Spectral problems in the Metropolis Algorithm
Gilles Lebeau
in collaboration with Persi Diaconis and Laurent Michel
Département de Mathématiques,
Université de Nice Sophia-Antipolis
Parc Valrose 06108 Nice Cedex 02, France
february 2016
G. Lebeau (Nice)
Metropolis
february 2016
1 / 51
The analysis of the rate of convergence of a Metropolis algorithm involves
the study of the spectral theory of the associated Markov operator. I will
illustrate what are these spectral properties on a simple variant of the
historical Metropolis chain related to hard spheres.
I will explain how these spectral properties are related to classical estimates
in PDE’s such as Weyl estimates, Sobolev inequalities, Fourier analysis...
I will describe some of the numerous new challenges and unsolved
problems.
And I will try to explain why I believe that the ”Metropolis Laplacian” is a
natural extension of the usual Laplacian in the framework of a metric and
measured space.
G. Lebeau (Nice)
Metropolis
february 2016
2 / 51
Some publications
-P. Diaconis and G. Lebeau Microlocal analysis for the Metropolis
algorithm , Mathematische Zeitschrift: Volume 262, Issue 2, (2009).
- G. Lebeau and L. Michel Semi-classical analysis of a random walk on a
manifold, Annals of Probability , Vol. 38, No. 1, (2010).
-P. Diaconis, G. Lebeau and L. Michel Geometric Analysis of the
Metropolis Algorithm in Lipschitz Domains, Inventiones 185 (2), (2011).
- P. Diaconis, G. Lebeau and L. Michel Metropolis algorithm on convex
polytops, Mathematische Zeitschrift, 272, (1), (2012).
- G. Lebeau and L. Michel Hypoelliptic random walks, Journal de l’Institut
Mathématique de Jussieu, (2014).
G. Lebeau (Nice)
Metropolis
february 2016
3 / 51
Hard discs in a square
NO
YES
NO
G. Lebeau (Nice)
Metropolis
february 2016
4 / 51
Outline
1
The Metropolis Algorithm
2
Random placement of non-overlapping balls
3
A local model in a bounded, connected, Lipschitz domain
4
Diffusion
5
Metropolis Laplacian
G. Lebeau (Nice)
Metropolis
february 2016
5 / 51
Outline
1
The Metropolis Algorithm
2
Random placement of non-overlapping balls
3
A local model in a bounded, connected, Lipschitz domain
4
Diffusion
5
Metropolis Laplacian
G. Lebeau (Nice)
Metropolis
february 2016
6 / 51
Metropolis Algorithm
R
Let (X , B, µ) be a σ-finite measure space. Let ρ(x) > 0, ρ(x)dµ(x) = 1,
be a probability density on X . The Metropolis algorithm gives a way of
drawing samples from ρ. It requires a symmetric
proposal density
R
p(x, y ) = p(y , x) ≥ 0 for all x, y , and p(x, y )µ(dy ) = 1 for all x. This
kernel allows us to define the Metropolis kernel
ρ(y ) , 1 µ(dy )
T (x, dy ) = m(x)δx + p(x, y )min
ρ(x)
Z
ρ(z) m(x) =
1−
p(x, z)µ(dz)
ρ(x)
{z:ρ(z)<ρ(x)}
G. Lebeau (Nice)
Metropolis
february 2016
(2.1)
7 / 51
Metropolis Algorithm
ρ(y ) T (x, dy ) = m(x)δx + p(x, y )min
, 1 µ(dy )
ρ(x)
Z
ρ(z) p(x, z)µ(dz)
m(x) =
1−
ρ(x)
{z:ρ(z)<ρ(x)}
(2.2)
Formula 2.2 has a simple algorithmic interpretation. From x, choose y
from the density p(x, y ). If ρ(y ) ≥ ρ(x), move to y . If ρ(y ) < ρ(x) flip a
coin with probability of heads ρ(y )/ρ(x). If it comes up heads, move to y .
If it comes up tails, stay at x. Observe that implementing this does not
require knowledge of the normalizing constant for ρ. This is a crucial
feature in applications where ρ(x) is given as Z −1 e −βH(x) with Z
unknownable in practice.
G. Lebeau (Nice)
Metropolis
february 2016
8 / 51
Metropolis Algorithm
R
Let T n (x, dy ) be the kernel of the iterate operator T n . Then A T n (x, dy )
is the probability to be in the set A after n steps of the walk. Under mild
conditions on p(x, y ), one has
kT n (x, dy ) − ρ(y )dµ(y )kTV
as n → ∞
∀x
The problem is to get estimates on this rate of convergence, and on the
spectral theory of the operator T acting as a self adjoint contraction on
L2 (ρdµ).
G. Lebeau (Nice)
Metropolis
february 2016
9 / 51
Outline
1
The Metropolis Algorithm
2
Random placement of non-overlapping balls
3
A local model in a bounded, connected, Lipschitz domain
4
Diffusion
5
Metropolis Laplacian
G. Lebeau (Nice)
Metropolis
february 2016
10 / 51
N-body configuration space
We suppose that Ω is a bounded, Lipschitz, quasi-regular, connected open
subset of Rd with d ≥ 2. Let N ∈ N, N ≥ 2 and > 0 be given. Let ON,
be the open bounded subset of RNd ,
n
o
N
ON, = x = (x1 , . . . , xN ) ∈ Ω , ∀ 1 ≤ i < j ≤ N, |xi − xj | > . (3.1)
Let ϕ(z)dz be the uniform probability on the unit ball of Rd , and let
Kh (x, dy ) be the Markov kernel
N
xj − yj
1 X
−d
δx1 ⊗· · ·⊗δxj−1 ⊗h ϕ
dyj ⊗δxj+1 ⊗· · ·⊗δxN ,
Kh (x, dy ) =
N
h
j=1
(3.2)
and the associated Metropolis operator on L2 (ON, )
Z
Th (u)(x) = mh (x)u(x) +
u(y )Kh (x, dy ),
(3.3)
ON,
Z
mh (x) = 1 −
Kh (x, dy ).
(3.4)
ON,
G. Lebeau (Nice)
Metropolis
february 2016
11 / 51
The operator Th is Markov and self -adjoint on L2 (ON, ). The
configuration space ON, is the set of N disjoint closed balls of radius /2
in Rd , with centers at the xj ∈ Ω. The topology of this set, and the
geometry of its boundary is generally hard to understand (!), but since
d ≥ 2, ON, is clearly non-void and connected for a given N if is small
enough. The Metropolis kernel Th is associated to the following algorithm:
at each step, we choose uniformly at random a ball, and we move its
center uniformly at random in Rd in a ball of radius h. If the new
configuration is in ON, , the change is made. Otherwise, the configuration
is kept as it started.
In order to study the random walk associated to Th , we prove that the
open set ON, is Lipschitz quasi-regular for > 0 small enough, and we
prove that the kernel of the iterated operator ThM (with M large, but
independent of h) admits a suitable lower bound, so that we will be able
to use some analytic tools on a simple ”model”.
G. Lebeau (Nice)
Metropolis
february 2016
12 / 51
Quasi-regular boundary
Definition
Let Ω be a Lipschitz open set of Rd . We say that ∂Ω is quasi-regular if
∂Ω = Γreg ∪ Γsing , Γreg ∩ Γsing = ∅ with Γreg the union of smooth
hypersurfaces, relatively open in ∂Ω, and Γsing a closed subset of Rd such
that
v ∈ H −1/2 (∂Ω) and support(v ) ⊂ Γsing =⇒ v = 0.
(3.5)
Observe that 3.5 is obviously satisfied if ∂Ω is smooth, since in that case
one can take Γsing = ∅. More generally, the boundary is quasi-regular if it
is ‘piece-wise smooth’ in the following sense: Ω is Lipschitz,
∂Ω = Γreg ∪ Γsing , Γreg ∩ Γsing = ∅, Γreg is a smooth hypersurface of Rd ,
relatively open in ∂Ω, and Γsing a closed subset of Rd such that
Γsing = ∪j≥2 Sj where the Sj are smooth disjoint submanifolds of Rd such
that codimRd Sj ≥ j and ∪k≥j Sk = Sj . This ‘piece-wise smooth’ condition
(often called “stratified”) is easy to visualize.
G. Lebeau (Nice)
Metropolis
february 2016
13 / 51
Proposition
There exists C > 0 such that for N < C , the set ON, is connected,
Lipschitz and with quasi-regular boundary.
The condition Nd < c, which says that the density of the balls is
sufficiently small, does not imply that the set ON, has Lipschitz regularity.
As an example, if Ω =]0, 1[2 is the unit square in the plane, then
1
is a
x = (x1 , . . . , xN ), xj = ((j − 1), 0), j = 1, . . . , N, with = N−1
configuration point in the boundary ∂ON, . However, ∂ON, is not
Lipschitz at x: otherwise, there would exist νj = (aj , bj ) such that
(x1 + tν1 , . . . , xN + tνN ) ∈ ON, for t > 0 small enough, and this implies
a1 > 0, aj+1 > aj and aN < 0 which is impossible.
G. Lebeau (Nice)
Metropolis
february 2016
14 / 51
For k ∈ N∗ denote B k = BRk (0, 1) the unit Euclidean ball and
1
ϕk (z) = vol(B
k ) 1Bk (z).
Lemma
Let be small. There exists h0 > 0, c0 , c1 > 0 and M ∈ N∗ such that for
all h ∈]0, h0 ], one has
x −y
M
−Nd
Th (x, dy ) = µh (x, dy ) + c0 h
ϕNd
dy ,
(3.6)
c1 h
where for all x ∈ ON, , µh (x, dy ) is a positive Borel measure.
This allows to reduce most of the analysis of the N-body problem in Rd to
a more simple model, of type 1-body problem in RNd .
G. Lebeau (Nice)
Metropolis
february 2016
15 / 51
Open problems
1. What can be said about the connected components of the configuration
space ON, even in the regime of low density Nd small ?
2. What is the geometric structure of the boundary ∂ON, of the
configuration space for N large ?
3. If K is a compact subspace of the configuration space ON, , is it true
that ThN (x, dy ) converges exponentially in total variation, with a rate
independent of x ∈ K , to the uniform probability on the h-connected
component C (h, K ) of K in ON, at least on compact subset of C (h, K )?
G. Lebeau (Nice)
Metropolis
february 2016
16 / 51
Outline
1
The Metropolis Algorithm
2
Random placement of non-overlapping balls
3
A local model in a bounded, connected, Lipschitz domain
4
Diffusion
5
Metropolis Laplacian
G. Lebeau (Nice)
Metropolis
february 2016
17 / 51
model
Let Ω be an open, bounded, connected, subset of Rd with Lipschitz
1
1B1 (z) so that
boundary. Let B1 be the unit ball of Rd and ϕ(z) = Vol(B
1)
R
ϕ(z)dz =
R 1. Let ρ(x) be a measurable bounded positive function on Ω
such that Ω ρ(x)dx = 1 and such that 1/ρ is bounded. For h ∈]0, h0 ], set
Kh,ρ (x, y ) = h−d ϕ(
x −y
ρ(y )
)min(
, 1)
h
ρ(x)
(4.1)
and let Th,ρ be the (local) Metropolis operator associated to these data,
that is
Z
Th,ρ (u)(x) = mh,ρ (x)u(x) +
Kh,ρ (x, y )u(y )dy
Ω
Z
(4.2)
mh,ρ (x) = 1 −
Kh,ρ (x, y )dy ≥ 0
Ω
G. Lebeau (Nice)
Metropolis
february 2016
18 / 51
model
1
2,3
4
G. Lebeau (Nice)
Metropolis
february 2016
19 / 51
model
Then the Metropolis kernel Th,ρ (x, dy ) = mh,ρ (x)δx=y + Kh,ρ (x, y )dy is a
Markov kernel, the operator Th,ρ is self-adjoint on L2 (Ω, ρ(x)dx), and thus
the probability measure ρ(x)dx on Ω is stationary. For n ≥ 1, we denote
n (x, dy ) the kernel of the iterate operator (T )n . For any x ∈ Ω,
by Th,ρ
h,ρ
n
Th,ρ (x, dy ) is a probability measure on Ω, and our main goal is to get
some estimates on the rate of convergence, when n → +∞, of the
n (x, dy ) toward the stationary probability ρ(y )dy .
probability Th,ρ
Observe that h is the size of each elementary step of the algorithm, and
we want to estimate the rate of convergence uniformly with respect to
h ∈]0, h0 ]
.
If h > 0 is fixed, estimates are easy: the problem is uniformity when h → 0.
G. Lebeau (Nice)
Metropolis
february 2016
20 / 51
The limit diffusion-operator
Let ν(ρ) be the best constant such that the following Poincaré inequality
holds true for all u in the Sobolev space H 1 (Ω)
Z
1
1
|∇u|2 (x)ρ(x)dx
(4.3)
kuk2L2 (ρ) − (u|1)2L2 (ρ) ≤
ν(ρ) 2(d + 2) Ω
For a smooth density ρ, ν(ρ) > 0 is the first non zero eigenvalue of the
unbounded self-adjoint operator Lρ acting on L2 (Ω, ρ(x)dx) with domain
D(Lρ ) = {u ∈ H 1 (Ω), 4u ∈ L2 (Ω) ∂n u|∂Ω = 0}
Lρ (u)(x) =
∇ρ
−1
(4u +
.∇u)
2(d + 2)
ρ
(4.4)
2(d
R + 2)Lρ2is the (positive) Laplacian associated to the Dirichlet form
Ω |∇u(x)| ρ(x)dx. It has a compact resolvant and we denote its spectrum
by ν0 = 0 < ν1 = ν(ρ) < ν2 < ... and by mj the multiplicity of νj . Observe
that m0 = 1 since Ker (L) is spanned by the constant function equal to 1 .
G. Lebeau (Nice)
Metropolis
february 2016
21 / 51
Spectral theory: easy facts
Since Th,ρ is self-adjoint on L2 (Ω, ρdx) and Markov, one has
Spect(Th,ρ ) ⊂ [−1, +1]
and since Th (1) = 1, 1 is an eigenvalue of Th . Moreover, since
Z
Kh,ρ (x, y )u(y )dy
Th,ρ (u)(x) = mh,ρ (x)u(x) +
Ω
R
and the operator u 7→ Ω Kh,ρ (x, y )u(y )dy is compact, the essential
spectrum of Th,ρ is equal to the closure of the range of the function
x 7→ mh,ρ (x). Since Ω is Lipschitz, there exists δ0 ∈]0, 1/2[ independent
of h such that 0 ≤ mh,ρ (x) ≤ 1 − δ0 for all x ∈ Ω. Therefore
The spectrum of Th,ρ in [−1, 0[∪]1 − δ0 , 1] is discrete
G. Lebeau (Nice)
Metropolis
february 2016
22 / 51
Spectral theory: easy facts
Moreover, there exists δ0 ∈]0, 1/2[, independent of h
Spec(Th,ρ ) ⊂ [−1 + δ0 , 1] for all h ∈]0, h0 ]. To see this point, write
Z
1
(u+Th,ρ u|u)L2 (ρ) =
Kh,ρ (x, y )|u(x)+u(y )|2 ρ(x)dxdy +2(mh,ρ u|u)L2 (ρ)
2 Ω×Ω
Therefore, it is sufficient to prove that there exists C0 > 0 such that the
following inequality holds true for all h ∈]0, h0 ] and all u ∈ L2 (Ω)
Z
x −y
)|u(x) + u(y )|2 dxdy ≥ C0 kuk2L2 (Ω)
h−d ϕ(
h
Ω×Ω
This is an easy exercise.
G. Lebeau (Nice)
Metropolis
february 2016
23 / 51
Spectral theory: easy facts
Finally, 1 is a simple eigenvalue of Th,ρ , since one has
Z
1
Kh,ρ (x, y )|u(x) − u(y )|2 ρ(x)dxdy
(u − Th,ρ u|u)L2 (ρ) =
2 Ω×Ω
Therefore, u = Th,ρ u implies |u(x) − u(y )| = 0 a.e. as soon as
|x − y | < h, and since Ω is connected, this shows u = Cte.
Thus one has the following lemma
Lemma
There exists δ0 ∈]0, 1/2[ independent of h such that Spect(Th,ρ ) is a
subset of [−1 + δ0 , 1]. The spectrum of Th,ρ is discrete in
[−1 + δ0 , 0[∪]1 − δ0 , 1]. 1 is a simple eigenvalue of Th,ρ and
Ker (Th,ρ − Id) is the one dimensional space spanned by constant
functions.
G. Lebeau (Nice)
Metropolis
february 2016
24 / 51
Spectral gap
We shall denote by g (h, ρ) the spectral gap of the Metropolis operator
Th,ρ . It is defined as the best constant such that the following Poincaré
inequality holds true for all u ∈ L2 (ρ) = L2 (Ω, ρ(x)dx)
kuk2L2 (ρ) − (u|1)2L2 (ρ) ≤
1
(u − Th,ρ u|u)L2 (ρ)
g (h, ρ)
(4.5)
or equivalently
Z
|u(x) − u(y )|2 ρ(x)ρ(y )dxdy
Z
1
Kh,ρ (x, y )|u(x) − u(y )|2 ρ(x)dxdy
≤
g (h, ρ) Ω×Ω
Ω×Ω
(4.6)
By the previous lemma, one has always g (h, ρ) > 0, but the previous
lemma doesn’t give a positive lower bound on g (h, ρ) > 0.
G. Lebeau (Nice)
Metropolis
february 2016
25 / 51
Spectral gap
Proposition
There exists positive constants C1 , C2 independent of h ∈]0, h0 ] such that
the spectral gap satisfies
C1 h2 ≤ g (h, ρ) ≤ C2 h2
Observe that the upper bound is obvious, since for any Lipschitz function
u, one has
(u − Th,ρ u|u)L2 (ρ) ≤ Ctek∇uk2L∞ h2
I will show how to get the lower bound in the next slides.
It is also obvious that limh→+∞ g (h, ρ) = 0, since for h large, the function
mh,ρ (x) takes values arbitrary close to 1.
Question: For a given density ρ, find h such that g (h, ρ) is maximal.
G. Lebeau (Nice)
Metropolis
february 2016
26 / 51
h-rough calculus
In all situations, the following theorem is a crucial step.
Theorem
There exists a family Ah,L , h ∈]0, h0 ] of linear operators such that:
1. For any q ∈ [1, ∞], Ah,L is uniformly in h bounded on Lq (Ω, ρdx).
2. There exists C > 0 such that for any u ∈ L2 (Ω, ρdx) such that
kuk2L2 + h−2 (u − Th,ρ u|u)L2 ≤ 1, one has
u = uL + uH ,
uL = Ah,L (u)
kuL kH 1 ≤ C ,
kuH kL2 ≤ Ch
In other words, uL = Ah,L (u) is the low frequency part of u, i.e. it
retains the behavior of u at scales ≥ h, and uH = (I − Ah,L )(u) is the
high frequency part of u. Observe that this theorem is a regularity
result. Observe also that as in the classical h-pseudodifferential calculus,
the cut-off on low frequencies u 7→ Ah,L (u) is bounded on all Lq spaces.
G. Lebeau (Nice)
Metropolis
february 2016
27 / 51
Dirichlet forms
For simplicity, I will now assume that the density ρ is given and smooth on
Ω. I erase the subscript ρ in all the notations.
Let Eh , Bh be the Dirichlet forms
Eh (u) = h−2 (u − Th u|u)L2 ,
Bh (u, v ) = h−2 (u − Th u|v )L2
Let E, B be the associated ”formal” limit objects when h → 0
Z
Z
1
1
2
|∇u| ρdx, B(u, v ) =
∇u∇v ρdx
E(u) =
2(d + 2) Ω
2(d + 2) Ω
Proposition
Let (f , u) ∈ H 1 × H 1 , (uh , vh ) ∈ H 1 × L2 . Assume that uh converges
weakly to u in H 1 and vh bounded in L2 . Then
lim Bh (f , uh + hvh ) = B(f , u)
h→0
G. Lebeau (Nice)
Metropolis
february 2016
28 / 51
The low frequency space
Take δ1 < δ0 . We know that the spectrum of Th in [1 − δ1 , 1] is discrete.
Let ej,h be the L2 normalized orthogonal eigenfunctions of Th with
eigenvalues 1 − h2 λj,h such that h2 λj,h ≤ δ1 .
Th (ej,h ) = (1 − h2 λj,h )ej,h ,
kej,h kL2 = 1
Definition
The low frequency space Eh,L is the finite dimensional space spanned by
the eigenfunctions ej,h , with h2 λj,h ≤ δ1 . The family (ej,h ) is an
orthonormal basis of Eh,L .
Observe that there exists C independent of ej,h such that one has the
crude bound
kej,h kL∞ ≤ Ch−d/2
This follows easily from the eigenfunction equation
Z
(1 − h2 λj,h − mh (x))ej,h (x) =
ej,h (y )Kh (x, dy )
Ω
G. Lebeau (Nice)
Metropolis
february 2016
29 / 51
The Weyl estimate
Observe that the ”formal” limit of h−2 (1 − Th ) is the Laplacian with
Neumann boundary condition
L=
∇ρ
−1
(4 +
.∇)
2(d + 2)
ρ
The spectrum of L is discrete, and the ”classical” Weyl estimate asserts
that the number of eigenvalues µ of L s.t. µ ≤ λ is equivalent to cλd/2 .
Lemma
For any 0 ≤ λ ≤ δ1 /h2 , the number of eigenvalues of Th in [1 − h2 λ, 1]
(with multiplicity) is bounded by C (1 + λ)d/2 . In particular,
dim(Eh,L ) ≤ Ch−d .
G. Lebeau (Nice)
Metropolis
february 2016
30 / 51
Abstract Weyl estimate
Lemma
Let s > 0 and Ah = A∗h ≥ 0, h ∈]0, 1] a family of non negative self-adjoint
bounded operators acting on L2 (M, µ). Assume that there exists a
constant C0 > 0 independent of h such that for all u ∈ L2 (M, µ) such that
((Id + Ah )u|u) ≤ 1, the following holds true:
∃(v , w ) ∈ H s ×L2 such that u = v +w , kv kH s ≤ C0 , kw kL2 ≤ C0 h. (4.7)
Let C1 <
1
.
4C02
There exists C2 > 0 independent of h such that
Spec(Ah ) ∩ [0, λ − 1] is discrete for all λ ≤ C1 h−2 and
#(Spec(Ah ) ∩ [0, λ − 1]) ≤ C2 < λ >dim(M)/2s ,
∀λ ≤ C1 h−2 . (4.8)
Here, #(Spec(Ah ) ∩ [0, r ]) is the number of eigenvalues
of Ah in the
√
interval [0, r ] with multiplicities, and < λ >= 1 + λ2 .
G. Lebeau (Nice)
Metropolis
february 2016
31 / 51
The Sobolev estimate
Lemma
There exist p > 2 and C independent of h ∈]0, h0 ] such that for all
u ∈ Eh,L , the following inequality holds true
kuk2Lp (M) ≤ C (Eh (u) + kuk2L2 ).
(4.9)
Proof Let u ∈ Eh,L such that Eh (u) + kuk2L2 ≤ 1. One has u = vh + wh
1
q
with kvh kH 1 ≤ C and kwh kL2 ≤ Ch. From the imbedding
P H ⊂ L (M) for
1 > d(1/2 − 1/q), we get kvh kLq ≤ C . One has u = λj,h ≤δ1 h−2 zj,h ej,h
P
with λj,h ≤δ1 h−2 |zj,h |2 ≤ 1. By Cauchy-Schwarz, the Weyl estimate and
kej,h kL∞ ≤ Ch−d/2 we get
kukL∞ ≤ Ch−d/2 (
X
|zj,h |2 )1/2 (dim(Eh,L ))1/2 ≤ Ch−d .
λj,h ≤δ1 h−2
From the rough calculus theorem, one has kvh kL∞ ≤ C kukL∞ ≤ Ch−d .
Thus we get kwh kL∞ = ku − vh kL∞ ≤ Ch−d . Since kwh kL2 ≤ Ch we
conclude
by interpolation.
G. Lebeau (Nice)
Metropolis
february 2016
32 / 51
The spectrum of Th,ρ near 1
Recall that ν(ρ) > 0 is the first non zero eigenvalue of
L=
−1
∇ρ
(4 +
.∇),
2(d + 2)
ρ
Neumann
The spectrum of L is denoted by 0 = ν0 < ν1 = ν(ρ) ≤ ν2 ≤ ... and mj is
the multiplicity of the eigenvalue νj .
Theorem
One has
limh→0 h−2 g (h, ρ) = ν(ρ)
(4.10)
Moreover, for any R > 0 and ε > 0, there exists h1 > 0 such that one has
Spec(
1 − Th
)∩]0, R] ⊂ ∪j≥1 [νj − ε, νj + ε],
h2
and the number of eigenvalues of
equal to mj .
G. Lebeau (Nice)
1−Th
h2
∀h ∈]0, h1 ]
(4.11)
in the interval [νj − ε, νj + ε] is
Metropolis
february 2016
33 / 51
Taylor expansion
The proof of the above theorem uses an elementary Taylor expansions
lemma:
Lemma
Let θ ∈ C ∞ (Ω) such that support(θ) ∩ Γsing = ∅ and ∂n θ|∂Ω = 0. Then
(Id − Th )(θ) = h2 L(θ) + r ,
kr kL2 ∈ 0(h5/2 )
This lemma implies that for any given interval I , and with
Iε = {x, dist(x, I ) ≤ ε} the number of eigenvalues of h−2 (Id − Th ) in Iε is
at least equal to the number of eigenvalues of L in I if h ≤ hε . Also, if
h−2 (Id − Th )(uh ) = νh uh , with νh → ν, then uh is compact in L2 , so we
may assume uh → u and we get (u|(L − ν)θ) = 0 for all θ ∈ C ∞ (Ω) such
that support(θ) ∩ Γsing = ∅ and ∂n θ|∂Ω = 0. With the quasi regularity of
the boundary, this implies that ν is an eigenvalue of L, and u an
eigenfunction. The equality of the dimensions follows by compacity of the
sequence uh .
G. Lebeau (Nice)
Metropolis
february 2016
34 / 51
Total Variation
The total variation distance between the two probability measure
n (x, dy ) and ρ(y )dy is defined by
Th,ρ
n
n
kTh,ρ
(x, dy ) − ρ(y )dy kTV = sup |Th,ρ
(x, A) −
A
Z
ρ(y )dy |
(4.12)
A
where the sup is taken over all Borel set A ⊂ Ω. Equivalently
Z
1
n
n
sup |T (g )(x) − g (y )ρ(y )dy |
kTh,ρ (x, dy ) − ρ(y )dy kTV =
2 kg k∞ ≤1 h,ρ
(4.13)
G. Lebeau (Nice)
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february 2016
35 / 51
Convergence in total variation distance
In the following theorem, we assume h0 small enough to have
g (h, ρ) < dist(Spect(Th ), −1).
Theorem
There exists a constant C , such that for all h ∈]0, h0 ] the following
estimate holds true for all integer n
supx∈Ω kThn (x, dy ) − ρ(y )dy kTV ≤ Ce −ng (h,ρ)
(4.14)
The above estimate is good for ”large n”. But it is not enough to answer
with some reasonable precision the question:
What is the minimal value of n to get
supx∈Ω kThn (x, dy ) − ρ(y )dy kTV ≤ 0.1
Observe that the following weaker L2 -estimate is obvious
kThn − Π0 kL2 ≤ e −ng (h,ρ)
G. Lebeau (Nice)
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february 2016
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Total variation
Let Π0 be the orthogonal projector in L2 (ρ) on the space of constant
functions
Z
Π0 (u)(x) = 1Ω (x) u(y )ρ(y )dy
(4.15)
Ω
Then
n
2supx∈Ω kTh,x
− ρ(y )dy kTV = kThn − Π0 kL∞ →L∞
(4.16)
Thus, we have to prove that there exist C , such that for any n and any
h ∈]0, h0 ], one has
kThn − Π0 kL∞ →L∞ ≤ Ce −ngh,ρ
G. Lebeau (Nice)
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(4.17)
february 2016
37 / 51
Total variation
Observe that since gh,ρ ' h2 , we may assume n ≥ Ch−2 . In order to prove
our estimate, we split Th in 2 pieces, according to the spectral theory.
Let 0 < λ1,h ≤ ... ≤ λj,h ≤ λj+1,h ≤ ... ≤ h−2 δ1 be such that the
eigenvalues of Th in the interval [1 − δ1 , 1[ are the 1 − h2 λj,h , with
associated orthonormals eigenfunctions ej,h
Th (ej,h ) = (1 − h2 λj,h )ej,h ,
(ej,h |ek,h )L2 (ρ) = δj,k
(4.18)
Then we write Th − Π0 = Th,1 + Th,2 with
Th,1 (x, y ) =
X
(1 − h2 λj,h )ej,h (x)ej,h (y )
(4.19)
λ1,h ≤λj,h ≤δ1 h−2
Th,2 = Th − Π0 − Th,1
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february 2016
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Th,2
One has
n
n
n
kL∞ →L∞ ≤ Ch−3d/2
− Π0 kL∞ →L∞ ≤ 2 + kTh,1
kTh,2
kL∞ →L∞ = kThn − Th,1
where the last estimate follows from the Weyl estimate and the crude L∞
bound on the eigenfunctions ej,h ∈ Eh,L , kej,h k ≤ Ch−d/2 .
Next we use Th = mh + Rh with kmh kL∞ →L∞ ≤ γ < 1 and
kRh kL2 →L∞ ≤ C0 h−d/2 and we iterate by Thp = mhp + Bp,h ,
n k
n
Bp+1,h = Thp Rh + Bp,h mh . Next, use kT2,h
L2 →L2 ≤ (1 − δ1 ) and
p+n
n
kT2,h
kL∞ →L∞ = kThp T2,h
kL∞ →L∞
n
n
≤ kmhp T2,h
kL∞ →L∞ + kBp,h T2,h
kL2 →L∞
(4.20)
≤ C γ p h−3d/2 + C0 h−d/2 (1 + γ + ... + γ p−1 )(1 − δ1 )n
Thus we get for some C > 0, µ > 0,
n
kT2,h
kL∞ →L∞ ≤ Ce −µn ,
∀h,
∀n ≥ 1/h
(4.21)
This is neglectible.
G. Lebeau (Nice)
Metropolis
february 2016
39 / 51
Nash inequality
From the Sobolev lemma, using the interpolation inequality
p
p−2
p−1
kuk2L2 ≤ kukLp−1
p kukL1 , we deduce the Nash inequality, with
1/D = 2 − 4/p > 0
2+1/D
kukL2
1/D
≤ C (Eh (u) + kuk2L2 )kukL1 ,
∀u ∈ Eh,L
(4.22)
For λj,h ≤ δ1 h−2 , one has h2 λj,h ≤ δ1 < 1, and thus for any u ∈ Eh,L , one
gets h2 Eh (u) ≤ kuk2L2 − kTh uk2L2 , thus we get from 4.22
2+1/D
kukL2
1/D
≤ Ch−2 (kuk2L2 −kTh uk2L2 +h2 kuk2L2 )kukL1 ,
G. Lebeau (Nice)
Metropolis
∀u ∈ Eh,L (4.23)
february 2016
40 / 51
Nash inequality
From the estimate on T2,h , there exists C2 such that for n ≥ h−1 one has
2
n k ∞
kT1,h
L →L∞ ≤ C2 and thus since T1,h is self adjoint on L
n k
−1
kT1,h
L1 →L1 ≤ C2 . Fix p ' h . Take g ∈ Eh,L such that kg kL1 ≤ 1 and
consider the sequence cn , n ≥ 0
n+p
cn = kT1,h
g k2L2
(4.24)
Then, 0 ≤ cn+1 ≤ cn and from 4.23, we get
1
1+ 2D
cn
1/D
n+p
≤ Ch−2 (cn − cn+1 + h2 cn )kT1,h
g kL1
1/D −2
≤ CC2
h
(4.25)
(cn − cn+1 + h2 cn )
Thus there exist A which depends only on C , C2 , D, such that for all
0 ≤ n ≤ h−2 , one has
Ah−2 2D
cn ≤ (
)
(4.26)
1+n
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february 2016
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Thus there exist C0 , such that for N ' h−2 , one has cN ≤ C0 . This
implies
N+p
kT1,h
g kL2 ≤ C0 kg kL1
(4.27)
and thus taking adjoints
N+p
kT1,h
g kL∞ ≤ C0 kg kL2
(4.28)
and so we get for any n and with N + p ' h−2
N+p+n
kT1,h
g kL∞ ≤ C0 (1 − h2 λ1,h )n kg kL2
(4.29)
And thus for n ≥ h−2 , since by definition h2 λ1,h = g (h, ρ)
n
kT1,h
kL∞ →L∞ ≤ C0 e −(n−h
−2 )h2 λ
1,h
= C0 e λ1,h e −ng (h,ρ)
(4.30)
QED
G. Lebeau (Nice)
Metropolis
february 2016
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Elementary Fourier Analysis
As a byproduct of the estimate (4.26), we get for g ∈ Eh,L by duality
−1
n+h
kTh,1
g kL∞ ≤ (
Ah−2 D
) kg kL2
1+n
With n ' h−2 < λ >−1 we get the following polynomial bound .
Lemma
There exists C independent of h such that for any eigenfunction
ej,h ∈ Eh,L , kej,h kL2 = 1, associated to the eigenvalue 1 − h2 λj,h of Th the
following inequality holds true
kej,h kL∞ ≤ C < λj,h >D .
(4.31)
Problem: In PDE’s, the natural estimate from Sobolev embeddings for
the eigenfunctions of L gives D = d/4. In fact, we have better L∞
estimates for the eigenfunctions of an elliptic operators that the one given
by Sobolev, namely, the Sogge estimates. So, what can be said about
the Lq norms , q > 2 of the Metropolis eigenvalues?
G. Lebeau (Nice)
Metropolis
february 2016
43 / 51
Elementary Fourier Analysis
Recall that νk is the kieme eigenvalue of the limit operator L with
Neumann boundary condition. Let Fk = Ker (L − νk ). Recall
mk = dim(Fk ) is the multiplicity of the eigenvalue νk of L. Let us denote
by Jk the set of indices j such that for h small, λj,h , the jeme eigenvalue
of the Metropolis is close to νk , and Fh,k = span(ej,h , j ∈ Jk ). By theorem
4.9 and his proof, the set Jk is independent of h ∈]0, hk ] for hk small, and
one has ](Jk ) = dim(Fh,k ) = mk for h ∈]0, hk ] . Let ΠFk and ΠFh,k be the
L2 -orthogonal projectors on Fk and Fh,k .
Lemma
For all f ∈ Fk one has
lim kf − ΠFh,k (f )kL∞ = 0.
(4.32)
h→0
This lemma is used to prove a weak convergence result of the Metropolis
walk to the continuous diffusion associated to the limit operator L with
Neumann boundary condition.
G. Lebeau (Nice)
Metropolis
february 2016
44 / 51
Outline
1
The Metropolis Algorithm
2
Random placement of non-overlapping balls
3
A local model in a bounded, connected, Lipschitz domain
4
Diffusion
5
Metropolis Laplacian
G. Lebeau (Nice)
Metropolis
february 2016
45 / 51
Convergence to the Wiener measure
bla-bla-bla....
G. Lebeau (Nice)
Metropolis
february 2016
46 / 51
Outline
1
The Metropolis Algorithm
2
Random placement of non-overlapping balls
3
A local model in a bounded, connected, Lipschitz domain
4
Diffusion
5
Metropolis Laplacian
G. Lebeau (Nice)
Metropolis
february 2016
47 / 51
A canonical random walk
Let (X , dist, µ) be a metric space equipped with a Borel measure µ , and
let h ∈]0, ∞[ be a parameter. We assume that µ satisfies
µ(B(x, h)) ∈]0, ∞[,
∀x ∈ X ,
∀h ∈]0, ∞[
(6.1)
The canonical random walk on (X , dist, µ) is associated to the Markov
kernel
Z
1
f (y )dµ
(6.2)
Th f (x) =
µ(B(x, h)) B(x,h)
This means that if the walk is at xn , then xn+1 is choosen randomly in the
ball B(x, h) equipped with the probability induced by µ.
Unfortunately, Th is not in general self-adjoint on L2 (X , dµ).
It is self-adjoint for the measure dνh = µ(B(x, h))dµ.
G. Lebeau (Nice)
Metropolis
february 2016
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The h-Laplace operator on functions
By definition, a 1-form on X at scale h is an antisymmetric function on
Diagh = {(x, y ) ∈ X × X , dist(x, y ) < h}.
If f (x) is a function on X , its h-differential is the 1-form
δh (f )(x, y ) = h−1 (f (x) − f (y ))
(6.3)
Let C (x, y , h) > 0 be a symmetric function on Diagh . One defines a
(1)
measure µh on Diagh by the formula
(1)
dµh (x, y ) = C (x, y , h)dµh (x)dµh (y )
(6.4)
The normalization factor C (x, y , h) is a parameter in the theory. The
(positive) Laplacian is then defined as usually by the formula
|4h | = δh∗ δh
G. Lebeau (Nice)
Metropolis
february 2016
49 / 51
Z
2
=
g (x, y )C (x, y , h)dµ(y )
h y ∈B(x,h)
Z
2
f (x) − f (y ) C0 (x, y , h)dµ(y )
|4h |(f )(x) = 2
h y ∈B(x,h)
δh∗ (g )(x)
If one defines Mh by the formula
Id − Mh
1
= |4h |
h2
2
(6.5)
one has obviously Mh (1) = 1, since |4h |(1) = 0, and
Z
Mh f (x) =
f (y )Mh (x, dy )
(6.6)
X
where for all x ∈ X , Mh (x, dy ) is the measure on X
Z
Mh (x, dy ) = 1 −
C0 (x, z, h)dµ(z) δx + 1B(x,h) C0 (x, y , h)dµ(y )
B(x,h)
G. Lebeau (Nice)
Metropolis
february 2016
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Metropolis
Now, observe that Mh is Markov iff for all x ∈ X , the measure Mh (x, dy ),
which has total mass 1, is positive. This means
Z
∀x ∈ X , F (x, h) =
C0 (x, z, h)dµ(z) ≤ 1
(6.7)
B(x,h)
A natural choice to get the inequalite 6.7 is to choose the symmetric
normalization factor
1
1
C0 (x, y , h) = min(
,
)
(6.8)
µ(B(x, h)) µ(B(y , h))
This means in that case that Mh , a reversible Markov kernel for the
measure µ, is exactly the Metropolis operator associated to the Markov
kernel Th !!!
The associated random walk is simple : if the walk is at xn , choose
1
dµ(z). If
z ∈ B(xn , h) uniform for the probability 1B(x,h) µ(B(x,h))
µ(B(z, h)) ≥ µ(B(xn , h)), go to z (xn+1 = z), else, flip a coin with
probability µ(B(z, h))/µ(B(xn , h)), go to z = xn+1 if you win, else, stay at
xn .
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Metropolis
february 2016
51 / 51