A bi-fidelity method for kinetic models with uncertainties - CIRM
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A bi-fidelity method for kinetic models with
uncertainties
Liu Liu
The Chinese University of Hong Kong
Collaborators: Xueyu Zhu (University of Iowa)
Lorenzo Pareschi (University of Ferrara)
Jean Morlet Chair Conference on
Kinetic Equations: From Modeling Computation to Analysis,
Mar 22-26, 2021
Outline • Background on Uncertainty Quantification (UQ) for kinetic equations • A bi-fidelity stochastic collocation method • Numerical examples for the Boltzmann and linear transport model • Error and convergence analysis • Conclusion
Kinetic equations
time
Continuum theory
(Euler, Navier-Stokes)
Kinetic theory bridges atomistic and
(Boltzmann) continuum models
Molecular dynamics
(Newton’s equation)
Quantum mechanics
(Schrodinger)
space
A hierarchy of multiscale modeling
Applications:
Rarefied gas: Boltzmann equation Nuclear reactor (neutron transport), radiative transfer
Plasma (Vlasov-Poisson, Landau, Fokker-Planck) Biology
Semiconductor device modeling Collective behavior in biological and social sciences, etc.
UQ for kinetic equations
UQ: Quantifying the uncertainties is important to assess, validate and improve the
underlying models.
Most modeling of physical systems contain various sources of uncertainties.
Kinetic equations: usually derived from N-body Newton’s second law, mean-field
limit etc, the modeling itself is uncertain.
Parametric uncertainty
Collision kernels are often empirical, due to incomplete knowledge of the
interaction mechanism; inaccurate measurement of the initial and boundary data,
etc.
Despite its popularity in solid mechanics, elliptic equations (Abgrall, Babuska,
Ghanem, Gunzburger, Hesthaven, Karniadakis, Mishra, Xiu, Webster, Schwab
etc), few efforts on UQ for kinetic equations have been done until recent years.
:
Stochastic Collocation
Stochastic Collocation
Np
{zj , u(zj )}j=1 ! uN (Z) = u(Z)
expensive to obtain for large N
Interpolation/quadrature rule based
on gPC expansion,
Non-intrusive
Np
{(xj , fj )}j=1 ! fe(X) ⇡ f (X) Fast convergence for smooth problems
However,
‣ The deterministic code is too computationally expensive
‣ Curse of dimensionality when d >>1
In general, this class of problem is challenging. There are attempts in:
‣ Sparse grid, sparse wavelet High cost: repetitive runs of deterministic solvers
basis, compressed sensing,
….
Problem setup
8
< ut (x, t, Z) = L(u), in D ⇥ (0, T ] ⇥ IZ ,
B(u) = 0, on @D ⇥ [0, T ] ⇥ IZ ,
:
u = u0 , in D ⇥ {t = 0} ⇥ IZ .
uL : low fidelity solution (cheap)
uH : high fidelity solution (expensive)
H
The goal: build a surrogate of u in a non-intrusive way
X m
v(x, t, z) = cn (z)uH (x, t, zn ), zn 2 IZ
n=1
Q1: How to choose zn intelligently?
Q2: How to compute cn without extensive sampling from the high- delity model?
A.Narayan, C.Gittelson, D.Xiu 14’ 6
fiPoint selection Problems of interest
We seek a solution, u(x, µ) , whi
Key idea: Explore the parameter space by using
The Reduced through a linear
theApproach:
Basis cheap Aparametrized
Motivation P
low-fi
model L(x, µ)u(x, µ) = f (x, µ) x
The Main Idea: Reduce µ) =
u(x,the Basis
g(x, µ) x
Search the space by the low-fi model via greedy algorithm:
We might expect a good approximation using a Galerkin
using solutions for “well chosen”The
Assumption: solution
sampling varies
of parameters
Lfunctions. L
zm = arg max d(u (z), Um 1)
dimensional manifold under par
z2
M
Candidate set: = {z1 , . . . , zM }
Low-fi approximation space: H
uu(µ(z
j )j
)
uu(µ)
H
(z)
X
L L L
Um 1 ( ) ={u (z1 ), . . . , u (zm 1 )}
Wednesday, November 2, 11
Enrich the space by finding the furtherest point away from
the space spanned by the existing set
The greedy choice is (almost) optimal (DeVore 13’).Lifting procedure
Construct {cn } as projection coefficients from low-fidelity model:
m
X
v(z)L = PU L ( ) uL (z) = cn (z)uL (zn )
n=1
Using the same coefficients, construct the approximation rule for
high-fidelity model m X
H H
v(x, t, z) = cn (z)u (zn )
n=1
This is in fact a lifting operator from low-fi space to high-fi space
Justification: e.g., scalar scaling, coarse/fine mesh
8Overview of Bi-fidelity algorithm
‣ Run the low-fi models at each point of to obtain uL ( ), U L ( )
select the most “important” m points —
Offline
Run the high-fi models on those m points to get u ( ) H
most expensive part
‣ Bi-fidelity approximation:
For any given z 2 Iz , compute low-fi projection coefficients
m
X
v(z)L = PU L ( ) uL (z) = cn (z)uL (zn ) Online
n=1
Apply the same coefficients to uH ( ) to get the bi-fi approximation
m
X
v(z)H = cn (z)uH (zn )
n=1
This approximation quality depends on how well the low-fi model approximates
the functional variation of high-fi model in the parameter space.
9The Boltzmann equation
is the probability density distribution function, modeling the probability
of finding a particle at time t, position x, velocity v
is the Knudsen number that characterizes the degree of rarefiedness
of the gas; ratio of mean free path and the characteristic length scale
kinetic regime; hydrodynamic regime
Q is a quadratic integral operator modeling binary interaction
between particles
Ludwig Boltzmann, 1872’Boltzmann collision operator
non-linear double integral collision operator; high-dimensional in physical space
elastic collisionsOur choice of the low-fidelity model Liu-X.Zhu (JCP 19’)
5.1 A double-peak initial data test
Example 1: double peak ini al data test
We first consider the following initial data to mimic the Karhunen-Loeve expan-
sion of the random field:
8 d
!
>
> 1 X 1
zk⇢
>
> ⇢
⇢0 (x, z ) = 2 + sin(2⇡x) + 0.2 sin[2⇡(k + 1)x] ,
>
> 3 2k
>
> k=1
>
>
>
> u = (0.2, 0),
>
< 0
d
! (5.3)
>
> 1 X 1
zk T
>
> T
T0 (x, z ) = 3 + cos(2⇡x) + 0.2 cos[2⇡(k + 1)x] ,
>
> 4 2k
>
> k=1
>
> ✓ ◆
>
> ⇢0 |v u0 | 2
|v + u0 | 2
>
: f0 (x, v, z) = exp( ) + exp( ) .
4⇡T0 2T0 2T0
The uncertain collisional cross section is given by
b(z) = 1 + 0.5z1b , (5.4)
Here z⇢ = z1⇢ , · · · , zd⇢1 , zT = z1T , · · · , zdT1 , and zb = z1b represent the random
variables in the collision kernel, initial density and temperature. Let the initial
distribution f0 follow a double-peak non-equilibrium initial data [26]. Set d1 = 7, thus
this is a d = 15 dimensional problem in the random space. We use the Boltzmann
equation as the high-fidelity model and the Euler system as the low-fidelity model,
set x = 0.01, t = 8 ⇥ 10 4 (in both the high- and low-fidelity models), Nvh = 16,
space: 1d velocity: 2d random space: 15d
and the final time t = 0.1.
In Figure 1, we consider the fluid regime with " = 10 4 . This figure shows the
tiFluid regimes
The smaller Knudsen number is, the As epsilon approaches to zero, the Euler (low-fid)
lower level errors saturate commits less modeling error, thus can
capture more information of the high-fid model.kineticKinetic
regime with " = 1. Fast convergence of
regime
number of high-fidelity runs is observed. Even
The fluid description breaks down in the
to the previous two tests, a space,
physical satisfactory accur
yet the BF approximation
can still capture important variations of
solution in the random the space is achieved in b
high-fidelity model in the random
space
the errors with Nvl = 8 is smaller than that
Figure 3, we plot the high-, low- and the cor
l
r = 20, Nv = 8) for a particular sample p
and bi-fidelity solutions match quite well, whe
inaccurate at some spatial points. This exam
in the kinetic regime, the fluid description br
bi-fidelity solution can still capture important
(Boltzman equation) in the random space.5.2 Sod shock tube test
Example 2: Sod Shock Tube Test
We next consider a more challenging problem where the initial data is discontin-
uous. Assume the random collision kernel in the form
The collision kernel
dX
1 +1
b zkb
b(z ) = 1 + 0.5 ,
2k
k=1
and the random initial distribution
0
⇢ |v u0 | 2
f 0 (x, v, z) = 0
e 2T 0 ,
2⇡T
where the initial data for ⇢0 , u0 and T 0 is given by
8 d1
>
> X z T
>
> ⇢ l = 1, u l = (0, 0), T l (z T
) = 1 + 0.4 k
, x 0.5,
< 2k
k=1
> X1 T d
>
> 1 1 z
>
: ⇢r = , ur = (0, 0), Tr (zT ) = (1 + 0.4 k
), x > 0.5.
8 8 2k
k=1
Here zb = z1b , · · · , zdb1 +1 and zT = z1T , · · · , zdT1 represent the random variables in
the collision kernel and initial temperature. Set d1 = 7, then the total dimension
d of the random space is 15. We use the Boltzmann equation as the high-fidelity
model, and solve it by x = 0.01, t = 8 ⇥ 10 4 , and Nvh = 24, until the final time
t = 0.15. We shall employ the Euler equation as the low-fidelity model, and solve it4
n Figure 1, we consider the fluid Fluid regime
regime with " = 10 . This figure shows the
n L2 errors of ⇢, u1 , T between the high- and bi-fidelity solutions with di↵erent
10
-1 10
-1
0.9
10
1
-1
1
1
0.9
0.9
drature points in velocity space. Here u1 in the figures below stands for the first
0.8
0.7
0.8
0.8
0.7 0.7
10 -2
ponent of the two-dimensional bulk velocity u. It is clear that the error decays
10
-2 0.6 -2
10
0.5
0.6 0.6
0.5
0.5
with the number of high-fidelity runs. In addition, when Nvl increases, the error
0.4
0.4
0.4
10 -3 0.3
10 -3 0.3
10 -3 0.3
ween the high- and bi-fidelity solutions decreases. This is expected because the
0.2
0.2
0.1 0.2
0.1
5 10 15 20 25 0 0.1 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
r equation solved by more quadrature points in velocity space can capture more
5 10 15 20 25 0
5
0.1 0.2
10
0.3
15
0.4 0.5
20
0.6
25
0.7 0.8 0.9
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
1.4
mation about the high-fidelity model. 1.2 1.4
1.4
1.2
n Figure 2, fluid regime is considered and we vary " from " = 10 2 to " =
10 -1
0.8
1
10 -1
1.2
1
1 0.8
10 -1
. The Euler equation
15-d random z needs onlyis10chosen as the low-fidelity model, solved by the same
-2
0.6
0.8 0.6
high-fidelity (Boltzmann) runs!
10
10 -2
0.4
0.4
0.6
0.2
0.2
10 -2
0.4
17
0
0
10 -3 10 -3 0.2
-0.2 -0.2
5 10 15 20 25 0 0.1 5 0.2 0.3 10 0.4 0.5
15 0.6 200.7 0.8 25 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0
10 -3
-0.2
10 0 5 10 15 20 25 10 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
1.6 1.6
0 1.4 1.4
10
10 -1
Low-fid costs about
-1
1.2 1.6 10 1.2
1
0.01 computation time
1 1.4
10 -1 0.8
1.2
0.8
of high-fid solver
-2 -2
10 10
0.6 0.6
1
0.4 0.4
0.8
-2 0.2
10 0.2
10 -3 10 -3
5 10 15 20 25 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
5 10 15 20 25 0 0.6 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
0.4Knudsen number " varying in space show in Figure. 6 and given by
Example
✓ 3: Mixed
◆ regime
✓ ◆
3 1 11 11
"(x) = 10 + tanh 1 (x 0.5) + tanh 1 + (x 0.5) . (5.5)
2
Knudsen number " varying in 2
space 2 by
show in Figure. 6 and given
✓ ◆ ✓ ◆
The random initial 1 and collision
3data 11 kernel are given by (5.3)
11 and (5.4). The total
"(x) = 10 + tanh 1 (x 0.5) + tanh 1 + (x 0.5) . (5.5)
2 2 2
dimension of the random space is d = 15.
The random initial data and collision kernel are given by (5.3) and (5.4). The total
dimension of the random
0.8 space is d = 15.
0.7
0.8
0.6
0.7
0.5
0.6
0.4
0.5
0.3 0.4
0.2 0.3
0.2
0.1
0.1
0
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Figure 6: The distribution of "(x) in (5.5).
Figure 6: The distribution of "(x) in (5.5).
All the
Allnumerical parameters
the numerical used
parameters in in
used temporal
temporaland
andspatial
spatial discretizations arethe
discretizations are theMacroscopic states 1
-1 1
10
1
10 -1
-1 0.9
10
0.9
0.9
0.8
0.8
0.8
0.7
0.7
-2
10 -2
10 0.7
10 -2 0.6
0.6
0.6
0.5
0.5
0.5
0.4 0.4
10 -3 10 -3
0.4
10 -3
5 10 15 20 25 05 0.1 10
0.2 0.3 150.4 0.5 20 0.6 0.7 25 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
5 10 15 20 25 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
-1
10 10 -1 0.3 0.3
-1
10 0.2 0.3
0.2
0.1 0.2
0.1
0 0.1
10 -2 0
-2
10
-0.1
0
-0.1
10 -2
-0.2
-0.1
-0.2
-0.3
-0.2
-0.3
-3
10
10
-3 -0.4
5 10 15 20 25 0 -0.3 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
-0.4
5 10 15 20 25 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
10 -3
-0.4
5 10 15 20 25 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
1.2
1.2
10 -1 1.1
-1
1.2
10 1.1
1
10 -1 1.1
1
0.9
10 -2
1
0.9
-2
10 0.8
0.9
10 -2 0.8
0.7
10 -3 0.8
0.7
0.6
5 10 15 20 25 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
10 -3
0.7
0.6Bi- delity method for LTE
Linear transport equation (LTE) under diffusive scaling:
The Goldstein-Taylor (GT)
model is:
Let then GT model becomes
Liu-Pareschi-X.Zhu (21’)
fiBi- delity method for LTE
Motivations:
We choose the GT equation as our low-fidelity model, which shares the same
limiting diffusion equation as the LTE, by letting
Random cross section
fiBi- delity approxima ons
fi
tiError plots Fast convergence, high accuracy, with low computational cost
Numerical Examples A mixed regime test
Hypocoercivity analysis
Hypocoercivity theory: an important tool to study the stability and long-time
behavior of the solution for kinetic equations
The study involves
(i) a degenerate dissipative kinetic operator;
(ii) a conservative operator , such that the combination of these operators
leads to a convergence towards the global equilibrium state;
and a Lyapunov functional (with mixed x, v derivatives) is needed to get a quantitive
convergence rate.
Many experts have contributed in this direction for deterministic models:
Villani, Mouhot, Neumann, Guo, Duan, Briant, Arnold, Desvillettes, Dolbeault,
Schmeiser, etc.
Local parameter sensitivity analysis studies the long-time behavior of the solution;
explore how the randomness of the “input” propagates in time and how it affects the
solution in the long time.Sensitivity analysis for the analytic solution Main results: Liu-Jin (SIAM MMS 18’) E.Daus-Jin-Liu (KRM 19’): improvement on the assumptions for random collision kernel
Error analysis for bi-fidelity method
In Gamba-Jin-Liu 19’, we use projection error of the greedy algorithm (Cohen-DeVore,
15’) and adapt the hypocoercivity analysis to conduct error analysis of bi-fidelity
method for a general class of multiscale kinetic problems.
Splitting the error:
Perturbative setting:
smallError analysis for bi-fidelity method error decays algebraically with respect to N convergence rate independent of the dimension of the random space uniform in the Knudsen number estimate error estimate valid in both kinetic and hydrodynamic regimes
Other relevant work in the multi-fidelity framework Dimarco-Pareschi (19’,20’) on multiscale control variate methods, Hu-Pareschi-Wang (20’) on MLMC method for the BGK model; Multi-fidelity moment method for the BGK model with uncertainties (with Wang-Li-Liang-Zhu 21’), …..
Conclusions Bi-fidelity stochastic collocation method accelerate the computation of multiscale kinetic equations with high-dimensional random parameters; The hypocoercivity analysis for multiscale kinetic problems with uncertainty provides a tool to conduct error analysis for the bi-fidelity method; practical error bound, etc. Future work: build a hierarchy of fidelity models extend to higher dimensions with more complex random variables and collision kernels study sharper error estimates for control variate variance reduction MLMC method study kinetic equations using deep learning approaches (Chen-Liu-Mu 21’, Liu- Zhang-Zeng 21’), etc.
Thank you for your attention!
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