Numerical Analysis of Quantum Graphs - UCLouvain › HHXIX › Plenaries › Benzi.pdf · Some...

Numerical Analysis of Quantum Graphs

Michele Benzi

Department of Mathematics and Computer ScienceEmory University

Atlanta, Georgia, USA

Householder Symposium XIX

Spa, Belgium8-13 June, 2014

1

Outline

1 Motivation

2 Basic definitions

3 Boundary conditions at the vertices

4 Discretization

5 Numerical experiments

6 Time-dependent problems

7 Conclusions and open problems

2

Acknowledgements

Joint work with Mario Arioli (Berlin)

Support: NSF, Emerson Center for Scientific Computation, BerlinMathematical School, TU Berlin

3

Outline

1 Motivation

2 Basic definitions


4 Discretization




4

Motivation

The purpose of this talk is to introduce the audience to a class ofmathematical models known as quantum graphs, and to describe somenumerical methods for investigating such models.

Roughly speaking, a quantum graph is a collection of intervals gluedtogether at the end-points (thus forming a metric graph) and a differentialoperator (“Hamiltonian") acting on functions defined on these intervals,coupled with suitable boundary conditions at the vertices.

5

Motivation (cont.)

Quantum graphs are becoming increasingly poular as mathematical modelsfor a variety of physical systems including conjugated molecules (such asgraphene), quantum wires, photonic crystals, carbon nanostructures, thinwaveguides, etc.

G. Berkolaiko and P. Kuchment, Introduction to Quantum Graphs, AmericanMathematical Society, Providence, RI, 2013.

6

Example: Graphene

7

Example: Carbon nanostructures

8

Example: Polystyrene

9

Motivation (cont.)

Other potential applications include the modeling of phenomena such asinformation flow, diffusion and wave propagation in complex networks(including social and financial networks), blood flow in the vascularnetwork, electrical signal propagation in the nervous system, traffic flowsimulation, and so forth.

10

Early structure of the Internet

11

Motivation (cont.)

While the theory of quantum graphs is very rich (well-posedness results,spectral theory, etc.), there is very little in the literature on the numericalanalysis of such models.

In this lecture we consider numerical methods for the analysis of quantumgraphs, focusing on simple model problems.

This is still very much work in progress!

12

Outline

1 Motivation

2 Basic definitions


4 Discretization




13

Combinatorial graphs

A combinatorial graph Γ is a pair (V, E) where V = {vj}Nj=1 is a set of

vertices, or nodes, and E = {ek}Mk=1 is a set of edges connecting the

vertices.

Each edge e can be identified by the couple of vertices that it connects(e = (vi, vj)).

The edges simply stand for some type of binary relation between pairs ofnodes: in particular, they are not endowed with any geometry.

Only undirected graphs are considered here. In some cases it will benecessary to assign a direction to the edges, but the choice will bearbitrary.

14

Some matrices associated with combinatorial graphs

Adjacency matrix A: symmetric N ×N vertex-to-vertex Booleanstructure, Aij = 1 iff edge (vi, vj) exists.Incidence matrix E: rectangular N ×M vertex-to-edge matrix. Eachcolumn corresponds to an edge e = (vi, vj) and has only two non zeroentries, 1 and −1, in position i and j (the sign is arbitrary and will beimmaterial for our purposes).Combinatorial Laplacian: LΓ = EET = DΓ −A, where DΓ is thediagonal matrix of degrees.

I E can be interpreted as a discrete divergence and ET as a discretegradient.

I If Γ is connected, Ker(ET ) is the 1D subspace of RN spanned bye = (1, 1, . . . , 1)T .

LΓ is a positive semidefinite singular matrix with Ker(LΓ) = span{e}

15

Example: graphene sheet

1 2 3 4 5 6 7 8

0

1

2

3

4

5

16

Example: graphene sheet (cont.)

17

Example: graphene sheet (cont.)

18

Beyond simple graphs

While combinatorial graphs (including weighted and directed ones) havelong been found extremely useful in countless applications, they are toosimple for modeling certain types of phenomena on networks.

In many cases, the interaction between pairs of nodes may be morecomplex than just a 0-1 relation.

19

Beyond simple graphs (cont.)

For instance, in physics and engineering applications the edges mayrepresent actual physical links between vertices, and these links willtypically be endowed with a notion of length.

Hence, communication between nodes may require some time ratherthan being instantaneous.

This simple observation is formalized in the notion of metric graph.

20

Metric graphs

A graph Γ is a metric graph if to each edge e is assigned a measure(normally the Lebesgue one) and, consequently, a length le ∈ (0,∞).

Thus, each edge can be assimilated to a finite interval on the real line(0, le) ⊂ R, with the natural coordinate s = se.

Note that we need to assign a direction to an edge in order to assign acoordinate to each point on e. For this we can take the (arbitrarily chosen)direction used in the definition of the incidence matrix of Γ.

In technical terms, a metric graph is a topological manifold (1D simplicialcomplex) having singularities at the vertices, i.e. it is not a differentiablemanifold (globally).

21

Metric graphs (cont.)

The points of a metric graph Γ are the vertices, plus all the points on theedges.

The Lebesgue measure is well-defined on all of Γ for finite graphs (the onlyones considered here). Thus, Γ is endowed with a global metric.

The distance between two points (not necessarily vertices) in Γ is thelength of the geodetic (shortest path) between them.

Note that Γ is not necessarily embedded in a Euclidean space Rn.

22

Metric graphs (cont.)

The edges may also have physical properties, such as conductivity,diffusivity, permeability etc., that are are not well represented by a singlescalar quantity, as in a weighted graph. Some of these quantities couldeven change with time.

In other words, interactions between nodes may be governed by laws,which could be described in terms of differential equations.

One can easily imagine similar situations also for other types of networks,including social or financial networks.

Formalization of this notion leads to the concept of quantum graph.

23

Hilbert spaces on Γ

To define a quantum graph, we need first to introduce certain Hilbertspaces of functions defined on a metric graph Γ.

Let L2(e) :={f : e→ R |

∫e |f |

2 ds <∞}

.

Definition: The space of square-integrable functions on Γ is defined as

L2(Γ) :=⊕e∈E

L2(e) .

In other terms,

f ∈ L2(Γ) iff ‖f‖2L2(Γ) =

∑e∈E

‖f‖2L2(e) <∞.

24

Hilbert spaces on Γ (cont.)

Next, consider the Sobolev space

H1(e) =

{f ∈ L2(e) |

∫e|f ′(s)|2 ds <∞

}.

Definition: The Sobolev space H1 on Γ is defined as

H1(Γ) =

(⊕e∈E

H1(e)

)∩ C0(Γ)

where C0(Γ) is the space of continuous functions on Γ.

In other terms,

f ∈ H1(Γ) iff f is continuous and ‖f‖2H1(Γ) =

∑e∈E

‖f‖2H1(e) <∞ .

25

Quantum graphs (def.)

Let H be a linear differential operator defined on suitable subspaceD(H) ⊂ L2(Γ). We will call H a Hamiltonian on Γ.

Definition: A quantum graph is a metric graph Γ together with aHamiltonian H and boundary (vertex) conditions that ensure that His self-adjoint.

Hence, a quantum graph is a triple (Γ,H, vertex conditions).

In some situations the definition may be altered to allow more general(non-self-adjoint, pseudo-) differential operators.

Remark: Although the concept goes back at least to G. Lumer (1980),the name “quantum graph" was introduced by T. Kottos and U. Smilansky(Phys. Rev. Lett. 79 (1997), pp. 4794–4797) and has since becomeuniversally adopted.

26

Hamiltonians

We will focus primarily on the simplest example of a Hamiltonian: the(negative) second derivative operator

u→ Hu = −d2u

ds2.

We will also consider Schrödinger operators

u→ Hu = −d2u

ds2+ V (s)u ,

where V is a potential, usually required to be bounded from below. Herewe assume u ∈ H2(e), ∀e ∈ E .More complicated operators can also be considered, for example the magneticSchrödinger operator

u→ Hu =

(1

i

dds−A(s)

)2

u+ V (s)u

as well as higher order operators, Dirac operators, pseudo-differential operators,etc.

27

Outline

1 Motivation

2 Basic definitions


4 Discretization




28

Examples of boundary conditions

In this talk we only consider the so-called Neumann-Kirchhoff conditions, aspecial case of δ-type conditions:

f(s) is continuous on Γ

∀v ∈ Γ∑

e∈Ev

dfdse

(v) = αvf(v)

Ev is the subset of the edges having v as a boundary point.

The αv’s are fixed real numbers.

29

Examples of boundary conditions (cont.)

The Hamiltonian is associated to the following quadratic form on H1(Γ):

h[f, f ] =∑e∈E

∫e

∣∣f ′(s)∣∣2 ds+∑v∈V

αv |f(v)|2 .

The case αv ≡ 0 corresponds to the Neumann-Kirchhoff conditions: f(s) is continuous on Γ

∀v ∈ Γ∑

e∈Ev

dfdse

(v) = 0

and the corresponding quadratic form reduces to

h[f, f ] =∑e∈E

∫e

∣∣f ′(s)∣∣2 ds.

30


The Neumann-Kirchhoff conditions are also called the standard vertexconditions. They are the natural boundary conditions satisfied by theSchrödinger operator.

The first condition expresses continuity, while the second can beinterpreted as conservation of current.

31


We observe that, on the other hand, the Dirichlet or Neumann boundaryconditions at the vertices are examples of decoupling conditions and are oflittle interest in this context, except for vertices of degree one.

For example, if we impose the vertex Dirichlet condition f(v) = 0 at eachvertex, the Hamiltonian is just the direct sum of the operators on eachedge e with Dirichlet conditions on the end; hence, the quantum graphdecouples into a set of independent intervals, and the topology of thegraph becomes irrelevant.

32

Outline

1 Motivation

2 Basic definitions


4 Discretization




33

Finite element discretization

Self-adjoint elliptic equations can be formulated as variational problems foran energy functional.

Given a function g ∈ L2(Γ), the minimum problem is

minu∈H1(Γ)

J(u) ,

where

J(u) =1

2

∑e∈E

∫e

{u′(s)2 + V (s)u(s)2

}ds−

∑e∈E

∫eg(s)u(s) ds.

We discretize the problem using 1D linear finite elements on each edgeand use a domain decomposition approach:

first we eliminate the unknowns associated with points inside the edges,

then we use the Neumann-Kirchhoff conditions and the values of thederivative at the vertices to form and solve a reduced-size linear system(Schur complement) for the values of the solution at the vertices.

34

Finite element discretization (cont.)

On each edge of the quantum graph it is possible to use the classical 1Dfinite-element method. Let e be a generic edge identified by two vertices,which we denote by va and vb.

The coordinate s will parameterize the edge such that for s = 0 we havethe vertex va and for s = è we have the vertex vb.

The first step is to subdivide the edge in ne intervals of length he. Thepoints {

sej

}n−1

j=1∪ {va} ∪ {vb}

form a chain linking va to vb lying on e.

35


Denoting by{ψe

j

}n+1

j=0the standard hat basis functions, we have

ψe0(s) =

{1− s

h if 0 ≤ s ≤ he

0 otherwise

ψej (s) =

{1− |sj−s|

h if sj−1 ≤ s ≤ sj+1

0 otherwise

ψene+1(s) =

{1− è−s

h if è − he ≤ s ≤ è

0 otherwise

. (1)

The functions ψej are a basis for the finite-dimensional space

V eh =

{uh ∈ H1(e) : uh|[se

j ,sej+1]

∈ P1, j = 0, . . . , n+ 1},

where P1 is the space of linear functions.36


In practice, we subdivide each edge, forming a chain made of nodes ofdegree 2, and we build the usual hat functions extending them to thevertices:

37


Globally, we construct the finite element space

Vh(Γ) =⊕e∈E

V eh ,

which is a finite-dimensional subspace of H1(Γ).

The continuity on Γ of the functions in Vh follows by construction: at eachvertex v we have dv (degree of the vertex v) linear functions that take thevalue 1 on v, each one belonging to an independent V e

h with e ∈ Ev.

Any function uh ∈ Vh(Γ) is then a linear combination of the ψej :

uh(s) =∑e∈E

n+1∑j=0

αejψ

ej (s).

38


The quadratic form h of the Hamiltonian operator can be tested on all theψ’s and we have the following finite dimensional (discrete) bilinear form:

hh[uh, ψek] =

∑e∈E

n+1∑j=0

αej

{∫e

dψej

dsdψe

k

dsds+

∫eV (s)ψe

jψek ds

}.

In both h and hh the Neumann-Kirchhoff conditions at each vertex are thenatural conditions and they are automatically satisfied.

39

Extended graph

The nodes on the edges will describe a chain path between two vertices.

We can then think of introducing a new (combinatorial) graph in whichthe nodal discretization points become additional vertices and the edgesare obtained by subdividing the edges of the original (metric) graph. Wecall this the extended graph associated with Γ and denote it by G.

Assuming for simplicity that all edges e ∈ E have equal length and thatthe same number n− 1 of internal nodes are used for each edge, the newgraph G will have (n− 1)×M +N vertices and n×M edges, where N isthe number of vertices and M the number of edges in Γ.

The extended graph can be huge, but it has a lot of structure.

40

Extended graph (cont.)

It is natural to order the vertices according to the original order of theedges so that the new (“subdomain") vertices on the edges are numberedcontiguously, and the vertices of the original graph (“separators") arenumbered last.

The resulting Gramian matrix H = (hh[ψej , ψ

ek]) is of the form

H =

[H11 H12

HT12 H22

]where H11 is a block diagonal symmetric and positive definite matrixwhere each diagonal block is of size n− 1 and tridiagonal, and H22 is adiagonal matrix with positive diagonal entries.

Important: We are assuming that the potential V (s) is positive.

41

A simple example

Figure: Example of a simple planar metric graph and of its incidence matrix.

42

A simple example

Figure: Example of the extension of the graph when a 4 nodes chain is addedinternally to each edge (left) and its incidence matrix (right).

42

A simple example

Figure: Pattern of the discrete Hamiltonian H where the red bullets correspondto the original vertices and the blue ones to the internal nodes on each edge.

42

Extended graph (cont.)

In the special case V = 0 (that is, H = − d2

ds2 , we obtain the discrete(negative) Laplacian (stiffness matrix) L on G.

When the same number of (equidistant) discretization points is used oneach edge of Γ, L coincides (up to the factor h−1) with the combinatorialgraph Laplacian LG .

Both the stiffness matrix L and the mass matrix M = (〈ψej , ψ

ek〉) have a

block structure matching that of H. For example, if V (s) = k (constant)then H = L + kM.

Minimization of the discrete quadratic form

Jh(uh) := hh[uh, uh]− 2〈gh, uh〉, uh ∈ Vh(Γ)

is equivalent to solving the extended linear system Huh = gh, of order(n− 1)M +N .

43

Solution of the extended linear system

The extended linear system can be solved efficiently by block LUfactorization, by first eliminating the interior edge nodes (this requiressolving, in parallel if one wishes, a set of M independent tridiagonalsystems of order n− 1), and then solving the N ×N Schur complementsystem

Suvh = gv

h −HT12H

−111 f

eh ≡ ch

for the unknowns associated with the vertices of Γ.

44

Solution of the extended linear system

The block LU factorization of H is given by

H =

[H11 H12

HT12 H22

]=

[H11 OHT

12 S

] [I H−1

11 H12

O I

].

Simple, yet crucial observation: the Schur complement

S = H22 −HT12H

−111 H12

is a sparse matrix.

45

Back to the simple example

Figure: The pattern of the Schur complement S.

46

Solution of the extended linear system (cont.)

Theorem: The nonzero pattern of the Schur complement

S = H22 −HT12H

−111 H12

coincides with that of LΓ, the (combinatorial) graph Laplacian of the(combinatorial) graph Γ.

In the special case V = 0, we actually have S = LΓ.

47


Note that S is SPD, unless V = 0 (in which case S is only positivesemidefinite).

For Γ not too large, we can solve the Schur complement system by sparseCholesky factorization with an appropriate reordering.

However, for large and complex graphs (for example, scale-free graphs),Cholesky tends to generate enormous amounts of fill-in, regardeless of theordering used.

Hence, we need to solve the Schur complement system by iterativemethods, like the preconditioned conjugate gradient (PCG) algorithm.

48


Preconditioning of matrices arising from complex graphs is an active areaof research.

Some of the techniques that work well for other types of problems (likeIncomple Cholesky Factorization) are useless here.

Here we consider two simple preconditioners:

diagonal scaling with D = diag(S)

a first degree polynomial preconditioner:

P−1 = D−1 + D−1(D− S)D−1 ≈ S−1.

Note: For the very sparse matrices considered here, each iteration of PCG withpolynomial preconditioning costs about the same as 1.5 iterations with diagonalpreconditoning.

49

Outline

1 Motivation

2 Basic definitions


4 Discretization




50

Numerical experiments

We present first results for a simple steady-state (equilibrium) problem

−d2u

ds2+ V u = g on Γ

with Neumann-Kirchhoff conditions at the vertices, for three differentchoices of Γ:

yeast, the PPI network of beer yeast (N = 2224, M = 6609)drugs, a social network of drug addicts (N = 616, M = 2012)pref2000, a synthetic scale-free graph constructed using thepreferential attachment scheme (N = 2000, M = 3974)

In each case we assume that all edges have unit length and we use n = 20interior discretization points per edge (h = 1

21).

For the potential we use V (s) = k(s− 12)2 and V (s) = k (const.) for

k = 0.1, 1, 10.51

PPI network of Saccharomyces cerevisiae (beer yeast)

52

Social network of injecting drug users in Colorado Springs

Figure courtesy of Ernesto Estrada.

53

Scale-free Barabási–Albert graph (pref)

54

Numerical experiments (cont.)

The sizes of the extended system Huh = gh and of the reduced systemSuv

h = ch are, respectively:

n = 134, 404, N = 2224 for yeast;n = 40, 856, N = 616 for drugs;n = 81, 480 and N = 2000 for pref2000.

The Schur complement can be formed efficiently since it is very sparse andwe know the location of the nonzero entries in advance.

Since the original graphs are very small, the Schur complement system isbest solved by sparse Cholesky factorization, but we also experiment withPCG. Without preconditioning, convergence can be slow.

For each problem we also need to solve M uncoupled tridiagonal systemsof order 20.

55


V (s) = k(s− 12)2

Problem k = 0.1 k = 1 k = 10It rel. error It rel. error It rel. error

yeast 13 3.3e-08 10 1.6e-08 8 4.8e-10drugs 10 1.4e-08 8 1.9e-08 6 4.4e-09

pref2000 9 2.9e-09 8 3.3e-10 6 1.4e-09

Results of running pcg (TOL =√eps) on Schur complement system,

diagonal preconditioner.

The “exact" solution is the one returned by backslash.

56


V (s) = k(s− 12)2

Problem k = 0.1 k = 1 k = 10It rel. error It rel. error It rel. error

yeast 7 5.9e-08 6 1.5e-09 5 1.3e-11drugs 6 3.9e-08 5 1.7e-09 4 1.2e-10

pref2000 5 4.5e-09 5 2.4e-11 4 3.1e-11


polynomial preconditioner.

57


V (s) = kProblem k = 0.1 k = 1 k = 10

It rel. error It rel. error It rel. erroryeast 36 7.7e-09 14 1.4e-08 5 6.2e-09drugs 42 6.9e-09 14 9.0e-09 5 4.9e-09

pref2000 23 3.5e-09 12 1.1e-08 5 2.5e-09


diagonal preconditioner.

58


V (s) = kProblem k = 0.1 k = 1 k = 10

It rel. error It rel. error It rel. erroryeast 20 4.0e-09 8 2.1e-08 4 5.6e-10drugs 24 4.6e-08 9 2.3e-09 4 4.3e-10

pref2000 13 9.2e-10 8 1.2e-09 4 2.2e-10


polynomial preconditioner.

59


Matrix S−1 for pref graph, k = 0.1.

60


Matrix S−1 for pref graph, k = 1.

61


Matrix S−1 for pref graph, k = 10.

62


N No prec. Diagonal Polynomial2000 83 23 135000 108 23 1310000 125 23 13

PCG iteration counts for pref graph, V (s) = k = 0.1, increasing N .

Note: Here h is constant, the size N of the graph Γ is increasing.The size of the extended graph G is n = 81, 480, n = 204, 360, andn = 409, 300, respectively.

63


h−1 No prec. Diagonal Polynomial21 83 23 1341 83 23 1381 82 22 12101 83 21 12

PCG iteration counts for pref graph, V (s) = k = 0.1, N = 2000.

Note: Here h is decreasing, the size N of Γ is fixed. The size of theextended graph G increases from 81,480 to 399,400 vertices.

With diagonal or polynomial preconditioning the solution algorithm isscalable with respect to both N and h for these graphs.

64

Outline

1 Motivation

2 Basic definitions


4 Discretization




65

The parabolic case

Among our goals is the analysis of diffusion phenomena on metric graphs.In this case, we assume that the functions we use also depend on a secondvariable t representing time, i.e.,

u(t, s) : [0, T ]× Γ −→ R.

A typical problem would be: given u0 ∈ H1(Γ) and a f ∈ L2([0, T ], L2(Γ)

)find u ∈ L2

([0, T ],H1(Γ)

)∩ C0

([0, T ];H1(Γ)

)such that

∂u

∂t− ∂2u

∂s2+mu = f on Γ

u(0, s) = u0,

where m ≥ 0.

Similarly, we can define the wave equation and the Schrödinger equationon Γ.

66

The parabolic case (cont.)

Space discretization using finite elements leads to the semi-discrete system

Mu̇h = Huh + fh, uh(0) = uh,0 ,

where uh = uh(t) is a vector function on the extended graph G, and themass matrix M and Hamiltonian H are as before.

A variety of methods are available for solving this linear system of ODEs:backward Euler, Crank-Nicolson, exponential integrators based on Krylovsubspace methods, etc.

Note that for large graphs and/or small h, this can be a huge system.

We have obtained some preliminary results using Stefan Güttel’s codefunm_kryl for evaluating the action of the matrix exponential on a vector.

67

Solution of diffusion problem on Γ for different times.

68

Solution of diffusion problem on Γ for different times.

68

Outline

1 Motivation

2 Basic definitions


4 Discretization




69

Summary

Quantum graphs bring together disparate areas: physics, graphtheory, PDEs, spectral theory, complex networks, finite elements,numerical linear algebra...On the surface just a huge set of “trivial" 1D problems, but thecomplexity of the underlying graph and the Neumann-Kirchhoffcoupling conditions make life interesting!Linear systems are huge but highly structured with much potential fororder reduction and parallelism.Challenges ahead include:

I Analyze PCG convergenceI Eigenvalue problemsI Hyperbolic problems (shocks)I Schrödinger, Dirac equations (important for graphene)I Non-self-adjoint and non-linear problems, non-local operators, etc.I Applications to real world problems

There is a lot of work to do in this area!70

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