Di usion equations: from Euclidean space to Graphsnguillen/697/697Week10.pdf · Mean Value Property...
Transcript of Di usion equations: from Euclidean space to Graphsnguillen/697/697Week10.pdf · Mean Value Property...
Diffusion equations:from Euclidean space to Graphs
MATH 697 AM:ST
November 7th, 2017
Warm upConvolutions and differentiability
Consider the following:
A function K : Rd 7→ R which is infinitely differentiable, with
K(y) ≥ 0 ∀ y, K(y) = 0 if |y| ≥ 1,
∫RdK(y) dy = 1,
and which is spherically symmetric, that is
K(y) = K(y′) if |y| = |y′|.
For δ > 0, define
Kδ(y) = δ−dK(δ−1y).
Warm upConvolutions and differentiability
Consider a domain D ⊂ Rd, and f : D 7→ R an integrablefunction, ∫
D|f(x)| dx <∞
For δ > 0 define the domain
Dδ = {x | d(x, ∂D) > δ}
Warm upConvolutions and differentiability
Lemma
For x ∈ Dδ, define
fδ(x) = (f ∗Kδ)(x)
Then, fδ is infinitely differentiable in Dδ, and
∂αfδ(x) = (f ∗ ∂αKδ)(x)
Minima of FunctionalsRECAP
Hilbert’s 19th Problem (1900)
Show that the minima for the functional
J (f) =
∫DL(∇f, f, x) dx
are always analytic functions of x (under certain specificconditions we will not specify).
Minima of FunctionalsRECAP
Hilbert’s 19th Problem (1900)
1. Schauder (1930’s): If there exists a C1,α minimizer, thenthis minimizer must be analytic.
2. De Giorgi (1957), Nash (1958) showed that there were C1,α
minimizers. Their method relied greatly on new regularityestimates for partial differential equations.
Minima of FunctionalsRECAP
The Dirichlet Energy
Given a bounded domain D ⊂ Rd with smooth boundary, and acontinuous function
φ : ∂D 7→ R
ProblemFind a function f : D 7→ R equal to φ on ∂D, minimizing
J (f) =1
2
∫D|∇f |2 dx
Minima of FunctionalsRECAP
The Dirichlet Energy
Theorem
If φ is differentiable, there is a unique continuous function
f : D 7→ R, f = φ on ∂D,
which is infinitely differentiable in the interior of D, and
∆f = 0.
Mean Value PropertyRECAPThe MVP
Theorem
Let f : D 7→ R be a continouous function which is twicedifferentiable and harmonic in its interior.
Then, if x0 is an interior point of D and r is strictly smallerthan the distance from x0 to ∂D, we have that
(Average of f over ∂Br(x0)) = f(x0).
Harmonic Functions and the MVPHigher differentiability of harmonic functions
Suppose we are given a continuous function f : D 7→ R which istwice differentiable and harmonic in D.
Lemma
If x ∈ D is a distance larger than δ from ∂D, then
f(x) = (f ∗Kδ)(x)
Harmonic Functions and the MVPHigher differentiability of harmonic functions
Theorem
A continuous function f : D 7→ R which is twice differentiableand harmonic in D is always infinitely differentiable.
Furthermore, for any index α there is a constant C(d, α) suchthat in Dδ we have the estimate
|∂αf(x)| ≤ C(d, α)δ−|α|‖f‖L1(D).
Harmonic Functions and the MVPStability properties of Harmonic Functions
An important consequence of the higher differentiability andthe estimate above, is the following stability property forharmonic functions (hinted at last time).
If a sequence of harmonic functions converge (locally) uniformlyto a function, then this function is itself harmonic and thederivatives also converge (locally) uniformly.
The Fractional Laplacian
Last time, we defined the fractional Laplacian of orderα ∈ [0, 2], by the formula
Lαf(x) := C(d, α)
∫Rd
f(y)− f(x)
|y − x|d+αdy
the constant C(d, α) is explicitly defined but we will notconcern ourselves with its exact form.
We observed that (−L1) = (−∆)12 , and that in general
−Lα = (−∆)α2
The Fractional Laplacian
Last time, we defined the fractional Laplacian of orderα ∈ [0, 2], by the formula
Lαf(x) := C(d, α)
∫Rd
f(y)− f(x)
|y − x|d+αdy
the constant C(d, α) is explicitly defined but we will notconcern ourselves with its exact form.
We observed that (−L1) = (−∆)12 , and that in general
−Lα = (−∆)α2
The Fractional Laplacian
The Nonlocal Dirichlet Energy: given f : Rd 7→ R
Jα(f) =1
2C(d, α)
∫Rd
∫Rd
|f(x)− f(y)|2
|x− y|d+αdxdy
Minimization of Jα leads to the equation Lαf = 0, just as forthe case of the Laplacian and the L2 norm of |∇f |.
GraphsVertices, Edges, and weights
A (finite) weighted graph G is most typically, described as
G = (V,E,w)
V = a (finite) set, the set of vertices
E = a subset of V × V, the set of edges
w : E 7→ R the weight function
It is said that x ∼ y if (x, y) ∈ E,
GraphsVertices, Edges, and weights
Some simplificationIt is usually preferable to think simply of the graph as being aset G (forget about distinguishing V ), coupled with a(non-negative) weight function w,
w : G×G 7→ R
with w(x, y) denoted sometimes as wxy for any x, y ∈ G.
One can think as E being the set of (x, y)’s such that wxy > 0.
GraphsExamples
1. Subsets of Rp.
Let V = {x1, . . . , xN} be a subset of Rp, let
w(xi, xj) = h(xi − xj)
A popular choice is for h to be spherically symmetric, e.g.
1
Zσe−|xi−xj |
2
σ , χBσ(0)(xi − xj)
where σ > 0 is a tuning parameter.
GraphsExample
2. Subsets of a metric space.
Let V = {x1, . . . , xN} be a subset of (X, ρ), a metric space.Then, define
w(xi, xj) = h
(ρ(xi, xj)
2
σ2
)Popular selections for h are
h(t) =√t
h(t) = e−t
h(t) = χ[0,1](t)
GraphsSetup
Given a vertex x, it’s degree is the number
d(x) =∑y∈V
wxy
The normalized weight function is then defined by
K(x, y) :=1
d(x)wxy
so that for fixed x, K(x, ·) is a probability distribution over G,∑y∈V
K(x, y) = 1.
The Dirichlet Problem on graphs andit’s use in Semi-supervised learning
MATH 697 AM:ST
November 9th, 2017
Warm up
A path in a graph G is a sequence of points xi (i = 1, . . . ,m)such that
xi ∼ xi+1 for i = 1, 2, . . . ,m− 1
Two points x, y ∈ G are said to be connected if there exists apath with x1 = x and xm = y.
A subset D of G is said to be connected if given two points inD then they can be connected by a path exclusively made outof points in D.
Warm up
We consider the Laplacian of a function f : G 7→ R
∆f(x) =∑y∈G
wxy(f(y)− f(x)).
Then, we observe the following:
f(x0) ≥ f(x) ∀ x ∈ G⇒ ∆f(x0) ≤ 0.
Moreover, if f(x0) > f(x1) for at least one x1 ∼ x0, then
∆f(x0) < 0.
Graph LaplaciansRECAP
We define various operators all with equal claims to be called aLaplacian.
First, the Combinatorial Laplacian (or just the Laplacian)
∆f(x) =∑y∈G
(f(y)− f(x))wxy
Graph LaplaciansRECAP
We define various operators all with equal claims to be called aLaplacian.
Secondly, we have the Random Walk Laplacian
∆f(x) =∑y∈G
(f(y)− f(x))K(x, y)
Graph LaplaciansRECAP
We define various operators all with equal claims to be called aLaplacian.
Last but not least, we have the Symmetric Laplacian
∆f(x) =1√d(x)
∑y∈G
wxy
(g(y)√d(x)
− g(x)√d(x)
)
Graph LaplaciansRECAP
The Combinatorial Laplacian (or just the Laplacian)
∆f(x) =∑y∈G
(f(y)− f(x))wxy
The Random Walk Laplacian
∆f(x) =∑y∈G
(f(y)− f(x))K(x, y)
The Symmetric Laplacian
∆f(x) =1√d(x)
∑y∈G
wxy
(g(y)√d(x)
− g(x)√d(x)
)
The (strong) Maximum Principle
Consider a function f : G 7→ R such that
∆f(x) =∑y∈G
wxy(f(y)− f(x)) = 0 ∀ x ∈ G.
Let M = maxx∈G
f(x), and let x0 ∈ G be such that
f(x0) = M
Claim:
If x is connected to x0 then we have f(x0) = M
The (strong) Maximum Principle
Theorem
Let G be a connected graph, then the only functions f : G 7→ Rwhich are harmonic in G are the constants.
In other words: for a connected graph, the kernel of the linearmap ∆ is one dimensional and given by the constant function.
To have any interesting harmonic functions, we must ask theybe harmonic only in some portion of G.
The Comparison Principle
Theorem
Let G be a graph, and D ⊂ G a connected subset of the graph.Then, if f1, f2 : G 7→ R are such that
∆f1 = ∆f2 = 0 in D, and f1 ≤ f2 in G \D
Then, we have
f1 ≤ f2 in D
Measuring Smoothness
Let f : G 7→ R, and define the norms
‖f‖L2 :=
(∑x∈G|f(x)|2
) 12
‖f‖Hw :=
∑x∈G|f(x)|2 +
∑x,y∈G
|f(x)− f(y)|2ωxy
12
as well as
‖f‖Hw :=
∑x,y∈G
|f(x)− f(y)|2ωxy
12
Measuring Smoothness
While ‖f‖L2 measures simply the average “size” of f , ‖f‖Hwmeasures the average size of its oscillations.
Observe that if ‖f‖Hw = 0, then f must be constant in eachconnected component of the graph.
Thus, the larger ‖f‖Hw is with respect to ‖f‖L2 , the more thefunction f is oscillating with respect to its size.
The Laplacian is a symmetric operator
We introduce an inner product for functions in G,
〈f, g〉 =∑x∈G
f(x)g(x)
Henceforth assume that G has symmetric weights, that is
wxy = wyx
The Laplacian is a symmetric operator
Let ∆ be the combinatorial Laplacian, and observe that
∑x∈G
∆f(x)g(x) =∑x∈G
∑y∈G
(f(y)− f(x))wxy
g(x)
The Laplacian is a symmetric operator
Then, expanding this sum, we have
〈∆f, g〉 =∑x,y∈G
f(y)g(x)wxy −∑x∈G
f(x)g(x)d(x)
where d(x) =∑y∈G
wxy. Since wxy = wyx, it follows that
〈∆f, g〉 = 〈f,∆g〉
Therefore, ∆ is symmetric with respect to this inner product.
The Laplacian is a symmetric operatorPositivity
One further expression, which is reminiscent of the Greenidentity, is the following (to be proved later)
−〈∆f, g〉 = −12
∑x,y∈G
wxy(f(y)− f(x))(g(y)− g(x))
It follows that for any f : G 7→ R,
−〈∆f, f〉 ≥ 0
The Laplacian is a symmetric operatorPositivity
Furthermore, if G is connected, then
−〈∆f, f〉 = 0
if and only if f is equal to a constant.
With this knowledge in hand, we obtain a good picture of theeigenfunction decomposition associated to ∆.
Eigenfunctions of ∆
There is a family of functions
{φn}Nn=0, φn : G 7→ R
as well as numbers 0 = λ0 ≤ λ1 ≤ λ2 ≤ . . . ≤ λN , such that
−∆φn(x) = λnφn(x) ∀ x ∈ G.
Moreover, the φn are orthonormal,
〈φn, φm〉 = δnm.
Eigenfunctions of ∆Decrasing Smoothness of φn
Observe that
‖φn‖1Hw =∑x,y∈G
wxy(φn(y)− φn(x))2
= −2〈∆φn, φn〉
Therefore,
‖φn‖1Hw = 2λn‖φn‖2L2
As the λn is increasing with n, we see that the φn becomeincreasingly more oscillatory as n gets larger and larger.
The Heat Equation
We consider the heat equation in all of G, with some initialdatum u0 : G 7→ R,
u = ∆u in G× (0,∞),
u = u0 at t = 0.
The Heat Equation
By the orthogonality of the φn, it follows that u0 can beexpressed as
u0 =
N∑n=0
αnφn where αn = 〈φn, u0〉.
Therefore, for t > 0, the solution to the heat equation is
u(t) =
N∑n=0
αne−tλnφn
The Heat Equation
By the orthogonality of the φn, it follows that u0 can beexpressed as
u0 =
N∑n=0
αnφn where αn = 〈φn, u0〉.
Therefore, for t > 0, the solution to the heat equation is
u(t) =
N∑n=0
e−tλn〈φn, u0〉φn
The Heat Equation
By the orthogonality of the φn, it follows that u0 can beexpressed as
u0 =
N∑n=0
αnφn where αn = 〈φn, u0〉.
Therefore, for t > 0, the solution to the heat equation is
u(x, t) =
N∑n=0
∑y∈G
e−tλnφn(y)u0(y)φn(x)
The Heat Equation
This leads to an analogue of the heat kernel H(t, x, y), namely,a function such that
u(x, t) =∑y∈G
H(t, x, y)u0(y)
This H(t, x, y) is given via the eigenfunction decomposition
H(t, x, y) =
N∑n=0
e−tλnφn(x)φn(y)
The Dirichlet Problem Revisited
Fix some subset D ⊂ G, and suppose we are given a function
g : G \D 7→ R
Then, we aim to find f : G 7→ R, such that
f = g in G \D,
and with f minimizing the discrete Dirichlet energy
J(f) :=1
2
∑x,y∈G
wxy(f(x)− f(y))2
The Dirichlet Problem Revisited
Let fs = f + sφ, then
d
dsJ(fs) =
d
ds
1
2
∑x,y∈G
wxy(fs(x)− fs(y)2
=∑x,y∈G
wxy(fs(x)− fs(y)(φ(x)− φ(y))
At s = 0, this results in∑x,y∈G
wxy(f(x)− f(y)(φ(x)− φ(y))
The Dirichlet Problem Revisited
∑x,y∈G
wxy(f(x)− f(y)(φ(x)− φ(y))
=
∑x,y∈G
wxy(f(x)− f(y))φ(x)
− ∑x,y∈G
wxy(f(x)− f(y))φ(y)
Since wxy = wyx means that∑
x,y∈Gwxy(f(x)− f(y))φ(y) =
∑x,y∈G
wxy(f(y)− f(x))φ(x)
The Dirichlet Problem Revisited
∑x,y∈G
wxy(f(x)− f(y)(φ(x)− φ(y))
= 2∑x,y∈G
wxy(f(x)− f(y))φ(x)
= −2∑x∈G
φ(x)∑y∈G
wxy(f(y)− f(x))
= −2∑x∈G
φ(x)∆f(x)
The Dirichlet Problem Revisited
In conclusion,
d
ds |s=0J(fs) = −2
∑x∈G
φ(x)∆f(x)
In particular, ∑x∈G
φ(x)∆f(x) = 0
for any function φ : G 7→ R with φ(x) = 0 if x ∈ G \D.
The Dirichlet Problem Revisited
There is a unique minimizer f , and it is characterized by{Lf(x) = 0 if x ∈ D,f(x) = g(x) if x ∈ G \D.
Compare with the Dirichlet problem for the Laplacian
min
∫D|∇f(x)|2 dx, f = g on ∂D.
and the one for the fractional Laplacian (α ∈ (0, 2)),
min
∫R2d
(f(x)− f(y)|2|x− y|−d−2α dx, f = g in Rd \D.
The Dirichlet Problem Revisited
There is a unique minimizer f , and it is characterized by{Lf(x) = 0 if x ∈ D,f(x) = g(x) if x ∈ G \D.
Compare with the Dirichlet problem for the Laplacian
min
∫D|∇f(x)|2 dx, f = g on ∂D.
and the one for the fractional Laplacian (α ∈ (0, 2)),
min
∫R2d
(f(x)− f(y)|2|x− y|−d−2α dx, f = g in Rd \D.
The Dirichlet Problem Revisited
Thanks to the variational characterization of harmonicfunctions and the comparison theorem, we have the following:
Theorem
Let D ⊂ G be a connected subset of G. Then, giveng : G \D 7→ R there exists one, and exactly one, function fwhich solves {
Lf(x) = 0 if x ∈ D,f(x) = g(x) if x ∈ G \D.
The Dirichlet Problem Revisited
The solution to {Lf(x) = 0 if x ∈ D,f(x) = g(x) if x ∈ G \D.
Is given by
f(x) =∑
y∈G\D
g(y)P (x, y)
for a certain kernel P (x, y) computable from the wxy.