Non-Intrusive Algorithms for Measure-Theoretic Propagation of - - PowerPoint PPT Presentation

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Non-Intrusive Algorithms for Measure-Theoretic Propagation of - - PowerPoint PPT Presentation

Jan 03, 2023 325 likes •719 views

Non-Intrusive Algorithms for Measure-Theoretic Propagation of Uncertainties: Errors, Opportunities, and Challenges T. Butler University of Colorado Denver Research supported by DOE DE-SC0009286 and NSF DMS-1228206 Collaborators: Don Estep,

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SLIDE 1

Non-Intrusive Algorithms for Measure-Theoretic Propagation of Uncertainties: Errors, Opportunities, and Challenges

  • T. Butler

University of Colorado Denver Research supported by DOE DE-SC0009286 and NSF DMS-1228206 Collaborators: Don Estep, Clint Dawson, Lindley Graham, Simon Tavener, Steven Mattis, Scott Walsh, Monty Vesselinov

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SLIDE 2

Inference for a deterministic model

Observation Space Prediction Space Space of Data and Parameters

Physics Model Solution Space

Observation Functionals Predication Functionals I n v e r s e P r

  • b

l e m f

  • r

P a r a m ete rs F

  • r

w a r d P r

  • b

l e m f

  • r

P r e d i c a t i

  • n
  • T. Butler

Measure-Theoretic UQ 2 / 34

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SLIDE 3

The mathematical model

Ingredients

Compact parameter domain Λ ⊂ Rn Model M(Y, λ) with solution Y = Y (λ) for λ ∈ Λ Quantities of interest (QoI) Q(λ) = Q(Y (λ)) ∈ Rd We assume that Q(λ) is differentiable. The specification of Λ is critical and should be determined by physical considerations. The set of all possible QoI D = Q(Λ) ⊂ Rd defines the observation space.

  • T. Butler

Measure-Theoretic UQ 3 / 34

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SLIDE 4

The core deterministic inverse problem

Given a Q ∈ D, find λ ∈ Λ with Q(λ) = Q.

Q(λ) Λ

The solution is generally a set of values. We call Q−1(Q(λ)) a generalized contour.

  • T. Butler

Measure-Theoretic UQ 4 / 34

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SLIDE 5

Measure theory and inverse problems

The core deterministic inverse problem imposes significant structure

  • n the solution of the stochastic inverse problem.

Measure theory is designed to handle the set-valued inverses of a map between measurable spaces in a natural way and is the basis for rigorous probability theory. Measure theory is ideally suited for the treatment of the stochastic inverse problem for a deterministic model.

  • T. Butler

Measure-Theoretic UQ 5 / 34

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SLIDE 6

Measure theory ingredients

Measurable space

A specified domain X A σ-algebra BX defining the collection of sets whose size can be measured and the operations on those sets (X, BX) defines a measurable space.

Measure space

A procedure for computing the measure µX of sets in the σ−algebra (X, BX, µX) defines a measure space.

  • T. Butler

Measure-Theoretic UQ 6 / 34

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SLIDE 7

Measure theory ingredients

X

How do we compute the µX-measure of this event?

  • T. Butler

Measure-Theoretic UQ 6 / 34

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SLIDE 8

Measure theory ingredients

X

We can approximate by using simpler sets from BX.

  • T. Butler

Measure-Theoretic UQ 6 / 34

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SLIDE 9

Measure theory ingredients

X

Only certain events in the approximation require refinement.

  • T. Butler

Measure-Theoretic UQ 6 / 34

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SLIDE 10

Inverse sensitivity analysis and generalized contours

The range of Q−1 is not Λ. The range of Q−1 is L whose individual points correspond to the natural set-valued inverses of Q in Λ that we call generalized contours. Properties like well-posedness are posed in L not in Λ. From this perspective, the inability to distinguish between representors in a set-valued solution is not ill-posedness.

  • T. Butler

Measure-Theoretic UQ 7 / 34

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SLIDE 11

Describing the space of contour manifolds D

Λ

L

Theorem

There exists a transverse parametrization (TP) representing L in Λ.

  • T. Butler

Measure-Theoretic UQ 8 / 34

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SLIDE 12

Solutions on (L, BL)

Theorem

A probability measure PD on (D, BD) corresponds to a unique probability measure PL on (L, BL).

D L

Λ

  • T. Butler

Measure-Theoretic UQ 9 / 34

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SLIDE 13

Natural but not desirable

The inverse solution on (L, BL) requires minimal assumptions. But . . . the physically meaningful space is Λ. The ideal inferential target is a probability measure PΛ on (Λ, BΛ). To construct PΛ from PL requires answering two (related) questions. How are events in BΛ related to events in BL?

◮ Ans: Use a transverse product σ-algebra.

How are measures on (Λ, BΛ) related to measures on (L, BL)?

◮ Ans: Use the Disintegration Theorem.

  • T. Butler

Measure-Theoretic UQ 10 / 34

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SLIDE 14

Structure of measures on Λ

The Disintegration Theorem implies that any PΛ can be written uniquely as the product of a marginal PL on a TP and conditional probabilities {Pℓ} on contours {Cℓ} for ℓ ∈ L. The Disintegration Theorem is like Fubini’s theorem where PΛ(A) is written as an iterated integral involving Pℓ and PL.

Theorem

Specifying Pℓ on generalized contours corresponding to ℓ ∈ L determines a unique probability measure on (Λ, BΛ).

  • T. Butler

Measure-Theoretic UQ 11 / 34

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SLIDE 15

The Standard Ansatz

L l

U n i f

  • r

m D e n s i t y A l

  • n

g G e n e r a l i z e d C

  • n

t

  • u

r s

The computational algorithm and BET code can treat any measure in the Ansatz or work directly with the contour events.

  • T. Butler

Measure-Theoretic UQ 12 / 34

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SLIDE 16

Solution of the example under the Ansatz

1

L

Λ

Λ Inverse density on L Inverse density on Λ

  • T. Butler

Measure-Theoretic UQ 13 / 34

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SLIDE 17

Simple function approximations of measures

Theorem

PΛ can be approximated using simple functions. PΛ is approximated on a partition of Λ taken as a subset of the generating sets to BΛ, BL, and {BCℓ, ℓ ∈ L}. This results in a direct discretization of the iterated integral in the Disintegration Theorem.

  • T. Butler

Measure-Theoretic UQ 14 / 34

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SLIDE 18

Approximations of events with random or regular sampling

  • T. Butler

Measure-Theoretic UQ 15 / 34

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SLIDE 19

General measure-theoretic algorithmic outline

Step 1: Define Voronoi tessellation {Vj}N

j=1 of Λ used to approximate

events in both BΛ and CΛ. Step 2: Define approximation to PD (or its density) on (D, BD) by computation of PD(Ii) for tessellation {Ii}M

i=1 of D.

Step 3: Use Ansatz to compute PΛ(Vj). Step 4: Compute PΛ(A) for events of interest A ∈ BΛ using PΛ(Vj).

  • T. Butler

Measure-Theoretic UQ 16 / 34

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SLIDE 20

Applying the algorithm/identifying errors

1 2 3 4 5 6 7 8 9 1

For A ∈ BΛ with µΛ(∂A) = 0, we estimate PΛ(A). The error in the µΛ-volume of a Voronoi coverage of Q−1(Ii) affects PΛ estimation.

  • T. Butler

Measure-Theoretic UQ 17 / 34

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SLIDE 21

Applying the algorithm/identifying errors

1 2 3 4 5 6 7 8 9 1 1 2 3 4 5 6 7 8 9 1

Theorem

As N, M → ∞, the counting measure converges to PΛ.

  • T. Butler

Measure-Theoretic UQ 17 / 34

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SLIDE 22

A priori optimal discretization

Convex sets are optimal

If contour events Q−1(Ij) are convex for 1 ≤ j ≤ M, then there exists a set of M samples

  • λ(j)M

j=1 such that PΛ(Q−1(Ij)) = PΛ,M(Q−1(Ij)).

This holds when dim(Λ) = +∞.

Partitioning contour events into approximate convex subsets

This can be generalized for specified ǫ > 0 to show the existence of N = O(M) samples

  • λ(i)N

i=1 such that

  • PΛ(Q−1(Ij)) − PΛ,N(Q−1(Ij))
  • < ǫ.

Finding the optimal discretization often requires a sensitivity analysis

  • n Q.
  • T. Butler

Measure-Theoretic UQ 18 / 34

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SLIDE 23

Multiple QoI

How does the geometric relationship between multiple QoI affect the solution of the stochastic inverse problem at a specified sample size?

Geometrically distinct QoI

The component maps of Q are geometrically distinct (GD) if the Jacobian of Q is full rank at every point in Λ

Theorem

If the component maps of Q are GD, then the generalized contours exist as n − d dimensional manifolds and a TP exists as a d dimensional manifold

  • T. Butler

Measure-Theoretic UQ 19 / 34

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SLIDE 24

The geometry matters

PΛ for fixed q1(λ) and varying q2(λ) with differing levels of skewness PΛ for different choices of q1(λ) and q2(λ) maintaining orthogonality of the contours.

  • T. Butler

Measure-Theoretic UQ 20 / 34

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SLIDE 25

Condition of the numerical solution

The skewness of an event determines the difficulty in computing accurate approximations using regular or uniform random sampling.

v2 v1 v1

Skewness

Skew(V, vi) = |vi| |v⊥

i |,

Skew(V ) = max Skew(V, vi)

  • T. Butler

Measure-Theoretic UQ 21 / 34

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SLIDE 26

Skewness and the number of samples

{xj} = a net of points in Λ.

Theorem

The number of samples needed to compute an accurate inverse solution is proportional to max

j

  • Skew(JQ|L(xj))

d−1

  • T. Butler

Measure-Theoretic UQ 22 / 34

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SLIDE 27

A posteriori error analysis

There is statistical/representation error from using finite N. There is numerical error in Q(λ(j)) for each j, and this deterministic error leads to possible misidentification of Vj ∈ Q−1(Ii). Deterministic error affects PΛ even in the limit of N.

Theorem

A computable a posteriori error estimate exists taking into account statistical and deterministic errors providing lower and upper bounds

  • n the probability of events.
  • T. Butler

Measure-Theoretic UQ 23 / 34

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SLIDE 28

Stochastic and deterministic error terms

Identify a computational algebra, ˜ BΛ ⊂ BΛ. We use the following notation: ˜ PΛ,N,h is the fully computable counting measure. ˜ PΛ,N is the counting measure with no deterministic error. For A ∈ ˜ BΛ, PΛ(A) − ˜ PΛ,N,h(A) =

  • PΛ(A) − ˜

PΛ,N(A)

  • I

−

  • ˜

PΛ,N,h(A) − ˜ PΛ,N(A)

  • II

. Term I is the error in approximating the probability of A with a counting measure. Term II is the error in using a numerical map to identify approximate Voronoi coverages of contour events.

  • T. Butler

Measure-Theoretic UQ 24 / 34

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SLIDE 29

An inverse problem for Manning’s n

Manning’s n is a parameter field that quantifies the surface roughness and is used to determine bottom stress in coastal hydrodynamics. We invert to determine Manning’s n from maximum water elevation measurements at a choice of fixed stations.

  • T. Butler

Measure-Theoretic UQ 25 / 34

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SLIDE 30

An inverse problem for Manning’s n

1 2 6 Open Ocean

We consider an idealized inlet with a jetty, sloping bathymetry, and an

  • pen ocean boundary at one end.
  • T. Butler

Measure-Theoretic UQ 26 / 34

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SLIDE 31

Defining Manning’s n field

Manning’s n is defined on a mesoscale using subgrid data of land classification type and local averaging (the community standard).

+ =

Λ is a generalized rectangle defined by physical data on the land classification types.

  • T. Butler

Measure-Theoretic UQ 27 / 34

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SLIDE 32

Relative skewness of two station pairs

q1 x q2 q1 x q6

Images of Λ

  • T. Butler

Measure-Theoretic UQ 28 / 34

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SLIDE 33

Effective support of inverse solutions

0.08 0.09 0. 1 0.11 0.12 0.13 0.14 0.11 0.12 0.13 0.14 0.15 0.16 0.17 0.18 0.19 λ 1 λ 2 100 200 300 400 500 600 0.08 0.09 0. 1 0.11 0.12 0.13 0.14 0.11 0.12 0.13 0.14 0.15 0.16 0.17 0.18 0.19 λ 1 λ 2 500 1000 1500 2000 2500

q1 x q2 q1 x q6

Inverse probability densities

  • T. Butler

Measure-Theoretic UQ 29 / 34

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SLIDE 34

Collaborative Research on Contaminant Transport

LANL UT-Austin CSU/UCD

Application & UQ Goals Model Development V&V Numerical Implementation Error Analysis UQ Theoretical UQ Analysis Methodology Algorithms UQ Formulation Analysis UQ Algorithms Error Analysis Models Numerics

  • T. Butler

Measure-Theoretic UQ 30 / 34

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SLIDE 35

LANL Chromium Site

50 ppb 1000 ppb

Model predicted plume shape (~2012) Cr6+ MCL 50 ppb

Sandia Canyon Mortandad Canyon

Vadose zone (~300 m)

Single-screen aquifer monitoring wells Two-screen aquifer monitoring wells

Current conceptual model for chromium migration in the subsurface is supported by multiple lines of evidence. Approximately 54,000 kg of Cr6+ released in Sandia Canyon between 1956 and 1972 (uncertain). Contaminant mass distribution in the subsurface is uncertain. Contaminant source location and mass flux at the top of the regional aquifer are unknown due to complex 3D pathways through the vadose zone. Limited remedial options due to aquifer depth and complexities in the subsurface flow.

  • T. Butler

Measure-Theoretic UQ 31 / 34

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SLIDE 36

End-to-End Quantification of Uncertainties

Successfully computed a probability measure quantifying uncertainties in source and transport parameters in a 3-D advection-dispersion- reaction model using available data from monitoring wells. Interrogated computed probability measure to analyze probability of failure

  • f various remediation

strategies to reduce contaminant concentrations below MCL at production wells at various future times.

No Remediation w1 w2 w3 w4 w5 w6 w7 w8 w9 w10 2 5 . t = 5y w1 w2 w3 w4 w5 w6 w7 w8 w9 w10 25.0 t = 10y w1 w2 w3 w4 w5 w6 w7 w8 w9 w10 25.0 t = 20y Strategy 1 w1 w2 w3 w4 w5 w6 w7 w8 w9 w10 25.0 w1 w2 w3 w4 w5 w6 w7 w8 w9 w10 w1 w2 w3 w4 w5 w6 w7 w8 w9 w10 Strategy 2 w1 w2 w3 w4 w5 w6 w7 w8 w9 w10 2 5 . w1 w2 w3 w4 w5 w6 w7 w8 w9 w10 2 5 . 2 5 . w1 w2 w3 w4 w5 w6 w7 w8 w9 w10 Strategy 3 w1 w2 w3 w4 w5 w6 w7 w8 w9 w10 2 5 . w1 w2 w3 w4 w5 w6 w7 w8 w9 w10 25.0 w1 w2 w3 w4 w5 w6 w7 w8 w9 w10 25.0 Strategy 4 w1 w2 w3 w4 w5 w6 w7 w8 w9 w10 25.0 2 5 . w1 w2 w3 w4 w5 w6 w7 w8 w9 w10 25.0 25.0 w1 w2 w3 w4 w5 w6 w7 w8 w9 w10 25.0 25.0 2 5 . 2 5 . Strategy 5 w1 w2 w3 w4 w5 w6 w7 w8 w9 w10 2 5 . w1 w2 w3 w4 w5 w6 w7 w8 w9 w10 2 5 . w1 w2 w3 w4 w5 w6 w7 w8 w9 w10

5 10 15 20 25 30 35 40 45 50 Concentration [mg/kg]

  • T. Butler

Measure-Theoretic UQ 32 / 34

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SLIDE 37

Final remarks

Robust convergence results hold for BΛ-consistent sampling with respect to µΛ. Changing to adaptive algorithms alters the a priori and a posteriori analyses in straightforward ways with respect to implicitly specified contour events. Current applications include Storm surge and coastal modeling Contaminant transport Geometric optimization General nonlinear ODEs (e.g., autocatalytic chemical and epidemic models) and PDEs (e.g., stationary Navier-Stokes flow past obstructions)

  • T. Butler

Measure-Theoretic UQ 33 / 34

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SLIDE 38

The BET Python Package

B E T

Open-source package for measure-theoretic inversion: https://github.com/UT-CHG/BET Documentation includes several introductory examples: http://ut-chg.github.io/BET/

Questions?

  • T. Butler

Measure-Theoretic UQ 34 / 34