Dynamically orthogonal field equations for stochastic flows with ...

Quantifying Uncertainty in Complex Turbulent Systems
Dynamically orthogonal field equations for stochastic flows
with applications
Part I
Themis Sapsis & Andrew Majda
New York University, CIMS
T. Sapsis, and P. Lermusiaux, Dynamically orthogonal field equations for continuous stochastic dynamical systems, Physica D, (2009).
T. Sapsis, Dynamically orthogonal field equations for stochastic fluid flows and particle dynamics, PhD Thesis, 2010, MIT
T. Sapsis, and P. Lermusiaux, Dynamical criteria for the evolution of the stochastic dimensionality in flows with uncertainty, Physica D, (2011).
T. Sapsis and A. Majda, Uncertainty Quantification in Reduced order subspaces, CIMS letcture notes, 2011
1

Large or ∞ dimensional systems with uncertainties
Examples of continuous systems
Geophysical fluid flows
Thermal transport in
heterogeneous media
Water waves
spectrum evolution
Examples of discrete systems
Systems biology
Structural systems
2
US power network

Challenges for continuous stochastic systems
Sources of uncertainty
• Initial and boundary conditions
• Limited observations and data
• Approximations on the modeling
• Internal instabilities (e.g. turbulence)
Computational Challenges
• Large or infinite dimensional phase-space
• Wide range of temporal and spatial scales
• Non-Gaussian, non-homogeneous statistics
• Transient and non-stationary dynamics
Re = 3000
Initial perturbation
__with energy
6
10 −
T = 30 22.5 50 27.5
What you see is a
1D family of solutions
Aim of this work
Develop new theory and tools for representing , evolving and describing stochastic
dynamical systems accurately and with low computational cost.
3

Some tools for high dimensional stochastic problems
Monte Carlo ‘based’ approaches
• Most widely used tool for realistic applications
• Can handle stochastic initial conditions, parametric uncertainties, random boundary conditions etc.
• Cost reduction with stochastic subspace reduction techniques (generation of optimal set of samples)
Ocean predictions and data assimilation: Lermusiaux, J. Comp. Phys., 2006; Lermusiaux, MWR, 1999.
Transport in porous media: B. Ganapathysubramanian & N. Zabaras, J. Comp. Phys., 2009.
• Spectrum evolution of nonlinear waves near coast
Ocean Waves Spectrum evolution: e.g. Dysthe et al, JFM, 2003
Slow convergence – vast computational cost for realistic simulations
Hopf Functional Equations (Hopf, 1952)
• Characteristic functional: contains the complete statistical information
• Hopf Functional Differential equations: Exact reformulation of the original problem
• Has been applied to a large series of problems including
Stochastic Navier Stokes, Hopf, J. Rat. Mech. & Anal., 1952.
Wave problems, Tatarskii, J. Exp. Theor. Phys., 1969.
Porous media, Beran, Statistical Continuum Theories, 1968.
• Infinite dimensional character of the functional equation
• Very far from practical applicability
• Need for representations of characteristic functional – very few available for simple cases
4

Some tools for high dimensional stochastic problems
Order Reduction Methods : Galerkin Projection (POD)
• Identification of an ‘optimal’ finite dimensional base
• Galerkin Projection of the original equations
• Derivation of low dimensional models that can capture the essential dynamical features (optimal base)
Ergodic properties for 2D Navier Stokes (W. E & J. Mattingly,2001)
• Respect dynamical symmetries
P. Holmes, J. Lumley, G. Berkooz: Turbulence, Coherent Structures, Dynamical Systems and Symmetry, 1996
• Poor performance for transient problems
• Need for a large set of realizations to extract optimal modes
• ‘Importance’ of a POD mode is not always connected with its energy
Polynomial Chaos Method (Ghanem & Spanos, 1991)
• Based on Wiener Polynomial Chaos Theory
• Fix a priori stochastic structure and evolve spatial structure
• Has been applied to both structural systems and fluids
Solid Mechanics, Ghanem & Spanos, 1991.
Stochastic Fluids, Xiu & Karniadakis, 2002.
• Slow convergence – exponential increase with order of PC method
• Poor performance for strongly non-Gaussian processes
• Difficulty to capture time dependent stochastic characteristics
5

Outline of the presentation
Part I
Part II
• Evolution of infinite-dimensional stochastic systems
– Problem formulation
– A unified overview of existing methods
– The dynamical orthogonality (DO) condition and the DO evolution equations
– Cost scaling of DO method – Curse of dimensionality
• Application: Linear systems & 2D Navier-Stokes
– Field equations for the evolution of stochastic Navier-Stokes
– Stochastic response of Navier-Stokes for various flow configurations
– Expansion-contraction of the stochastic subspace dimensionality
• UQ in reduced order spaces
– Information theoretic framework for UQ in reduced order subspace
– Application to linear systems: the effect of non-normal operators
– Application to quadratic systems: triad systems and Lorenz 96
• Reduced-order stochastic dynamics for quadratic systems
– Energy transport properties in quadratic systems
– Volume and length deformations in phase space in connection to energy flow from the mean field
– Nonlinear dimensionality of the support of the probability measure
– Illustration in specific cases
6

Problem Setup
Statement of the problem: A stochastic PDE
( ,; t ω)
∂u x
∂t
L
( t ω)
= ⎡⎣u x,; ; ω⎤⎦
u( x, t ; ω) = u ( x ; ω)
x∈
D B ⎡u| ∂D
⎤= h[ ∂D;
ω]
0 0
⎣
x∈
D
⎦
L[ g;ω ]
h
( x )
u0 ;ω
[ ∂D;
ω]
Nonlinear differential operator (possibly with stochastic coefficients)
Stochastic initial conditions (given full probabilistic information)
Stochastic boundary conditions (given full probabilistic information)
Goal: Evolve the full probabilistic information describing u( x,;
t ω)
An important representation property for the solution
( ,; t ω) ( , t) Y ( t; ω) ( , t)
u x u x u x
= +∑
s
i=
1
i
i
Advantage: Finite Dimensionality of Stochastic Space
Disadvantage: Redundancy of representation
7

For a stochastic field
Geometrical picture of KL expansion
Solution of the eigenvalue problem
( )
Provides principal directions ui
x
of the infinite-dimensional space
over which probability is distributed
∫ uu
( )( ( ) ( ))
u( x ;ω)
we have C ( , ) = E ω ⎡
⎢
( ; ω) − ( ; ω)
; ω − ;
uu
x y u x u x u y u y ω
⎣
2
( ) ( ) d = λ ( )
C x y u x x u y
ui 1
,
i i i
( x)
Probability measure
T
⎤
⎥⎦
( )
For each principal direction ui
x
the associated eigenvalue λ i
defines the spread of the probability
ui
3
( x)
For infinite-dimensional probability measures (case of stochastic fields) we have
Countable infinity of principal directions
ui 2
( x)
However for most of them (infinite)
the spread of the probability measure can be neglected!
8

Geometrical picture of KL expansion
Based on this property we define the stochastic subspace
( ) u ( x )
{ }
V σ , t = span , t | λ > σ
s cr i i cr
V ⊥
s
Probability measure
V s
Finite dimensional
V s
σcr
> λ←
u
i
1
( x,
t)
u
i
2
( x,
t)
( ,; t ω) ( , t) Y ( t; ω) ( , t)
u x u x u x
= +∑
• All the information in V ⊥
s
is captured by u( x,t)
• All the stochastic information is captured by the stochastic coefficients Y ( t;
ω)
s
i=
1
i
i
s=
dimV
s
i
9

A unified overview of existing methods (in fluids)
[C. Rowley,
Oberwolfach, 2008]
[Lermusiaux & Robinson,
Deep Sea Research, 2001]
( ,; t ω) ( , t) Y ( t; ω) ( , t)
u x u x u x
= +∑
10
s
i=
1
Uncertainty propagation via generalized
Polynomial-Chaos Method
Meecham & Siegel, Phys. Fluids, 1964
Xiu & Karniadakis, J. Comp. Physics, 2002
Knio & Maitre, Fluid Dyn. Research, 2006
i
Uncertainty propagation via Monte Carlo method
Lermusiaux & Robinson, Deep Sea Research, 2001
Lermusiaux, J. Comp. Phys., 2006
i
Redundant Representation
Uncertainty propagation via POD method
According to Lumley (Stochastic tools in Turbulence, 1971) it was introduced
independently by numerous people at different times, including Kosambi (1943),
Loeve (1945), Karhunen (1946), Pougachev (1953), Obukhov (1954 ).
B. Ganapathysubramanian & N. Zabaras, J. Comp. Phys., (under review)
[Xiu & Karniadakis,
J. Comp. Physics, 2002]

The dynamical orthogonality (DO) condition
(Sapsis & Lermusiaux, Physica D, 2009)
V ⊥
s
Probability measure
2
( x,
+δ )
ui t t
u
i
2
( x,
t)
How is the probability measure evolving?
Probability measure constrained to move inside
V s
How is evolving?
Through its basis elements
u
i
( x,
t)
1
V s
u
i
1
( x,
t)
( x,
+δ )
ui t t
V
s
completely controlled by Y ( t;
ω)
that can evolve in two ways:
V ⊥
s
1) A basis element can vary towards causing modification of .
2) A basis element can vary towards another direction of leaving invariant !
Additionally, the second kind of variation can be covered by the evolution of Yi
( t;
ω)
Source of Redundancy
∂ui
( x,
t)
∫ uj
( x ) x=
= =
Constrain each basis element
to vary normal to Vs
∂t
V
s
V s
, t d 0 for all i 1,..., s and j 1,..., s
V s
i
11

The DO representation
( ,; t ω) ( , t) Y ( t; ω) ( , t)
u x u x u x
= +∑
Separate deterministic from stochastic part
Without loss of generality ( )
12
s
i=
1
Y t; ω = 0
Evolving the finite dimensional subspace
Restrict evolution of to be normal to V i.e.
∫
∂u
i
( x,
t)
∂t
( )
i
i
u x, t dx= 0 for all i= 1,..., s and j = 1,..., s
j
i
( , t)
u x s
• No restrictions on the probabilistic structure of the stochastic coefficients Y ( t;
ω)
• Dynamically evolving stochastic subspace
• DO condition preserves orthonormality of basis elements ui
( x,
t)
d
∂uj
( x, t) ∂ui
( x,
t)
( ) ( ) ( )
( )
∫ ∫ ∫
ui x, t uj x, t dx = ui x, t dx + uj
x, t dx=
0
dt ∂t ∂t
T. Sapsis, and P. Lermusiaux, Dynamically orthogonal field equations for continuous stochastic dynamical systems, Physica D, 238 (2009) 2347.
Vs
i
i

Dynamically Orthogonal Evolution Equations
Theorem 1 (Sapsis & Lermusiaux, Physica D, 2009): For a stochastic field
described by the evolution equation
( ,; t ω)
∂u x
= L ⎡ ( ,; t ω)
; ω⎤
∂t
⎣u x ⎦
, t ; ω = ; ω , x∈
D
u( x ) u ( x )
⎡u( t ω) ⎤ h( t ω)
0 0
assuming a response of the form
, x∈
following closed set of evolution equations
D
B ⎣ ξ,; ⎦= ξ,; , ξ∈∂D
( ,; t ω) ( , t) Y ( t; ω) ( , t)
u x u x u x
= +∑
s
i=
1
i
i
we obtain the
SDE describing
evolution of
stochasticity inside
V s
Family of PDEs
describing evolution of
stochastic subspace V s
dY
j
( t;
ω)
dt
( x,
t)
∫
ω
{ L⎡⎣ ( ,; ω) ⎤⎦−
⎡L⎡ ⎣ ( ,; ω)
⎤⎦
⎤} j ( ,)
= u y t E
⎣
u y t
⎦
u y t dy
D
∂uj
ω
⎡ ⎤
= E ⎡Yi t t ⎤
YY
− E
i j
⎢ k
t Yi t t d ⎥ YY i j k
t
t ⎣ ⎣u x
∂
⎦⎦
C ∫ u y ⎣u y ⎦ y C u x
⎣D
⎦
−
( ) ( ) 1 ω
−
; ω L⎡ , ; ω ⎤ ( , ) ( ; ω) L⎡ ( , ; ω) ⎤
1
( , )
1
( t ) E ω
−
ω ⎡Y ( t ω) h( t ω) ⎤
B ⎡
⎣uj ξ,; ⎤= ⎦ ⎣ i
; ξ, 0; ⎦CYY,
ξ∈∂D
i j
PDE describing
evolution of
mean field
( , t)
∂u x
∂t
L
( t ω)
ω
= E ⎡
⎣
⎡⎣u x,; ⎤⎦
⎤
⎦
, x∈
D
( t ω) E ω h( t ω)
B ⎡⎣u
ξ,; ⎤⎦= ⎡⎣ ξ,; ⎤⎦,
ξ∈∂D
T. Sapsis, and P. Lermusiaux, Dynamically orthogonal field equations for continuous stochastic dynamical systems, Physica D, 238 (2009) 2347.
13

Stochastic initial conditions
( , t ; ω) = ( ; ω)
0 0
Formulation of initial conditions
u x u x x∈
D
Formulation in stochastic subspace terms
Initial condition for the mean field equation
( 0 0 )( 0( ) 0( ))
( ) ⎡ ( ) ( )
C , E ω uu
x y = u x; ω −u x; ω u y; ω −u y;
ω
⎣
∫ uu
( , t ; ω) E ω
( ; ω)
u x
0
= ⎡⎣u0
x ⎤⎦
Initial condition for the basis of the stochastic subspace
V
2
( ) ( ) d = λ ( )
C x y u x x u y
,
i i i
Initial condition for the stochastic coefficients
i
( ; ω) ⎡ ( ; ω) − ( ; ω) ⎤ ( , )
Y t
( ) span ( )
{ | }
s cr i i cr
∫ ⎣ 0 0 ⎦
D
= u y u y u y t dy
i
Computation of eigenvalues/eigenvectors
σ = u x λ > σ Selection of Stoch. Subspace dimensionality
T
⎤
⎦
14

POD & PC methods from DO equations
SDE describing
evolution of
stochasticity inside
V s
dY
j
( t;
ω)
dt
∫
ω
{ L⎡⎣ ( ,; ω) ⎤⎦−
⎡L⎡ ⎣ ( ,; ω)
⎤⎦
⎤} j ( ,)
= u y t E
⎣
u y t
⎦
u y t dy
D
Family of PDEs
describing evolution of
stochastic subspace V s
( x,
t)
∂uj
ω
⎡ ⎤
= E ⎡Yi t t ⎤
YY
− E
i j
⎢ k
t Yi t t d ⎥ YY i j k
t
t ⎣ ⎣u x
∂
⎦⎦
C ∫ u y ⎣u y ⎦ y C u x
⎣D
⎦
−
( ) ( ) 1 ω
−
; ω L⎡ , ; ω ⎤ ( , ) ( ; ω) L⎡ ( , ; ω) ⎤
1
( , )
1
( t ) E ω
−
ω ⎡Y ( t ω) h( t ω) ⎤
B ⎡
⎣uj ξ,; ⎤= ⎦ ⎣ i
; ξ, 0; ⎦CYY,
ξ∈∂D
i j
PDE describing
evolution of
mean field
( , t)
∂u x
∂t
L
( t ω)
ω
= E ⎡
⎣
⎡⎣u x,; ⎤⎦
⎤
⎦
, x∈
D
( t ω) E ω h( t ω)
B ⎡⎣u
ξ,; ⎤⎦= ⎡⎣ ξ,; ⎤⎦,
ξ∈∂D
Choosing a priori the stochastic subspace
POD equations.
V s
using POD methodology we recover
Choosing a priori the statistical characteristics of the stochastic coefficients
we recover the PC equations.
Yj
( t;
ω)
T. Sapsis, and P. Lermusiaux, Dynamically orthogonal field equations for continuous stochastic dynamical systems, Physica D, 238 (2009) 2347.
15

Dynamically orthogonal field equations: Some properties
1) DO equations are exact; no approximations have been made and the operator
is a general nonlinear differential operator (with polynomial nonlinearities).
2) Equations are consistent with the representation constraint
∫
∂u
i
( x,
t)
∂t
( )
3) They are in resolved form.
u x, t dx= 0 for all i= 1,..., s and j = 1,..., s.
j
( )
4) By choosing a priori the fields u x,
t we recover POD method.
5) By choosing a priori the stochastic processes Y t;
ω we recover the
various versions of Polynomial-Chaos Method.
6) The consistency with the second constraint also guarantees preservation
of orthonormal properties for u x,
t .
i
i
( )
( )
7) Preserve energy (quadratic functionals, e.g. energy) for conservative systems.
i
L[
u]
16

Application to linear systems
We first apply the DO methodology to linear systems described by the equation
( ,; t ω)
∂u x
= L0( x, t) + L1( x, t) u+ Zk( t; ωσ ) k( x,
t)
, x∈
D
∂t
u( x, t0; ω) = u0( x ; ω)
, x∈
D B ⎡⎣u( ξ,; t ω) ⎤⎦= h( ξ,; t ω)
, ξ∈∂D
To obtain the reduced order set
∂ u =
∂t
( , ) + ( , )
L x t L x t u
0 1
( t ω) E ω h( t ω)
B ⎡⎣u
ξ,; ⎤⎦= ⎡⎣ ξ,; ⎤⎦,
ξ∈∂D
dY
∂
j
( t;
)
ω = YL u . u
dt
j 1 l j
u
j
= L ( x , t) u − L ( x , t)
u . u
1
( t ) E ω
−
ω ⎡Y ( t ω) h( t ω) ⎤
∂t
( )
1 j 1 l l
B ⎡
⎣uj ξ,; ⎤= ⎦ ⎣ i
; ξ, 0; ⎦CYY,
ξ∈∂D
i j
17

Application to 2D Navier-Stokes flows
Stochastic Navier-Stokes equations
Zero-mean stochastic forcing
∂u 1 = −∇ p+ Δ u − u. ∇ u − f k ˆ × u + τ( x , t) + ϕ( x , t; ω) ≡ L[ u ], x ∈ D,
ω∈Ω
∂t
Re
Deterministic forcing
∇ . u = 0
Expansion of forcing term
Representation of flow
Representation of pressure
( x,; t ) Z ( t; ) ( x,
t)
ϕ ω = ∑ r
ω ϕr
r=
1
( ,; t ω) ( , t) Y ( t; ω) ( , t)
u x u x u x
= +∑
Expanded form of the system operator
L
R
s
i=
1
i
i
Inner-product (2D)
T
( t) ( )
u, u = ∫ u x, u x,
t dx
1 2 1 2
D
( x,; ω) = ( x, ) + ( ; ω) ( x, ) − ( ; ω) ( ; ω) ( x,
)
0
s s s
∑
p t p t Y t p t Y t Y t p t
[ u] =−∇ p+ Δu−u. ∇u− fk× u+
τ ( x t)
i i i j ij
i= 1 i= 1 j=
1
+
∑∑
R
∑
r=
1
r
( ; ω) ( x,
)
Z t b t
1 ˆ ,
Re
R
⎡ 1 ˆ ⎤
+ Yi ⎢
Δui −u i. ∇u−u. ∇ u
i+ fk× ui −YY ⎡ ⎤
i j i
∇
j
+ Zr t; r
, t
Re
⎥ ⎣u . u ⎦ ∑ x
⎣
⎦
r=
1
r
( ωϕ ) ( )
18

Application to 2D Navier-Stokes flows
Equations for the stochastic pressure
Δ p0
= div ( −u. ∇u − fkˆ
× u +τ ( x,
t)
)
Δ p = div u. ∇ u , i, j=
1,..., s
r
ϕr( x )
( )
ij i j
Equation for the mean flow
Equations for the stochastic coefficients
Equations for the DO modes
( u . u u. u + kˆ
u )
Δ p = div − ∇ − ∇ f × , i=
1,..., s
i i i i
Δ b = div , t , r = 1,..., R.
∂u 1 = −∇ p0
+ Δ u − u. ∇ u − f k ˆ × u + τ ( x , t) − C ⎡
( ) ( )
−∇p ⎤, .
Yi t Yj t ij i
∇
j
∈ D
∂t
Re ⎣ +u . u ⎦ x
dYi
dt
∂ui
∂t
( ( ) ( ) ) ϕ
Y t Y t ( ) ( ω)
1
= Δu −u . ∇u −u. ∇ u + fkˆ × u , u Y − u . ∇u , u Y Y − C + x, t , u Z t;
m m m m i m m n i m n r i r
Re
m n
= Qi − Qi, um um, x∈
D, i=
1,..., s
1 ˆ
−
Q =−∇ p + Δu −u . ∇u −u. ∇ u + fk × u −C M −∇p
+ u . ∇u
Yi t Yj t Yj t Ym t Yn
t
Re
1
( ) ( ) ( ) ( ) ( ) [ ]
i i i i i i mn m n
−
+ C C −∇ b + , t ⎤.
Yi t Yj t Yj t Zr
t ⎣
⎦
1
( ) ( ) ( ) ( )
⎡
r
ϕr( x )
i= 1,..., s, ω ∈Ω.
19

Spatial discretization
Numerical scheme
Finite-volume. Advection scheme used: Central difference or Total variation diminishing scheme.
Time discretization
Combination with projection methods allows for the solution of only 1+ s pressure equations.
Stochastic ODE solution
5
( )
Using Monte-Carlo method and O 10 samples (small dimensionality of SODE).
Integration using a 4 th order Runge-Kutta method.
Computational time
( )
4
4
Order of hours on a single processor for O 10 spatial nodes and O 10 time steps.
Convergence
Comparison with direct Monte-Carlo method.
Second order convergence in space and first order convergence in time.
( )
20

An idealized wind-driven ocean circulation model
∂ u = 0, v = 0
∂y
∂u 1 = −∇ p+ Δ u − u. ∇ u − f k ˆ × u + τ( x , t) + ϕ( x , t; ω)
, x ∈ D,
ω∈Ω
∂t
Re
∇ . u = 0
u = 0
v = 0
u = 0
v = 0
τ
( x t)
∂ u = 0, v = 0
∂y
Spatially symmetric forcing
⎛ a ⎞
, = ⎜ − cos2 πy,0 ⎟ ,
⎝ 2π
⎠
Corriolis coefficient
3
a =10
f = f0 +β0y
U
Ro = = 1.4×
10
Lf
−4
3
Re =10
Non-dimensional numbers
based on realistic parameters
L
3
= 10 km T = 4.46
y
Application of a very
small stochastic
perturbation at t = 0
Simmonet & Dijkstra, JPO, 2002
21

An idealized wind-driven ocean circulation model
u( x,t)
vorticity
u
( g, t) , u( g,
t)
( ω)
ω 2
E ⎡
⎣Yi
t; ⎤
⎦, i=
1,...,5
( x )
u1 ,t
vorticity
f
( y t)
Y 3 3 ,
( x )
u2 ,t
vorticity
f
( y t)
Y 4 4 ,
22

Flow behind a cylinder (Re = 3000)
Re = 3000
Initial perturbation
__with energy
6
10 −
T = 30 22.5 50 27.5
23

Cost scaling with stochastic dimensionality
Direct approach - ‘Curse’ of dimensionality
Assuming that N degrees of freedom are involved in each stochastic dimension the storage cost will be
Sdirect
s
( )
= O N , where s is the stochastic dimension.
Exponential growth of the number of unknowns with respect to the stochastic dimensionality of the problem.
Polynomial Chaos method
Assuming that p is the order of the polynomial-chaos approximation and s is
the stochastic dimensionality we have the storage and computational cost
where q is the nonlinearity order of the system operator. (Xiu & Karniadakis, 2003).
DO method
( ) (
q
)
p
p
⎡ ⎤
PC
SPC
= O Ns , C = O ⎣Ns
⎦ .
In this case the storage and computational cost will be
(
q
)
( ) [ ]
S = O Ns , C = O Ns .
DO
For Navier-Stokes we have q = 2.
DO
In DO the storage cost grows linearly and the computational cost scales with the order of nonlinearity
24
Lid-driven cavity flow computational cost
Numerical exponent = 1.986
T. Sapsis, and P. Lermusiaux, Dynamical criteria for the evolution of the stochastic dimensionality in flows with uncertainty, Physica D, (2011).

Evolution of stochastic dimensionality
V ⊥
s
Probability measure
V s
σcr
> λ
ui 1
( x)
ui 2
( x)
• In the context of the DO equations the size of the stochastic subspace
remains invariant.
V s
• For transient stochastic phenomena the dimensionality of the stochastic
subspace may vary significantly with time.
We need criteria to evolve the dimensionality of the stochastic subspace
This is a particularly important issue for stochastic systems with
deterministic initial conditions (initial dimensionality is zero)
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Criteria for dimensionality reduction / increase
Dimensionality reduction
C YY
Comparison of the minimum eigenvalue of the correlation matrix .
i
j
λ
⎡C
⎤< σ
⎣ ⎦
min YY i j
2
cr
pre-defined value
Removal of the corresponding direction from the stochastic subspace.
Dimensionality increase
Comparison of the minimum eigenvalue of the correlation matrix .
i
( , t)
λ ⎡
min
C ⎤
YY
>Σ
⎣ i j⎦
u x Vs
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2
cr
C YY
pre-defined value
Addition of a new direction in the stochastic subspace .
How do we choose this new direction ?
Same problem when we start with deterministic initial condition
(dimension of stochastic subspace is zero)
T. Sapsis, and P. Lermusiaux, Dynamical criteria for the evolution of the stochastic dimensionality in flows with uncertainty, Physica D, (2011).
i
j

Analytical criteria for selection of new directions
Theorem 2 (Sapsis & Lermusiaux, 2010): For a stochastic field described by the equation
∂u( x,;
t ω)
= L ⎡ ( ,; t ω)
; ω⎤
∂t
⎣u x ⎦
, x∈
D
and with current state at t = t c
described by
( , t ; ω) ( , t ) Y ( t ; ω) ( , t )
u x u x u x
= +∑
c c i c i c
i=
1
the maximum variance growth rate of a stochastic perturbation in
s
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V ⊥
s
will be given by
Frechet Derivative
L u( g,;
t ω)
( )
δ u
where ϑ
i ( y, t)
, i=
1,..., m is a finite basis that approximates V ⊥ s .
The corresponding direction of maximum growth is given by
where v, i 1,..., m
i
⎡Aij
+ Aji
⎤ δ ⎡ ⎤
ρ⎡tc; ( , t; ω) max λk , Aij ϑj ( , t
⎣ ⎦
⎣ u x ⎤⎦ = ⎢ ⎥ ≡ )
⎡ϑi
, t ⎤d
k 2
∫ y ⎣ y ⎦ y
⎣ ⎦
D
ϑ
m
( x, t) vϑ
( x,
t)
= ∑
i=
1
= is the eigenvector associated with .
The above analysis adds the direction with the maximum potential growth
which is not included in the stochastic subspace.
T. Sapsis, and P. Lermusiaux, Dynamical criteria for the evolution of the stochastic dimensionality in flows with uncertainty, Physica D, (2011).
i
i
ρ

An idealized wind-driven ocean circulation model
Number of required modes to capture variance as low as
10 −6
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QUESTIONS ??
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