Variance reduction and Brownian Simulation Methods

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Variance reduction and Brownian Simulation Methods

• Yossi Shamai
• Raz Kupferman
• The Hebrew University

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(No Transcript)
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Dumbbell models
All (incompressible) fluids are governed by
mass-momentum conservation equations
u(x,t) velocity ?(x,t) polymeric stress
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Dumbbell models
The polymers are modeled by two beads connected
by a spring (dumbbell) . The conformation is
modeled by an end-to-end vector q.
less affect
more affect
?(q,x,t) pdf.
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The (random) conformations are distributed
according to a density function ?(q,x,t), which
satisfies an evolution equation
advection
deformation
diffusion
The stress is an ensemble average of polymeric
conformations,
g(q) q?F(q)
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Conservation laws (macroscopic dynamics)
The stress
Polymeric density distribution (microscopic
dynamics)

• Problem high dimensionality

• Assumption 1-D

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Closable systems
In certain cases, a PDE for ?(x,t) can be
derived, yielding a closed-form system for
u(x,t), ?(x,t).
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Outline
1. Brownian simulation methods 2. Some
mathematical preliminaries on spatial
correlations 3. A variance reduction mechanism in
Brownian simulations 4. Examples
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Brownian simulations
The average stress ?(x,t) is an expectation with
respect to a stochastic process q(x,t) with PDF
?(q,x,t)
PDE
SPDE
q(x,t) is simulated by a collection of
realizations qi(x,t). The stress is approximated
by an empirical mean
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• A reminder real-valued Brownian motion
• B(t) is a random function of time.
• Almost surely continues.
• Independent increments.
• B(t)-B(s) N(0,t-s).

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Spatial correlations
B(x,t) is characterized by the spatial
correlation function
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Spatial correlations (cont.)

• An L2 - function is a correlation function iff
• a. c(x,x) 1.
• b. It has a square root in L2

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Spatial correlations (cont.)

• No spatially uncorrelated L2-valued Brownian
  motion.

• Spatially uncorrelated noise has meaning only
  in a discrete setting. It is a sequence of
  piecewise constant standard Brownian motions,
  uncorrelated at any two distinct steps, that
  converges to 0.

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Spatial correlations (cont.)
Spatial correlations can be alternatively
described by Correlation operators

• C is nonnegative, symmetric and trace class.

• No Id-correlated Brownian motion (trace Id 8 ).

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SDEs versus SPDEs
SDEs (Stochastic Differential Equations)
Itos integral
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SDEs versus SPDEs
SDEs
PDE (Fokker-Plank)
SDE

• q(x,t) has spatial correlation.

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Brownian simulationsunifying approach

• Equivalence class insensitive to spatial
  correlations.

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Brownian simulation methods
The stochastic process q is simulated by n
realizations driven by i.i.d Brownian motions.
Expectation is approximated by an empirical mean
with respect to the realizations
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Brownian simulation methods
The approximation
The system
?
?
Correlation affects approximation but not the
exact solution
CONNFFESSIT (Calculations of Non Newtonian Fluids
Finite Elements and Stochastic Simulation
Techniques) - Piecewise constant uncorrelated
noise (Ottinger et al. 1993) BCF - Spatially
uniform noise (Hulsen et al. 1997)
Error reduction ?
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Goals
The error of the Brownian simulations is

• Prove that e(n,t)?0.
• Reduce the error by choosing the spatial
  correlation of the Brownian noise
• Step 1. Express e(n,t) as a function F(c).
• Step 2. Minimize F(c).

The idea of adapting correlation to minimize
variance first proposed by Jourdain et al. (2004)
in the context of shear flow with a specific FEM
scheme.
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Example
An integral-type system
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Results
n 2000 with spatially uniform noise ( c(x,y)
1 ).
Brownian simulation
Brownian simulation
The stress
Stress
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Results
n 2000 with piecewise constant uncorrelated
noise.
The Brownian simulation at t20 (dotted curve)
noisy simulations
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Error analysis
We want to analyze the error of the Brownian
simulations
Lets demonstrate the analysis for semi-linear
system
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Closable systems
In certain cases, a PDE for ?(x,t) can be
derived, yielding a closed-form system for
u(x,t), ?(x,t).
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Error analysis for Semi-linear systems
We want to estimate the error of the Brownian
simulations
In semi-linear systems, the stress field ?(x,t)
satisfies a PDE

• Linearize (properly)

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? Linearized system
and
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Theorem 1. To leading order
where
and k is a kernel function determined by the
parameters.
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Error analysis for Closable systems
Theorem 1. To leading order in n,

• F is convex
• In principle, the analysis is the same
• Proof is restricted to closable systems

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The optimization problem
Minimize F(c) over the domain S c(x,y) c
has a root in L2, c(x,x) 1

• In general, there is no minimizer

• Difficulties
• A. Infinite dimensional optimization problem.
• B. S is not compact.

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Finite dimensional approximations
1. Set a natural k. 2. Discretize the problem to
a k-point mesh
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The F-D optimization problem

• We want to minimize F(ck) over Sk.
• ck(x,y) is indexed by k2 mesh points (xi ,xj)
  (matrix).
• Symmetric Positive-Semi-Definite.
• ck (xi ,xi) 1.

The F-D optimization problem is Minimize
F(A), A is k-by-k symmetric
PSD Subject to Aii 1, i1,,k
F is convex ? SDP algorithms (Semi-Definite
Programming)
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So what did we do?

• Developed a unifying approach for a variance
  reduction mechanism in Brownian simulations.
• Formulated an optimization problem (in infinite
  dimensions).
• Showed that it is amenable to a standard
  algorithm (SDP).

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Example 1
A linear advection-dissipation equation in 0,1.
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The error is
Variance independent of correlations (no
reduction) Insights the dynamics (advection and
dissipation) do not mix different points in
space. Thus, the error only sees diagonal
elements of the correlations, which are fixed by
the constraints.
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Example 2
An integral-type system (x?0,1)
Closable
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Results
n 2000 with spatially uniform noise ( c(x,y)
1 ). (BCF)
Brownian simulation
Brownian simulation
The stress
Stress
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Results
n 2000 with piecewise constant uncorrelated
noise (CONNFFESSIT).
The Brownian simulation at t20 (dotted curve)
noisy simulations
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Why?
? the optimal error is obtained by taking c ? 0
(CONNFESSIT).
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Example 3 1-D planar Shear flow model.
(Jourdain et al. 2004)
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To leading order, the error of the Brownian
simulations is

• C - the spatial correlation operator.

• K(t) - a nonnegative bounded operator.

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So is CONFFESSIT always optimal?

• No!
• We can construct a problem for
• which e(n,t) n-1(const TrK(t)C)
• for K(t) bounded and not PSD.
• Theorem. If the semi-groups are Hilbert-Schmidt
  (they have L2-kernels) then CONNFFESSIT is
  optimal.

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Some further thoughts

• The spatial correlation of the initial data
  q(x,0) may also be considered.
• Non-closable systems?
• Gain insights about the optimal correlation by
  understand relations between type of equation and
  optimal correlation.