Bayesian Data Assimilation using Stochastic Spectral ... - MUCM

**Bayesian** **Data** **Assimilation** **using**

**Stochastic** **Spectral** Methods

Richard Dwight, Alex Loeven, Hester Bijl

24-8-2010

Delft

University of

Technology

Challenge the future

Overview

1. Probabilistic collocation (PC) for uncertainty

quantification (UQ).

2. Probabilistic collocation with MCMC for **Bayesian**

inference.

3. Model problem: Poisson’s equation with unknown

dissipation coefficient.

1. Model-as-prior approach for handling model

discrepancy.

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Uncertainty quantification problem

Main goals:

• Cheap propogation of uncertainties.

• No modification of the code (non-intrusive).

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Probabilistic collocation

Central idea:

• Use input pdf-weighted Gauss quadrature rule in probability

space to approximate:

• Ivo Babuška, Fabio Nobile, and Raúl Tempone, “A **Stochastic** Collocation Method for Elliptic

Partial Differential Equations with Random Input **Data**” SIAM J. Numer. Anal. Volume 45,

Issue 3, pp. 1005-1034 (2007).

• G.J.A. Loeven, "Efficient Uncertainty Quantification in Computational Fluid Dynamics”,

PhD Thesis, TU Delft, 2010 (http://repository.tudelft.nl/, search Loeven).

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Probabilistic collocation - Derivation

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Probabilistic collocation - Derivation

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Probabilistic collocation - Derivation

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Probabilistic collocation - Derivation

Complete decoupling in probability space.

Given u k the full stochastic solution is:

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Probabilistic collocation - Support

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Case study: NASA’s rotor

• 36 blades, aspect ratio of 1.19, tip-clearance of 0.365 mm

• Grid of 600,000 cells around 1 blade with periodic b.c.’s

• RANS + SA turbulence model

• Experimental data from Reid and Moore

- Reid and Moore, Design and overall performance of four highly loaded, high-speed inlet

stages for an advanced high-pressure-ratio core compressor, NASA-TP-1337, 1978

- Dunham, CFD validation for propulsion system components, AGARD report 355, 1998.

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Variability in the flow field

• Mean static pressure is similar to a deterministic simulation

• Standard deviation show largest variation near the shock

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Compressor maps show mainly the effect

of the uncertainty in the mass flow

• Bars show ±s around the mean

• Small effect on the efficiency and pressure ratio

• Large effect on the mass flow

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Probabilitic collocation for **Bayesian**

Inference

Experiment **Data**

Numerics Model

For example:

= Measured C L on an airfoil

where

is a complete CFD solver.

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Model and **Data**

Using Bayes:

Need a statistical model for

So:

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Approximation of Model by PC

The full model is too expensive, use PC based on prior:

No accurate error estimate for Gauss rules (unlike

for Gauss-Kronrod rules).

But since we‘re already in a **Bayesian** framework, use

a poor error estimate , and:

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Poisson Problem with Uncertain

Dissipation Coefficient

Application: Oil exploration **using** Very Low

Frequency electromagnetic waves (a diffusive

process).

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Poisson problem test-case – 4d

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MAP Estimator – Results

Adjoint-based optimization

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Metropolis-Hastings - Results

10000 samples – including 5000 burn-in

80% confidence interval

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PC Posterior Approximations

Marginals of variables 3 and 4 (step and wave amplitudes)

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PC Posterior Approximations

Lower and upper step position – no PC error in model

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PC Posterior Approximations

Lower and upper step position – with PC error in model

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PC(1) with MC sampling – Results

80% confidence interval

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PC(2) with MC sampling – Results

80% confidence interval

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PC(3) with MC sampling – Results

80% confidence interval

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PC(4) with MC sampling – Results

80% confidence interval

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PC(5) with MC sampling – Results

80% confidence interval

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Conclusions

Method # Model evals.

Optimization ~50-100

MCMC ~5000-10000

PC(2) in 4d + MCMC 81

PC(3) in 4d + MCMC 256

PC(M) in N-dimensions (M+1) N

• Probabilistic collocation can be an accurate surrogate –

informed by a Baysian prior.

• Incorperating surrogate error into the statistical model is

inexpensive (especially appropriate for Gaussian process

surrogates).

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Treating modelling error

Model-as-Prior approach

Fundamental problem with statistical model:

Ignores modelling error:

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Model-as-Prior approach

Idea: If m is trivial, model discrepancy will be zero.

So redefine parameters to include flow solution:

Now , is a good model.

a new Prior

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Model-as-Prior approach

Old Modified

**Stochastic** parameters w,

Model C L(w, ), R(w, ) = 0 C L(w, )

Prior P() ~ N(, 1) P 0[ ||R(w, )|| ]

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Model-as-Prior approach

Bayes:

Only problem: we have many millions of parameters.

Simplify:

• Assume Gaussian:

• Only find maximum of

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Model-as-Prior approach

Bayes:

This has a maximum when:

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Model-as-Prior approach

Does anybody recognise:

Least-squares Finite-Element Methods (LS-FEM)

A modified LS-FEM functional:

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New LS-FEM Solver

• Arbitrary governing equations

• High-order, 2d/3d

• Parallel

• hp-adaptitivity

• Steady/unsteady

• C++, deal.II FEM library

Adaptivity + parallel

Unsteady + parallel

Burgers

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3d

Rectangular Channel Test-Case

Problem setup

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Rectangular Channel Test-Case

Poor quality solution

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Rectangular Channel Test-Case

Inverse problem: unknown inflow

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Goal: To combine PIV and CFD

Experiment

• Measurement noise

• Systematic error

• Parametric uncertainty

• Limited data

• Cheap topology changes

Simulation

• Discretization error

• Model discrepancy

• Parametric uncertainty

• Complete data

• Cheap shape changes

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Conclusions

• LS-FEM + data is theoretically sound (Bayes)

• LS-FEM weights correspond to standard-deviations

• Potential framework for experiment-simuation

intergration.

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Thank you

for your attention

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