

2

4x 10

-5

Coefficients of the solution in the gamblet basis

c(6)j

Operator Compression

Throw 99% of the coefficients

u

Gamblets behave like wavelets but they are adapted to the PDE and can compress its solution space

Gamblet compression

Compression ratio = 105Energy norm relative error = 0.07

Fast gamblet transform

Nesting A(k) = (R(k,k+1))TA(k+1)R(k,k+1)

Level(k) gamblets and stiffness matrices can be computedfrom level(k+1) gamblets and stiffness matrices

Well conditioned linear systems

ψ(k)i = ψ

(k+1)(i,1) +

Pj C

(k+1),χi,j χ

(k+1)j C(k+1),χ = (B(k+1))−1Z(k+1)

LocalizationZ(k+1)j,i := −(e

(k+1)j − e(k+1)j− )TA(k+1)e

(k+1)(i,1)

Underlying linear systems have uniformly bounded condition numbers

The nested computation can be localized without compromising accuracy or condition numbers

O(N ln3dN) complexity

Theorem

Localizing (ψ(k)i )i∈Ik and (χ

(k)i )i to subdomains of size

≥ CHk ln 1Hk

≥ CHk(ln 1Hk

+ ln 1² )

Cond. No (B(k),loc) ≤ C

°°u− u(1),loc −Pq−1k=1(u

(k+1),loc − u(k),loc)°°a≤ ²

The number of operations to compute gamblets andachieve accuracy ² is O

¡N ln3d

¡max( 1

², N1/d)

¢¢(and O

¡N lnd(N1/d) ln 1

²

¢for subsequent solves)

Theorem

Complexity

O(N ln3dN) O(N lnd+1N)GambletTransform

LinearSolve

u(3) − u(2)8× 10−3

u(2) − u(1)0.03

u(1)0.14

...

u(1)

ϕi, Ah,Mh χ

(q)i , B

(q)ψ(q)i , A(q) u(q) − u(q−1)

ψ(q−1)i , A(q−1) χ

(q−1)i , B(q−1) u(q−1) − u(q−2).

..

...

χ(3)i , B(3)ψ

(3)i , A(3) u(3) − u(2)

ψ(2)i , A(2) χ

(2)i , B(2) u(2) − u(1)

ψ(1)i , A(1)

ψ(1)i

χ(2)i

χ(3)i

ψ(1)i

ψ(2)i

ψ(3)i

Parallel operatingdiagram

both in space and in

frequency

Numerical Homogenization

HMM

Harmonic Coordinates

[Gloria 2010]

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

[Chu-Graham-Hou-2010] (limited inclusions)

[Babuska-Lipton 2010] (local boundary eigenvectors)[Efendiev-Galvis-Wu-2010] (limited inclusions or mask)

[Malqvist-Peterseim 2012] Volume averaged interpolation

Flux norm Berlyand, Owhadi 2010; Symes 2012

Arbogast, 2011: Mixed MsFEM

Kozlov, 1979

Localization

[Fish - Wagiman, 1993]

Projection based method

[Owhadi-Zhang 2011] (localized transfer property)

Statistical approach to numerical approximation

Some high level remarks

What is the worst?

Au†: Unknown element of A

Robust Optimization worst case

Φ : A −→ Ru −→ Φ(u) Quantity of Interest

What is Φ(u†)?

infu∈AΦ(u) ≤ Φ(u†) ≤ supu∈A Φ(u)

A

Game theoretic worst case

AMax Min

Confidence error

Robust Optimization worst caseFailure is not an option. You want to always be right.

Game theoretic worst case

Interpretation depends on the choice of loss function.

You want to be right with high probability.

Quadratic errorYou want to be right on average.Well suited for numerical computation where you need to keep computing with partial information (e.g. invert a 1,000,000 by 1,000,000 matrix)

Non Bayesian

Bayesian

Complete class theorem

Risk

Prior

EstimatorNon cooperative Minmax loss/error

cooperative Bayesian loss/error

Over-estimate risk

Under-estimate risk

Can we approximate the optimal prior?

Numerical robustness of Bayesian inference

Can we numerically approximate the prior when closed form expressions are not available for posterior values?

• Brittleness of Bayesian Inference under Finite Information in a Continuous World. H. Owhadi, C. Scovel and T. Sullivan. Electronic Journal of Statistics, vol 9, pp 1-79, 2015. arXiv:1304.6772

• Brittleness of Bayesian inference and new Selberg formulas. H. Owhadi and C. Scovel. Communications in Mathematical Sciences (2015). arXiv:1304.7046

• On the Brittleness of Bayesian Inference. H. Owhadi, C. Scovel and T. Sullivan. SIAM Review, 57(4), 566-582, 2015, arXiv:1308.6306

• Qualitative Robustness in Bayesian Inference (2015). H. Owhadi and C. Scovel. arXiv:1411.3984

Positive Negative• Classical Bernstein Von Mises• Wasserman, Lavine, Wolpert (1993)• P Gustafson & L Wasserman (1995)• Castillo and Rousseau (2013)• Castillo and Nickl (2013)• Stuart & Al (2010+). • ….

• Freedman (1963, 1965)• P Gustafson & L Wasserman (1995)• Diaconis & Freedman 1998• Johnstone 2010• Leahu 2011• Belot 2013

Robustness of Bayesian conditioning in continuous spaces

10.000 children are given one pound of play-doh. On average, how much mass can they put above awhile, on average, keeping the seesaw balanced around m?

Paul is given one pound of play-doh. What can you say about how much mass he isputting above a if all you have is the belief thathe is keeping the seesaw balanced around m?

Brittleness of Bayesian Inference under Finite Information in a Continuous World. H. Owhadi, C. Scovel and T. Sullivan. Electronic Journal of Statistics, vol 9, pp 1-79, 2015. arXiv:1304.6772

What is the least upper bound on

If all you know is ?

Answer

Theorem

Reduction calculus with measures over measures

QΨ

QΨ−1 ⊂M(Q)M(A) ⊃Theorem

supπ∈Ψ−1Q

Eμ∼π Φ(μ)

supQ∈Q

hEq∼Q

£sup

μ∈Ψ−1(q)Φ(μ)

¤i=

M(X ) ⊃ Polishspace

Brittleness of Bayesian Inference under Finite Information in a Continuous World. H. Owhadi, C. Scovel and T. Sullivan. Electronic Journal of Statistics, vol 9, pp 1-79, 2015. arXiv:1304.6772

What is the worst with random data? A

A

Frequentist/Concentration of measure worst case

Theorem

N. Fournier and A. Guillin, On the rate of convergence in Wasserstein distance of the empirical measure, Probability Theory and Related Fields, (2014), pp. 1-32.

A

The extreme points of the Prokhorov, Monge-Wasserstein and Kantorovich metric balls about a measure whose support has at most n points, consist of measures whose supports have at most n+2 points.

• D. Wozabal. A framework for optimization under ambiguity. Annals of Operations Research, 193(1):21—47, 2012.

• P. M. Esfahani and D. Kuhn. Data-driven distributionally robust optimization using the Wasserstein metric: performance guarantees and tractable reformulations. arXiv:1505.05116, 2015.

• Extreme points of a ball about a measure with finite support (2015). H. Owhadi and Clint Scovel. arXiv:1504.06745

Reduction calculus of the ball about the empirical distribution

QuestionGame/Decision Theory + Information Based Complexity

Turn the process of discovery of scalable numerical solversinto an algorithm

Worst case calculus

?

P. L. Chebyshev1821-1894

M. G. Krein1907-1989

A. A. Markov1856-1922

The truncated moment problem

³EX∼μ[X],EX∼μ[X2], . . . ,EX∼μ[Xk]

´Ψ

Study of the geometry of Mk := Ψ¡M([0, 1])

¢

Finite dim.Infinite dim.

³EX∼μ[X],EX∼μ[X2], . . . ,EX∼μ[Xk]

´Ψ Mk := Ψ¡M([0, 1])

¢

Finite dim.

Finite dim.Infinite dim.

Ψ

t1 t2 tj tN

Index

Theorem

t1 t2 tj tN

t1 t2 tj tN

Upper

Lower

Sn(α,β, γ) =Qn−1j=0

Γ(α+jγ)Γ(β+jγ)Γ(1+(j+1)γ)Γ(α+β+(n+j−1)γ)Γ(1+γ)

Selberg Identities

Sn(α,β, γ) :=R[0,1]n

Qnj=1 t

α−1j (1− tj)β−1|∆(t)|2γdt .

∆(t) :=Qj<k (tk − tj)

Brittleness of Bayesian inference and new Selberg formulas. H. Owhadi and C. Scovel. Communications in Mathematical Sciences (2015). arXiv:1304.7046

Forrester and Warnaar 2008

The importance of the Selberg integral

Used to prove outstanding conjectures inRandom matrix theory and cases of the Macdonald conjectures

Central role in random matrix theory, Calogero-Sutherland quantum many-body systems, Knizhnik-Zamolodchikov equations, and multivariable orthogonal polynomial theory

Index

Theorem

t1 t2 tj tNt∗

New Reproducing Kernel Hilbert Spaces and Selberg Integral formulas related to the Markov-Krein representations of moment spaces.

Ψ ³EX∼μ[X],EX∼μ[X2], . . . ,EX∼μ[Xk]

´RImΣt−1 ·Qm

j=1 t2j (1− tj)2∆4m(t)dt = Sm(5,1,2)−Sm(3,3,2)

2RImΣt−1 ·Qm

j=1 t2j ·∆4m(t)dt = m

2 Sm−1(5, 3, 2)

Sn(α,β, γ) =Qn−1j=0

Γ(α+jγ)Γ(β+jγ)Γ(1+(j+1)γ)Γ(α+β+(n+j−1)γ)Γ(1+γ)

∆m(t) :=Qj<k (tk − tj)¡

Σφ¢(t) :=

Pmj=1 φ(tj), t ∈ Im

I := [0, 1]

ZIm−1

hk(t)ΣQj(t)

m−1Yj0=1

t2j0 ·∆4m−1(t)dt = V ol(M2m−1)(2m−1)!(m−1)!(k + 2)!

(8k + 4)(k − 2)!δjk .

Theorem

ej(t) :=X

i1<···<ijti1 · · · tij

Πn0 : n-th degree polynomials which vanish on the boundary of [0, 1]Mn ⊂ Rn: set of q = (q1, . . . , qn) ∈ Rn such that there exists a probabilitymeasure μ on [0, 1] with Eμ[Xi] = qi with i ∈ 1, . . . , n.

Consider the basis of Π2m−10 consisting of the associated Legendre polyno-mials Qj , j = 2, .., 2m − 1 of order 2 translated to the unit interval I. Fork = 2, .., 2m− 1 define

ajk :=(j + k + k2)Γ(j + 2)Γ(j)

Γ(j + k + 2)Γ(j − k + 1), k ≤ j ≤ 2m− 1

hk(t) :=2m−1Xj=k

(−1)j+1ajke2m−1−j(t, t) .

Then for j = k mod 2, j, k = 2, .., 2m− 1, we have

Bi-orthogonal systems of Selberg Integral formulas

Collaborators

Research supported by Air Force Office of Scientific Research

U.S. Department of Energy Office of Science, Office of Advanced Scientific Computing Research, through the Exascale Co-Design Center for Materials in Extreme Environments

National Nuclear Security Administration

Clint Scovel (Caltech), Tim Sullivan (Warwick), Mike McKerns (Caltech), Michael Ortiz (Caltech), Lei Zhang (Jiaotong), Leonid Berlyand (PSU), Lan HuongNguyen, Paul Herve Tamogoue Kamga.

DARPA EQUiPS Program (Enabling Quantification of Uncertainty in Physical Systems)

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