# numerical algorithms as discrete-time control systems .numerical algorithms as discrete-time control

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Numerical Algorithms as Discrete-Time ControlSystems

Jens Jordan

University of Wurzburg,Universite de Liege

Dynamical systems and Computation Day p.1/23

Numerical algorithms as discrete-time control systems

Contents:

Dynamical systems and Computation Day p.2/23

Numerical algorithms as discrete-time control systems

Contents:

Introduction

Dynamical systems and Computation Day p.2/23

Numerical algorithms as discrete-time control systems

Contents:

Introduction

Examples

Dynamical systems and Computation Day p.2/23

Numerical algorithms as discrete-time control systems

Contents:

Introduction

Examples

Reachable Sets

Dynamical systems and Computation Day p.2/23

Numerical algorithms as discrete-time control systems

Contents:

Introduction

Examples

Reachable Sets

Controllability

Dynamical systems and Computation Day p.2/23

Numerical algorithms as discrete-time control systems

Contents:

Introduction

Examples

Reachable Sets

Controllability

Current and Future Work

Dynamical systems and Computation Day p.2/23

Introduction

Iterative Algorithms : M M iteration map

xt M, xt xt+1 M

Shifted Iterative Algorithms : MU M iteration map, U set of shift parameters

xt M, ut U (xt, ut) xt+1 M

Dynamical systems and Computation Day p.3/23

Introduction

Iterative Algorithms : M M iteration map

xt M, xt xt+1 M

Shifted Iterative Algorithms : MU M iteration map, U set of shift parameters

xt M, ut U (xt, ut) xt+1 M

Dynamical systems and Computation Day p.3/23

Introduction

Iterative Algorithms : M M iteration map

xt M, xt xt+1 M

Shifted Iterative Algorithms : MU M iteration map, U set of shift parameters

xt M, ut U (xt, ut) xt+1 M

Example: Shifted Inverse Iteration on Sn1.A Rnn, M = Sn1, U = R \ (A).

Iteration Step

: MU M, (x, u) =(A uI)1x

(A uI)1x

Dynamical systems and Computation Day p.3/23

Introduction

Iterative Algorithms : M M iteration map

xt M, xt xt+1 M

Shifted Iterative Algorithms : MU M iteration map, U set of shift parameters

xt M, ut U (xt, ut) xt+1 M

Observation: (M,U , ) describes a discrete-time control System:

x0 M

xt+1 = (xt, ut)

Dynamical systems and Computation Day p.3/23

Introduction

Motivation:

Dynamical systems and Computation Day p.4/23

Introduction

Motivation:

Control theoretical description of numerical algorithms

Dynamical systems and Computation Day p.4/23

Introduction

Motivation:

Control theoretical description of numerical algorithms

Better understanding of algorithms

Dynamical systems and Computation Day p.4/23

Introduction

Motivation:

Control theoretical description of numerical algorithms

Better understanding of algorithms

Optimization of shift strategies using control theoretical tools

Dynamical systems and Computation Day p.4/23

Introduction

Motivation:

Control theoretical description of numerical algorithms

Better understanding of algorithms

Optimization of shift strategies using control theoretical tools

Justify existing shift strategies

Dynamical systems and Computation Day p.4/23

Introduction

Motivation:

Control theoretical description of numerical algorithms

Better understanding of algorithms

Optimization of shift strategies using control theoretical tools

Justify existing shift strategies

Creation of new algorithms

Dynamical systems and Computation Day p.4/23

Examples

Example I: Shifted Inverse Iteration on Projective Spaces

A Fnn

M = FPn1 = {Set of one dimensional subspaces of Fn}

U = F \ (A)

: MU M defined by

(X , u) = (A uI)1X

Questions and remarks:

Can we find feedback controls to reach eigenvectors by arbitrary

initial points?

Can we find feedback controls to reach specific eigenvectors by

arbitrary initial points?

Dynamical systems and Computation Day p.5/23

Examples

Example I: Shifted Inverse Iteration on Projective Spaces

A Fnn

M = FPn1 = {Set of one dimensional subspaces of Fn}

U = F \ (A)

: MU M defined by

(X , u) = (A uI)1X

Questions and remarks:

Can we find feedback controls to reach eigenvectors by arbitrary

initial points?

Can we find feedback controls to reach specific eigenvectors by

arbitrary initial points?

Dynamical systems and Computation Day p.5/23

Examples

Example II: Shifted Inverse Iteration on Grassmann Manifolds

A Rnn

M = Grass(p, n) = {Set of p-dimensional subspaces of Rn}

U set of feedback maps F : ST(p, n) Rpp

U = {X ST(p, n), M GLp(R) : F (XM) = M1F (X)M}

: MU M defined by procedure

1) Choose X ST(p, n) such that X = X

2) Solve AX+ X+F (X) = X

3) (X , F ) := X+ := X+

Dynamical systems and Computation Day p.6/23

Examples

Example II: Shifted Inverse Iteration on Grassmann Manifolds

Questions and remarks

The choice F = R with R(X) = (XT X)1XT AX leads to

Grassmann Rayleigh Quotient Iteration (Absil, Mahony, Sepulchre,

Van Dooren, 2002).

Does the Grassmann Rayleigh Quotient Iteration has global

convergence properties?

Can we find feedback laws for global convergence?

Can we find feedback laws for global convergence to a specific

eigenspace?

Dynamical systems and Computation Day p.7/23

Examples

Example III: Shifted Inverse Iteration on Flag Manifolds

A Fnn

M = Flag(Fn) = {V = (V1, . . . , Vn) |Vi Vi+1, dimF Vi = i}

U = F \ (A)

: MU M defined by

(V, u) =(

(A uI)1V1, . . . , (A uI)1Vn

)

Questions and Remarks

Algorithm is closely related to the QR algorithm (on isospectral

manifolds).

Can we steer to arbitrary eigenflags?

Dynamical systems and Computation Day p.8/23

Examples

Example III: Shifted Inverse Iteration on Flag Manifolds

A Fnn

M = Flag(Fn) = {V = (V1, . . . , Vn) |Vi Vi+1, dimF Vi = i}

U = F \ (A)

: MU M defined by

(V, u) =(

(A uI)1V1, . . . , (A uI)1Vn

)

Questions and Remarks

Algorithm is closely related to the QR algorithm (on isospectral

manifolds).

Can we steer to arbitrary eigenflags?

Dynamical systems and Computation Day p.8/23

Examples

Example IV: Shifted QR Algorithm on Isospectral Manifolds

A Cnn

M = {QAQ |Q Un(C)}

U = C \ (A)

: MU M defined by

(X, u) = (X uI)Un(C)(X uI)(X uI)Un(C)

where (X uI)Un(C) is the unitary factor of the QR decomposition

of (X uI).

Dynamical systems and Computation Day p.9/23

Examples

Example V: Shifted Inverse Iteration on Hessenberg Manifolds

A Fnn regular

M = HessA(Fn) = {V = (V1, . . . , Vn) Flag(F

n) |AVi Vi+1}

U = F \ (A)

: MU M defined by

(V, u) =(

(A uI)1V1, . . . , (A uI)1Vn

)

Questions and Remarks

Algorithm is closely related to the QR algorithm on Hessenberg

Matrices.

Can we steer to arbitrary Hessenberg eigenflags?

Dynamical systems and Computation Day p.10/23

Examples

Example V: Shifted Inverse Iteration on Hessenberg Manifolds

A Fnn regular

M = HessA(Fn) = {V = (V1, . . . , Vn) Flag(F

n) |AVi Vi+1}

U = F \ (A)

: MU M defined by

(V, u) =(

(A uI)1V1, . . . , (A uI)1Vn

)

Questions and Remarks

Algorithm is closely related to the QR algorithm on Hessenberg

Matrices.

Can we steer to arbitrary Hessenberg eigenflags?

Dynamical systems and Computation Day p.10/23

Examples

Example VI: Restart-Shifts of Krylov Methods

M = Grass(p, n)

U = Rk[t]

x0 Rn \ {0}, K(x0) = x0, Ax0, . . . , A

p1x0 M

: MU M defined by

(K(x), u) = u(A)K(x) = K(u(A)x)

Questions and Remarks

Find shifts to approximate specific eigenspaces (Beattie, Embree,

Sorensen, Rossi).

Dynamical systems and Computation Day p.11/23

Examples

Example VI: Restart-Shifts of Krylov Methods

M = Grass(p, n)

U = Rk[t]

x0 Rn \ {0}, K(x0) = x0, Ax0, . . . , A

p1x0 M

: MU M defined by

(K(x), u) = u(A)K(x) = K(u(A)x)

Questions and Remarks

Find shifts to approximate specific eigenspaces (Beattie, Embree,

Sorensen, Rossi).

Dynamical systems and Computation Day p.11/23

Reachable Sets

System = (M,U , ); x0.

Definition: Reachable set of x0 M

R(x0) := {x wich can be reached from x0 in finite many steps}

Definition: k N, k : MUk M

1(x, u) = (x, u)

k(x, u1, . . . , uk) = ((x, u1, . . . , uk1), uk)

Proposition:

R(x0) := {x M|N N, u UN : x = N (x0, u)}

Dynamical systems and Computation Day p.12/23

Reachable Sets

System = (M,U , ); x0.

Definition: Reachable set of x0 M

R(x0) := {x wich can be reached from x0 in finite many steps}

Definition: k N, k : MUk M

1(x, u) = (x, u)

k(x, u1, . . . , uk) = ((x, u1, . . . , uk1), uk)

Proposition:

R(x0) := {x M|N N, u UN : x = N (x0