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
Inference for a deterministic model Prediction Space Observation Space Observation Predication Functionals Functionals I n v e n r s o e i t P a r Solution Space c o i d b e l e r m P r f o o f r m P Physics Model a e l r b a o m r e t e P d rs r a w r o F Space of Data and Parameters T. Butler Measure-Theoretic UQ 2 / 34
The mathematical model Ingredients Compact parameter domain Λ ⊂ R n Model M ( Y, λ ) with solution Y = Y ( λ ) for λ ∈ Λ Quantities of interest (QoI) Q ( λ ) = Q ( Y ( λ )) ∈ R d 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) ⊂ R d defines the observation space. T. Butler Measure-Theoretic UQ 3 / 34
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
Measure theory and inverse problems The core deterministic inverse problem imposes significant structure on 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
Measure theory ingredients Measurable space A specified domain X A σ -algebra B X defining the collection of sets whose size can be measured and the operations on those sets ( X, B X ) defines a measurable space. Measure space A procedure for computing the measure µ X of sets in the σ − algebra ( X, B X , µ X ) defines a measure space. T. Butler Measure-Theoretic UQ 6 / 34
Measure theory ingredients X How do we compute the µ X -measure of this event? T. Butler Measure-Theoretic UQ 6 / 34
Measure theory ingredients X We can approximate by using simpler sets from B X . T. Butler Measure-Theoretic UQ 6 / 34
Measure theory ingredients X Only certain events in the approximation require refinement. T. Butler Measure-Theoretic UQ 6 / 34
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
Describing the space of contour manifolds D L Λ Theorem There exists a transverse parametrization (TP) representing L in Λ . T. Butler Measure-Theoretic UQ 8 / 34
Solutions on ( L , B L ) Theorem A probability measure P D on ( D , B D ) corresponds to a unique probability measure P L on ( L , B L ) . D L Λ T. Butler Measure-Theoretic UQ 9 / 34
Natural but not desirable The inverse solution on ( L , B L ) requires minimal assumptions. But . . . the physically meaningful space is Λ . The ideal inferential target is a probability measure P Λ on (Λ , B Λ ) . To construct P Λ from P L requires answering two (related) questions. How are events in B Λ related to events in B L ? ◮ Ans: Use a transverse product σ -algebra. How are measures on (Λ , B Λ ) related to measures on ( L , B L ) ? ◮ Ans: Use the Disintegration Theorem. T. Butler Measure-Theoretic UQ 10 / 34
Structure of measures on Λ The Disintegration Theorem implies that any P Λ can be written uniquely as the product of a marginal P L on a TP and conditional probabilities { P ℓ } on contours {C ℓ } for ℓ ∈ L . The Disintegration Theorem is like Fubini’s theorem where P Λ ( A ) is written as an iterated integral involving P ℓ and P L . Theorem Specifying P ℓ on generalized contours corresponding to ℓ ∈ L determines a unique probability measure on (Λ , B Λ ) . T. Butler Measure-Theoretic UQ 11 / 34
The Standard Ansatz L s l r u o t n o C d e z i l a r e n e G g n o l A t y s i e n D o r m U n i f 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
Solution of the example under the Ansatz 1 0 L Λ Λ Inverse density on Λ Inverse density on L T. Butler Measure-Theoretic UQ 13 / 34
Simple function approximations of measures Theorem P Λ can be approximated using simple functions. P Λ is approximated on a partition of Λ taken as a subset of the generating sets to B Λ , B L , and {B C ℓ , ℓ ∈ L} . This results in a direct discretization of the iterated integral in the Disintegration Theorem. T. Butler Measure-Theoretic UQ 14 / 34
Approximations of events with random or regular sampling T. Butler Measure-Theoretic UQ 15 / 34
General measure-theoretic algorithmic outline Step 1: Define Voronoi tessellation {V j } N j =1 of Λ used to approximate events in both B Λ and C Λ . Step 2: Define approximation to P D (or its density) on ( D , B D ) by computation of P D ( I i ) for tessellation { I i } M i =1 of D . Step 3: Use Ansatz to compute P Λ ( V j ) . Step 4: Compute P Λ ( A ) for events of interest A ∈ B Λ using P Λ ( V j ) . T. Butler Measure-Theoretic UQ 16 / 34
Applying the algorithm/identifying errors 1 9 8 7 6 5 4 3 2 1 0 For A ∈ B Λ with µ Λ ( ∂A ) = 0 , we estimate P Λ ( A ) . The error in the µ Λ -volume of a Voronoi coverage of Q − 1 ( I i ) affects P Λ estimation. T. Butler Measure-Theoretic UQ 17 / 34
Applying the algorithm/identifying errors 1 1 9 9 8 8 7 7 6 6 5 5 4 4 3 3 2 2 1 1 0 0 Theorem As N, M → ∞ , the counting measure converges to P Λ . T. Butler Measure-Theoretic UQ 17 / 34
A priori optimal discretization Convex sets are optimal If contour events Q − 1 ( I j ) are convex for 1 ≤ j ≤ M , then there exists a � λ ( j ) � M j =1 such that P Λ ( Q − 1 ( I j )) = P Λ ,M ( Q − 1 ( I j )) . set of M samples 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 � λ ( i ) � N N = O ( M ) samples i =1 such that � � � < ǫ . � P Λ ( Q − 1 ( I j )) − P Λ ,N ( Q − 1 ( I j )) Finding the optimal discretization often requires a sensitivity analysis on Q . T. Butler Measure-Theoretic UQ 18 / 34
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
The geometry matters P Λ for fixed q 1 ( λ ) and varying q 2 ( λ ) with P Λ for different choices of q 1 ( λ ) and q 2 ( λ ) differing levels of skewness maintaining orthogonality of the contours. T. Butler Measure-Theoretic UQ 20 / 34
Condition of the numerical solution The skewness of an event determines the difficulty in computing accurate approximations using regular or uniform random sampling. v 1 v 1 v 2 Skewness Skew ( V, v i ) = | v i | Skew ( V ) = max Skew ( V, v i ) i | , | v ⊥ T. Butler Measure-Theoretic UQ 21 / 34
Skewness and the number of samples { x j } = a net of points in Λ . Theorem The number of samples needed to compute an accurate inverse solution is proportional to � � d − 1 max Skew ( J Q | L ( x j )) j T. Butler Measure-Theoretic UQ 22 / 34
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 V j ∈ Q − 1 ( I i ) . Deterministic error affects P Λ 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 on the probability of events. T. Butler Measure-Theoretic UQ 23 / 34
Stochastic and deterministic error terms Identify a computational algebra, ˜ B Λ ⊂ B Λ . We use the following notation: ˜ P Λ ,N,h is the fully computable counting measure. ˜ P Λ ,N is the counting measure with no deterministic error. For A ∈ ˜ B Λ , � � � � P Λ ( A ) − ˜ P Λ ( A ) − ˜ P Λ ,N,h ( A ) − ˜ ˜ P Λ ,N,h ( A ) = P Λ ,N ( A ) − P Λ ,N ( A ) . � �� � � �� � I 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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