Estimating Iceberg Drag Coefficients Using Bayesian …reu/iceberg.pdf · Real Iceberg Model Recall...
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Aaron Alphonsus, Javier Salazar, Chi Zhang August 8, 2018 Dartmouth College, Cold Regions Research and Engineering Laboratory (CRREL)
Transcript of Estimating Iceberg Drag Coefficients Using Bayesian …reu/iceberg.pdf · Real Iceberg Model Recall...
Aaron Alphonsus, Javier Salazar, Chi ZhangAugust 8, 2018
Dartmouth College, Cold Regions Research and Engineering Laboratory (CRREL)
Motivation
Source: Soderman/NLSI Staff
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Bayesian Inference Framework
Goal: Infer coefficients θ⃗ given data x⃗
d⃗xdt = u⃗
mdu⃗dt = F⃗(θ)
Prior: Reasonable assumptions based on past experienceand knowledge.
Likelihood: A function describing the compatibility of theobserved data with the model.
Posterior: Result of updating the prior given the new data.
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Forward Model
F⃗(θ) = ma⃗ F⃗(θ) = mdu⃗dt u⃗ =d⃗xdt
Damped Harmonic Oscillator:
F⃗(θ) = −θ1F⃗spring − θ2F⃗damping
Iceberg Model:
F⃗(θ) = θ1F⃗water + θ2F⃗air + F⃗coriolis
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Prior & Likelihood
Prior: A distribution allowing only non-negative values forθ1 and θ2.
Likelihood: A function showing model-data mismatch for eachgiven θ.
π(d|θ) ∝ exp(L(∥d− G(θ)∥2))
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Bayes’ Rule
Posterior ∝ Prior · Likelihoodπ(θ|d) ∝ π(θ)π(d|θ)
Goal: Generate samples from the posterior distributionMethod: Markov chain Monte Carlo (MCMC) sampling
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Markov chain Monte Carlo (MCMC)
Metropolis (1953) & Hastings (1970)
θt+1 only depends on θt
3 step algorithm (for t = 1→ ∞):
1. Propose new point:θ̂ ∼ q(·|θt)
2. Compute acceptance rate α:
0 ≤ α(θt, θ̂) ≤ 1
3. Accept / Reject:
θt+1 =
{θ̂ with probability αθt otherwise
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MCMC - Visualization
Source: The University of British Colombia, Ricky Chen
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Harmonic Oscillator
Recall the forward model:
d⃗xdt = u⃗
F⃗(θ) = −θ1F⃗spring − θ2F⃗damping
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Data with Additive Noise
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Sampled Posterior Distribution
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Posterior Predictive Distribution
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Real Iceberg Model
Recall the forward model:d⃗xdt = u⃗
mdu⃗dt = θ1F⃗water + θ2F⃗air + F⃗coriolis
F⃗air(x, y, t) = |⃗vair − v⃗ice|(⃗vair − v⃗ice)
F⃗water(x, y, t) = |⃗vwater − v⃗ice|(⃗vwater − v⃗ice)
Source: Mountain(1980)
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Sample Forward Model Run
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Approximate Prior v/s Posterior Predictive
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Simplified Iceberg Model
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Data with Additive Noise
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Sampled Posterior Distribution
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Posterior Predictive Distribution
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Conclusion
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Questions
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Damping Ratios of Oscillatory Systems
Source: Stuart Aitken, University of Leeds
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Posterior Predictive Distribution
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