uncertainty quantification for plasma- and pwi- models · 2018-07-19 · uncertainty quantification...
TRANSCRIPT
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
Overview
Motivation Approaches to Uncertainty Quantification Example: Vlasov-Poisson-system Sensitivity Analysis Emulators: Gaussian Processes Example: SOLPS-data Challenges
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
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
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
5
Assessment:
Motivation
6
Motivation
Recognized in many other (engineering) fieldsExample: V&V process flowchart from the ASME Solid Mechanics V&V guide (2006).
May be to simplistic...
7
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
8
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
9
UQ-Model System: Vlasov-Poisson
• Phase-space distribution function f(x,v) of collisionless plasma:
q
10
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?
11
Methods in Uncertainty Quantification
Methods applied in Uncertainty Quantification
➢ Sampling
➢ Spectral Expansion ➢ Galerkin Approach➢ Stochastic Collocation➢ Discrete Projection
➢ Surrogate models (eg. Gaussian processes)
12
➢ 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
13
➢ 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
14
➢ 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
15
➢ 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:
16
➢ 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
17
➢ Spectral Expansion – Non-intrusive I
Methods in Uncertainty Quantification
Stochastic Collocation• Collocation Approach• No need to reformulate equations• Pseudospectral approach:
18
➢ 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:
19
➢ 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
20
➢ 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
21
➢ Covariance matrix: correlation length essential:
no white noise!
Methods in Uncertainty Quantification
22
➢ I) Polynomial chaos expansion – Galerkin approach:
Methods in Uncertainty Quantification
• Let M ~ GP(μ, C)• Apply truncated Karhunen-Loeve expansion
• Express problem in c :
23
➢ I) Covariance matrix:
Methods in Uncertainty Quantification
Eigenvectors Eigenvalues
24
➢ Expansion of random field in KL-representation
Methods in Uncertainty Quantification
25
➢ Expansion of random field in KL-representation
Methods in Uncertainty Quantification
26
➢ Expansion of random field in KL-representation
Methods in Uncertainty Quantification
27
➢ Expansion of random field in KL-representation
Methods in Uncertainty Quantification
28
➢ Expansion of random field in KL-representation
Methods in Uncertainty Quantification
29
➢ Random samples of random field in KL-representation
Methods in Uncertainty Quantification
30
➢ Vlasov-Poisson with fluctuating external E-field: Uncertainty?
➢ Now multivariate Gauss-Hermite integration:
Result
Example: M=2, P=3
31
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
32
Sensitivity Assessment: Example
➢ I) Polynomial chaos expansion
33
➢ Vlasov-Poisson with fluctuating external E-field: all gentle...
➢ Initial E-field amplitude more important than E-field variation
Result
34
➢ Vlasov-Poisson with fluctuating external E-field
Result
35
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)
36
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
37
38
39
40
41
SOLPS-Application
SOLPS-Data base: - 1500 parameters- collected by D. Coster
Idea: exploit for initialization, scans
42
SOLPS-Data
• Result of GP-interpolation• Color code: normalized variance
43
SOLPS-Application
Input-space locations with largest information gain:
44
SOLPS-Application
Input-space locations with largest information gain:
Need to define region of interest and 'integral' measures
45
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...
46
SOLPS-Application
• Present data base “insufficient” …• Physic based line-scans (power, density, temperature...)• Does not cover space (eg. approx. Latin hypercube)
47
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?
48
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
49
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
50
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
51
The End
Thank you!
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, ...
53
Gaussian processes
➢ Gaussian process based emulators
54
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))
55
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)
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
57
58
59