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

Download Numerical Algorithms as Discrete-Time Control Systems .Numerical Algorithms as Discrete-Time Control

Post on 28-Jun-2018

213 views

Category:

Documents

1 download

Embed Size (px)

TRANSCRIPT

  • 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

Recommended

View more >