Uncertainty Quantification for Plasma- and PWI- models · 2018-07-19 · Uncertainty Quantification...
Transcript of Uncertainty Quantification for Plasma- and PWI- models · 2018-07-19 · Uncertainty Quantification...
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Uncertainty Quantification forPlasma- and PWI- models
U. von Toussaint, R. Preuss, D. CosterMax-Planck-Institut für Plasmaphysik, Garching
EURATOM Association
UQ-Workshop, Stony Brook, 7.-9. November 2015
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Overview
Motivation Approaches to Uncertainty Quantification Example: Vlasov-Poisson-system Sensitivity Analysis Emulators: Gaussian Processes Example: SOLPS-data Challenges
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Preliminaries
Two types of uncertainties (lack of knowledge) are to be distinguished: Isolable uncertainties eg.
- electron mass- cross-sections from first principles - ...
non-isolable uncertainties- most non-trivial simulations eg. climate- or plasma-
simulation output- complex/integrated data analysis
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Verification, Validation and Uncertainty Quantification in a scientific software/modeling context:
Simulations provide approximate solutions to problems for which we do not know the exact solution.
This leads to three more questions:• How good are the approximations?• How do you test the software?• 'Predictive' power?
Motivation
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Different Assessment Techniques for Different Sources of Uncertainty or Error
Problem:
• Model(s) not good enough - Validation• Numerics not good enough
• Algorithm is not implemented - Code verification correctly
• Algorithm is flawed - Code verification• Problem definition not good enough → UQ
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Assessment:
Motivation
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Motivation
Recognized in many other (engineering) fieldsExample: V&V process flowchart from the ASME Solid Mechanics V&V guide (2006).
May be to simplistic...
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Motivation: Design Cycle
physical model of system
mathematical model
numerical model
computer simulation
validation
Simulation-based decision making e.g. design, control, operations,
material selection, planning
scalable algorithms & solvers
data/observations
(multiscale) models
advanced geometry &discretization schemesparameter inversion
data assimilationmodel/data error controluncertainty quantification
visualizationdata mining/science optimization
stochastic modelsuncertainty quantification
verificationapproximation / error control
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Motivation
• Systematic Uncertainty Quantification in Plasma Physics
• Often still at the very beginning (parameter scans)
• Importance increasingly realized ('shortfall'): V&V&UQ
• connection with 'surrogate' models
Setting up a reference case based on relevant, non-trivial and well
understood system in plasma physics:
Vlasov-Poisson-Model
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UQ-Model System: Vlasov-Poisson
• Phase-space distribution function f(x,v) of collisionless plasma:
q
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UQ-Model System: Vlasov-Poisson
• Phase-space distribution function f(x,v,t) of collisioness plasma:
System details:• 1+1-dimensional (x,v-space)• Solver: Semi-Lagrangian-solver• Boundary conditions: periodic• Negligible B-field contributions• Static ion-background• External random E-field contribution E0(x)
Effect of the random E-field?
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Methods in Uncertainty Quantification
Methods applied in Uncertainty Quantification
➢ Sampling
➢ Spectral Expansion ➢ Galerkin Approach➢ Stochastic Collocation➢ Discrete Projection
➢ Surrogate models (eg. Gaussian processes)
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➢ Sampling approach
Use random sample S={x(i), i=1,...,N}
to compute distribution p(y) and moments :
Methods in Uncertainty Quantification
⟨ y j⟩=1N∑i=1
N
M (x i) j
var ( y )=⟨ y2⟩−⟨ y ⟩2
+ higher order terms….
Advantages: - Includes all correlations (→ verification)- Parallel & straightforward
Downside: - sample point density: curse of dimensions: ρ~ρ0-Dim
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➢ Sampling approach
Example: Vlasov-Poisson, random external field E=E0+N(μ=0,σ=0.1E
0)
t=0.4 s t=2.8 s
Results
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➢ Sampling approach
Example: Vlasov-Poisson, random external field E=E0+N(μ=0,σ=0.1E
0)
t=0.4 s t=2.8 s
Results
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➢ Spectral Expansion (Polynomial chaos expansion)
Methods in Uncertainty Quantification
• Consider a probabilistic model for the uncertain input parameters
• Then, under weak assumptions, the random model response Y=Y(X) can be expressed in an orthonormal basis:
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➢ Spectral Expansion
Two different approaches:
➢ Intrusive methods depend on the formulation and solution of a
stochastic version of the original model
➢ Nonintrusive methods require multiple solutions of the original
(deterministic) model only
Methods in Uncertainty Quantification
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➢ Spectral Expansion – Non-intrusive I
Methods in Uncertainty Quantification
Stochastic Collocation• Collocation Approach• No need to reformulate equations• Pseudospectral approach:
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➢ Spectral Expansion – Non-intrusive I
➢ Computation of multi-dim integrals:
- exploit structure (Gaussian p(ξ)): Gauss-Hermite quadrature
Methods in Uncertainty Quantification
with
Mean <Y>=a0 and variance σ2:
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➢ Spectral Expansion – Non-intrusive I
Methods in Uncertainty Quantification
Example: Vlasov-Poisson, random external field E=E0+N(μ=0,σ=0.1E
0)
t=0.4 s t=2.8 s
4-th order expansion in excellent agreement with MC in fraction of computing time
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➢ Spectral Expansion – Non-intrusive II
Methods in Uncertainty Quantification
Now: Random field M(x,t) instead of random variable(s)
Infinite number of random variables ??
Example: E-field fluctuation
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➢ Covariance matrix: correlation length essential:
no white noise!
Methods in Uncertainty Quantification
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➢ I) Polynomial chaos expansion – Galerkin approach:
Methods in Uncertainty Quantification
• Let M ~ GP(μ, C)• Apply truncated Karhunen-Loeve expansion
• Express problem in c :
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➢ I) Covariance matrix:
Methods in Uncertainty Quantification
Eigenvectors Eigenvalues
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➢ Expansion of random field in KL-representation
Methods in Uncertainty Quantification
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➢ Expansion of random field in KL-representation
Methods in Uncertainty Quantification
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➢ Expansion of random field in KL-representation
Methods in Uncertainty Quantification
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➢ Expansion of random field in KL-representation
Methods in Uncertainty Quantification
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➢ Expansion of random field in KL-representation
Methods in Uncertainty Quantification
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➢ Random samples of random field in KL-representation
Methods in Uncertainty Quantification
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➢ Vlasov-Poisson with fluctuating external E-field: Uncertainty?
➢ Now multivariate Gauss-Hermite integration:
Result
Example: M=2, P=3
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Sensitivity Assessment
➢ I) Polynomial chaos expansion
Sobol indices represent the fraction of the model variance that can be assigned to the respective input variables
• Partial variance of model:
• The Sobol indices are normalized:
with
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Sensitivity Assessment: Example
➢ I) Polynomial chaos expansion
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➢ Vlasov-Poisson with fluctuating external E-field: all gentle...
➢ Initial E-field amplitude more important than E-field variation
Result
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➢ Vlasov-Poisson with fluctuating external E-field
Result
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Conclusion I
➢ Spectral expansion approach
- intrusive methods: best (only?) suited for new codes
- non-intrusive methods: general purpose approach
- selection of collocation points
- sparse methods for larger problems possible
- influence of input parameter combinations as
byproduct: Sobol decomposition
➢ Proof of principle for 1+1 Vlasov-Poisson Equation
➢ Validated against MC-approach (and other test cases)
➢ Some implementations: eg. DAKOTA
• Best suited for medium number of dimensions (O(10)) and
moderately expensive simulators (forward models)
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Emulators
x Code y( x)
A simulator is a model of a real process Typically implemented as a computer code Think of it as a function taking inputs x and giving
outputs y:
An emulator is a statistical representation of this function Expressing knowledge/beliefs about what the output
will be at any given input(s) Built using prior information and a training set of
model runs
What if conditions for Spectral expansion do not hold? Use of emulators as surrogate for simulators
Focus on Gaussian Processes
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SOLPS-Application
SOLPS-Data base: - 1500 parameters- collected by D. Coster
Idea: exploit for initialization, scans
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SOLPS-Data
• Result of GP-interpolation• Color code: normalized variance
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SOLPS-Application
Input-space locations with largest information gain:
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SOLPS-Application
Input-space locations with largest information gain:
Need to define region of interest and 'integral' measures
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SOLPS-Application
• GP-algorithm established
- Mid size problems: 1000 data points, ~50 dimensions- many areas of application, ie. fitting of MD-potentials to
DFT-data
but...
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SOLPS-Application
• Present data base “insufficient” …• Physic based line-scans (power, density, temperature...)• Does not cover space (eg. approx. Latin hypercube)
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SOLPS-Application
• GP-algorithm now in routine application• Present data base insufficient
Semi-automated data base generation (eg. Bayesian Experimental Design):
• Design Cycle:
- Determine best location(s) for next simulation(s): Utility
- Recompute uncertainty estimates
- Check for design criteria: exit?
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SOLPS-Application
• GP-algorithm now in routine application• Present data base insufficient
Semi-automated data base generation (eg. Bayesian Experimental Design):
• Design Cycle:
- Determine best location(s) for next simulation(s): Utility
- Recompute uncertainty estimates
- Check for design criteria: exit?• Challenges: - adequate coverage of relevant input space
- code convergence
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Gaussian Processes: Challenges
• Scaling: N3 : not yet prohibitive
• Correlated output
- Standard approach: independent scalar response variablesDrawback: Prediction not satisfactory: co-variance
- Difficulty: Cokriging extremely expensive
• Phase transitions: 'global' scale of covariance matrix
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Conclusion II
• Gaussian Processes are powerful tool for high-dimensional interpolation → fast emulators → UQ
• Analytical formulas for mean and variance → exp. Design• Best suited for scalar output
• Automated experimental design cycle: - works on test cases- At present: too much human intervention needed for
plasma codes- Problems appear solvable
• Correlated output: research and tests ongoing
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The End
Thank you!
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GEM-SA course - session 2 52
References
1. O'Hagan, A. (2006). Bayesian analysis of computer code outputs: a tutorial. Reliability Engineering and System Safety 91, 1290-1300.
2. Santner, T. J., Williams, B. J. and Notz, W. I. (2003). The Design and Analysis of Computer Experiments. New York: Springer.
3. Rasmussen, C. E., and Williams, C. K. I. (2006). Gaussian
Processes for Machine Learning. Cambridge, MA: MIT Press.
Please note: Some introductory slides of this presentation are adapted from C.E. Rasmussen, A.O. Hagan, B.Sudret,Y. Mazouk, ...
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Gaussian processes
➢ Gaussian process based emulators
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Uncertainty Quantification
➢ Quantifying uncertainty in computer simulations
➢ Sensitivity Analysis (which parameters are most important?)
➢ Variability Analysis (intrinsic variation associated with physical system)
➢ Uncertainty Analysis (degree of confidence in data and models)
Uncertainty quantification in computer simulations is an active&recent
research area
➢ Meteorology
➢ Geology
➢ Engineering (FEM-codes)
➢ Military Research (Accelerated Strategic Computing Initiative (ASCI
(2000))
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Computer Experiments
➢ Taxonomy of tasks in UQ
1) Interpolation/Prediction
2) Experimental Design
3) Uncertainty/Output Analysis
4) Sensitivity Analysis
5) Calibration
6) Prediction
7) Robust inputs
Most tasks have 'natural' solutions by approximating y(xc,x
e) by a
fast predictor (metamodel, simulator, emulator)
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Predicting Data:
• Infer tN+1 given tN :
– Simple, because conditional distribution is also a Gaussian
–
– Use incremental form of
)],(),...,,([ 111 NNNT xxkxxkk ++=
),( 11 ++= NN xxkκ
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