numerical schemes from the perspective of …bayen.eecs.berkeley.edu/sites/default/files/cd-rom...

PreliminariesTwo Motivating Problems

Illustrative Example: Heat EquationGeneralizationsGoing Forward

Numerical Schemes from the Perspective ofConsensus

Exploring Connections between Agreement Problems and PDEs

Yusef Sha�

Department of Electrical Engineering and Computer SciencesUniversity of California, Berkeley

April 27, 2010

Yusef Sha� Numerical Schemes from the Perspective of Consensus



Characterizing Numerical Schemes through Consensus

Given a general PDE, how can we analyze it from the p.o.v. ofLinear Systems Theory?

Particularly interested in parabolic and elliptic PDEs thatinvolve the Laplace operator

Auxx + Buxy + Cuyy + . . . = 0, where B2 − 4AC ≤ 0For example, heat equation: ut = auxx

Why the Laplace operator?

Rich theory of graphs utilizing discrete Lapace matrixConcise framework through which to understand manyproblems in discrete distributed control

Question: which classes of PDE control problems can be

studied as consensus problems and how can this

framework help us?




Outline of this talk

General overview of the objectives of the project

Mathematical Preliminaries

Two Motivating Problems

Consensus on NetworksControl of Spatially Varying Interconnected Systems

Illustrative example studying a PDE system as a consensusproblem

Broader discussion of the framework and potential futuredirections




Outline

1 Preliminaries

2 Two Motivating Problems

3 Illustrative Example: Heat Equation

4 Generalizations

5 Going Forward




Numerical Methods and Control

Numerical Analysis well-understood, lends itself to solution of�real� ODE/PDE systems

Simplest example: First-order Euler explicit forward di�erence(from Taylor series)

u = f (t, u)ut+h = ut + h(t, ut)




How Robust is the Scheme?

Stability: Backward or implicit di�erence schemes stable forany h

Sti� Systems[1]

Accuracy vs. e�ciency: take into account more terms, tightergranularity in step size

Choice may depend on operating constraints, e.g., online oro�ine




Numerical Schemes as Linear Systems

For systems, linear discretization results in an LTI system:

u = Su + Q, u, Q ∈ Rn, S ∈ Rn×n

Standard linear systems tests apply for stability.

See if S is Hurwitz, etc.




Some Working De�nitions for Graphs

De�nition

A vertex v is a point in n dimensional space.

De�nition

An edge is an ordered pair (v1, v2) where vi are vertices connectedby a segment.

De�nition

A graph G = G (V ,E ) is a collection of vertices V connected bythe set of edges E .




Graph Models[3]

Cycle Randomly Generated Graph




The Graph Laplacian

De�nition

The degree of a vertex is the number of edges of which it is amember.

De�nition

A graph Laplacian is a square matrix de�ned as follows[2]:

[L]ij =

di : i = j

−1 : ∃eij = (vi , vj) ∈ E

0 : otherwise(1)




Laplacian Spectral Analysis

Laplacian is symmetric positive semide�nite

At least one eigenvalue at zero, with corresponding eigenvector[1 . . . 1]T

Spectrum of Laplacian describes network dynamics

Fiedler eigenvalue[1]: smallest positive eigenvalue a measure ofconnectedness




The Agreement Problem

Given a collection of agents that can communicate

Characterize conditions required for agreement on a parameter

Describe convergence properties, e.g., asymptotic, exponential,. . .




Outline

1 Preliminaries



4 Generalizations

5 Going Forward




Consensus Problem on Networks

Olfati-Saber, et al., 2002-2007.

Given a collection of nodes, model their interaction by

x = −Lx , continuous case

x(k + 1) = (I − εL)x(k), discrete case

Theorem

For a continuous-time system, global exponential consensus is

reached with speed at least as fast as λ2(L). For a discrete-time

system, global exponential consensus is reached with speed at least

as fast as 1− ελ2(L), provided that ε < 1

dmax, where dmax is the

maximum degree of any vertex of L.




Consensus Problem on Networks, II

Fascinating emergent behavior: group of nodes behaves like acontinuum reaching a steady-state value

Applications[3]

Formation controlCoupled oscillatorsFlockingSmall world networksDistributed sensor fusion




Control of Spatially Varying Systems[3]

D'Andrea and Dullerud, 2002-2003.

Spatially varying system

De�nes necessary and su�cient LMI conditions for stabilization

Example application: discretization of the heat equation in twodimensions

Our goal: understand discretization from the perspective ofconsensus




Control of Spatially Varying Systems[4], II

Finite linear connection Finite planar connection1

1D'Andrea, Langbort, and Chandra, 2003.Yusef Sha� Numerical Schemes from the Perspective of Consensus



Outline

1 Preliminaries



4 Generalizations

5 Going Forward




Heat Equation on the Unit Circle

Heat equation on the circle[4] parametrized by x ∈ [0, 1) withθ = 2πx : ut = auxx

Separation of variables readily yields:

u(x , t) = (T0 ∗ Ht)(x)

=∞∑

n=−∞ane−4π2n2te2πjnx

Ht(x) is the heat kernel for the unit circle and an are theFourier coe�cients of T0

Eigenvalues given by 4π2n2




Spatial Representation[3]

Circle2 (1D Periodic BC)

2D'Andrea and Dullerud, 2003.Yusef Sha� Numerical Schemes from the Perspective of Consensus



Three point stencil

1D Laplacian: second-order centered di�erence scheme.

(ut)i =a

∆x2(ui+1 − 2ui + ui−1)

Resulting space discretization: �rst order linear ODE

Same dynamics as the consensus problem!




Laplacian Graph Structure and Interpretation

To impose periodic boundary conditions[2], note that u0 = uN . Thesystem representations are

−L =a

∆x2

−2 1 0 . . . 0 11 −2 1 0 . . . 0

0. . .

. . .. . .

. . ....

... 0. . .

. . .. . . 0

0...

. . . 1 −2 11 0 . . . 0 1 −2

u = −Lu

u(k + 1) = (I − εL)u(k).




Spectral Analysis

Using discrete Fourier transform, with the eigenfunctionsf (j)(x) = e Ikj i∆x , we get

λ(φm) =2a

∆x2(cosφm − 1) ,

with φm = Ikm∆x = 2mπN

, m = 0, . . .N − 1. So the eigenvaluesrange in value from − 4a

∆x2to zero. Let a = 1 and ∆x = 1

N. Then:

λ(L) =2

∆x2

(1− cos

2mπ

N

)≈ 4m2π2.




Spectral Analysis, II

Eigenvalues of discretization converge to eigenvalues ofcontinuous operator.

λ2(L) ≈ 4π2

Space-discretized ODE will reach globally exponentialconsensus value with a speed at least as great as 4π2.

Standard ODE time integration methods chosen based onlocation of eigenvalues

Key idea: The original PDE system, discretized, is

exactly the consensus problem for a cycle topology.




Outline

1 Preliminaries



4 Generalizations

5 Going Forward




Di�erent Schemes

Use di�erent discretizations

5, 7, 9 point stencils

Algebraic connectivity improvesComputational cost increasesFor a large grid, stencil size will be less important

Modi�ed 3 point stencil: look two steps behind and ahead

Doesn't have much e�ect on connectivity




Extensions to higher dimensions

Same idea applicable to higher spatial order Laplacian terms

ut = a∆u

Apply higher order spatial discretizations

2D case: typical choice 5 or 9 point scheme

(ut)i, j = a∆x2

(ui+1, j + ui−1, j + ui, j+1 + ui, j−1 − 4ui, j)




Visual Representation[3]

Torus3 (2D Periodic BC)

3D'Andrea and Dullerud, 2003.Yusef Sha� Numerical Schemes from the Perspective of Consensus



Stencil Grids

One spatial dimension Two spatial dimensions




Classes of Systems Amenable to Consensus Graph Analysis

Periodic Boundary Conditions

Neumann Boundary Conditions

Can control through the boundaries

Control terms can also be applied at each actuator: turns outthat if there are at least two independent inputs (outputs), LTIsystem is controllable (observable).

General LTI form:u = −Lu + Gr




Outline

1 Preliminaries



4 Generalizations

5 Going Forward




Horizons

Other Boundary Conditions: cannot be directly represented asgraph Laplacian

Potential future work in this direction: how can these systemsbe understood as consensus problems?Can use linear state feedback to get to Laplacian form, butmay not be realistic

Designing better interconnection structures through graphtheory

Boyd et al.: Convex optimization to design fastest mixingMarkov chians, minimize resistance in a network, etc.




Summary

Many second-order PDEs can be analyzed in a consensusframework

Consensus a useful paradigm for distributed control

Example: Heat Equation

Laplacian graph theory and consensus:

Facilitate better numerical schemesAid in understanding complex continuum dynamics




Additional Reading

R. L. Burden and J. D. Faires. Numerical Analysis, 2005.

F. Chung, Spectral Graph Theory, 1997

R. D'Andrea and G. Dullerud. "Distributed Control Design forSpatially Interconnected Systems," 2003.

R. D'Andrea, C. Langbort, and R. Chandra. "A State SpaceApproach to Control of Interconnected Systems," 2003.




Additional Reading, II

M. Fiedler. "Algebraic Connectivity of Graphs," 1973.

C. Hirsch. Computational Fluid Dynamics, 2007.

R. Olfati-Saber, et al. "Consensus and Cooperation inMultiagent Systems," 2007.

E. M. Stein and R. Shakarchi. Introduction to Fourier Analysis,2003.




Questions

Thank You

Any Questions?


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