COMPLICACIONES DE LA CATETERIZACIÓN VENOSA CENTRAL 42

We now combine the MM algorithm described in Section 3.2 with the Golub-Kahan lower bidiagonalization in Section 3.3. We describe the application of the modulus-based iterative method to Tikhonov regularization with nonnegativity constraint (3.1). We discuss how the computational effort for large-scale problems can be reduced by using a Krylov subspace method with a fixed Krylov subspace. Finally, we comment on how to exploit the BCCB structure of A in image deblurring applications.

Application of ℓ ≪ min{m, n} steps of Golub-Kahan lower bidiagonalization, i.e., of Algo- rithm 3.3, to A with initial vector u0 = bδ/kbδk gives the decompositions

AVℓ = Uℓ+1Bℓ+1,ℓ, AtUℓ = VℓBℓ,ℓt , (3.15) where Uℓ+1 = [u1, u2, . . . , uℓ+1] ∈ Rm×(ℓ+1) and Vℓ = [v1, v2, . . . , vℓ]∈ Rn×ℓhave orthonor- mal columns, Uℓ ∈ Rm×ℓ is made up of the first ℓ columns of Uℓ+1, Bℓ+1,ℓ ∈ R(ℓ+1)×ℓ is lower bidiagonal with positive diagonal and subdiagonal entries, and Bℓ,ℓis the leading ℓ×ℓ submatrix of Bℓ+1,ℓ.

3.4. Krylov subspace methods for nonnegative Tikhonov regularization 23

We assume for now that ℓ is chosen small enough so that the decompositions (3.15) with the stated properties exist. As stated in the previous Section we note that the columns of Vℓspan the Krylov subspace Kℓ(AtA, Atbδ).

We first rewrite the minimization problem (3.1)

min x≥0  Ax − bδ 2+ α_kxk2  = min x≥0  A √ αIn  x₋  bδ 0  2 = min x≥0 ˜Ax_{− ˜b}δ 2, (3.16)

where we assume that α > 0. Then the matrix ˜A ∈ R(m+n)×n _{is of full column rank and} the minimization problem (3.16) satisfies the conditions in Section 3.2. Therefore, the iterates determined by Algorithm 3.1 will converge.

When the matrix A is large and without exploitable structure, the computations with Algo- rithm 3.1 with D = µInmay be expensive. In particular, factoring the matrix µIn+ ˜AtA˜in order to solve the linear systems of equations with this matrix required by Algorithm 3.1 may be unattractive or infeasible. We are interested in trying to reduce the computational effort required for solving these linear systems of equations. One way to achieve this is to solve them by the conjugate gradient method. It is convenient to use the CGLS implementation [16]. This solution approach is discussed in [122] and also illustrated in Section 3.5.

We now describe an alternative way to reduce the computational effort. We first determine an initial reduction of A to a small bidiagonal matrix with the aid of Golub-Kahan lower bidiagonalization. The Krylov solution subspace generated by this reduction method then is reused for all linear systems of equations with the matrix µIn+ ˜AtA˜that have to be solved. Substituting x = Vℓy, y ∈ Rℓ, into (2.10) and determining an approximate solution by a Galerkin method gives the equation

V_ℓt(AtA + αIn)Vℓy= VℓtAtbδ,

which, with the aid of the decompositions (3.15), can be expressed as

(B_ℓ+1,ℓt B_ℓ+1,ℓ+ αI_ℓ)y =

e

1

Atbδ _. (3.17)

Here and below

e

_jdenotes the jth column of an identity matrix of appropriate order. The re- duced Tikhonov equations (3.17) are the normal equations associated with the least-squares problem min y∈Rℓ  B_√ℓ+1,ℓ αIℓ  y₋√α

e

_{ℓ+2 A}tbδ . (3.18)

We solve the latter instead of (3.17) for y = yα for reasons of numerical stability. For each fixed α > 0 the least-squares problem (3.18) can be solved in only O(ℓ) arithmetic floating- point operations [60] for details on the solution of least-squares problems of the form (3.18). We turn to the determination of α > 0 by the discrepancy principle. Substituting xα = Vℓy into (2.12) and using (3.15) gives the reduced problem

where y solves (3.18).

Proposition 3.5. Introduce the function

φℓ(α) =kbδk2

e

t1(α−1Bℓ+1,ℓBℓ+1,ℓt + Iℓ+1)−2

e

1. (3.20) Then the solution α > 0 of

φℓ(α) = τ2δ2 (3.21)

determines a solution y = yαof (3.18) that solves (3.19). The vector xα,ℓ= Vℓyαsatisfies(2.12).

Proof. It follows from (3.15) that

Atbδ= AtUℓ

e

11kbδk = v1

e

t1Bℓ+1,ℓt

e

1kbδk.

Substituting this expression and the solution of (3.17) into the left-hand side of (3.19) gives kBℓ+1,ℓ(Bℓ+1,ℓt Bℓ+1,ℓ+ αIℓ)−1

e

1kAtbδk −

e

1kbδk k2

= _kBℓ+1,ℓ(Bℓ+1,ℓt Bℓ+1,ℓ+ αIℓ)−1Bℓ+1,ℓt

e

1−

e

1k2kbδk2. (3.22) The identity

Bℓ+1,ℓ(Bℓ+1,ℓt Bℓ+1,ℓ+ αIℓ)−1Bℓ+1,ℓt − Iℓ+1 =−(α−1Bℓ+1,ℓBℓ+1,ℓt + Iℓ+1)−1

can be shown, e.g., by multiplication by Bℓ+1,ℓB_ℓ+1,ℓt + αIℓ+1 from the right-hand side. Sub- stitution into (3.22) gives

kBℓ+1,ℓyα−

e

1kbδk k2 =kbδk2

e

t1(α−1Bℓ+1,ℓBℓ+1,ℓt + Iℓ+1)−2

e

1,

This shows (3.20). The fact that the vector xµ,ℓsatisfies (2.12) follows from (3.15) and (3.19).

Proposition 3.6. Let φℓ(α) be defined by(3.20). Then the function ν → φℓ(1/ν) is strictly decreas- ing and convex for ν > 0. Moreover,

lim

α→∞φℓ(α) =kb δ_k2_.

In particular, Newton’s method applied to the solution of the equation φℓ(1/ν) = τ2δ2 with initial approximate solution ν0to the left of the solution converges monotonically and quadratically.

Proof. The decrease, convexity, and limit follows from the representation

φℓ(1/ν) =kbδk2

e

t1(νBℓ+1,ℓBℓ+1,ℓt + Iℓ+1)−2

e

1.

Newton’s method converges monotonically and quadratically for decreasing convex func- tions when the initial iterate is smaller than the solution. The initial iterate ν0 can be chosen to be zero with lim νց0φℓ(1/ν) =kb δ_k2_{, lim} νց0 d dνφℓ(1/ν) =−2kb δ_k2_kBt ℓ+1,ℓ

e

1k2.

In actual computations it typically suffices to choose ℓ ≪ min{m, n}. We apply MM Algo- rithm 3.1 to the reduced Tikhonov minimization problem (3.17). Thus, we replace At_{Ain the}

3.4. Krylov subspace methods for nonnegative Tikhonov regularization 25

algorithm by

Tℓ,α= Bℓ+1,ℓt Bℓ+1,ℓ+ αIℓ. (3.23) Since the matrix Bℓ+1,ℓis small, we can easily determine its largest singular value σmax. Typ- ically, zero is a quite sharp lower bound for the smallest singular value. The largest eigen- value of Tℓ,αis σ2max+ µand the smallest eigenvalue is bounded below by, and is generally close to, α. Hence, we will use the relaxation parameter

µ =p(σ2

max+ α)α (3.24)

for the algorithm; cf. (3.8). This yields the following scheme.

Algorithm 3.4(Krylov subspace Modulus Method). Choose the number of Golub-Kahan lower

bidiagonal steps, ℓ, and compute the decompositions(3.15). Determine a regularization parameter α that satisfies(3.21) as described in Proposition 3.6. Compute the solution y = yαof (3.18) and define the initial approximate solution x0 = PΩ(Vℓyα) of (3.1). Determine the largest singular value of the matrix Bℓ+1,ℓand define the relaxation parameter(3.24). Let Tℓ,αbe given by(3.23).

ˆ

bδ=

e

1 Atbδ y0 = Vℓtx0 ˜

y0 = Vℓt|Vℓy0|

for k = 0, 1, 2, . . . until convergence

y_k+1= (µI_ℓ+ T_ℓ,α)−1(µI_{ℓ− Tℓ,α})˜y_k+ ˆbδ ˜

yk+1= V_ℓt|Vℓyk+1|

end

x= Vℓy˜k+1+|Vℓy˜k+1|

The above algorithm computes the magnitude of every entry of an n-vector at each step. Therefore, a transformation from the ℓ-dimensional subspace, where the vectors yk live, to Rn_{is required. Every step demands the solution of a linear system of equations of the form}

(µI_ℓ+ T_ℓ,α) z = d

for some vector d. The solution z can be computed by solving a least-squares problem anal- ogous to (3.18).

We remark that the Krylov subspace K(At_{A, A}t_bδ_{)is invariant under shifts of A}t_{Aby a mul-} tiple of the identity, i.e.,

Kℓ(AtA, Atbδ) =Kℓ(AtA + µIn, Atbδ).

It follows that the shifted matrix µIℓ+ Tℓ,αin Algorithm 3.4 corresponds to the shifted matrix µIn+ AtA + αIn
.

We described above how to determine the regularization parameter α by first reducing equa- tion (2.12) to an equation with a small matrix (3.19). When restoring images, we sometimes may impose periodic boundary conditions without affecting the quality of the computed restoration significantly. This yields a BCCB blurring matrix A ∈ Rn×n_{, which can be diago-} nalized by the unitary Fourier matrix F defined in (2.7),

Here the matrix Σ is diagonal, possibly with complex entries, and the superscript ∗ _de- notes transposition and complex conjugation; see, e.g., [84] for details. We can transform the Tikhonov minimization problem (2.10) to a minimization problem with a diagonal matrix, and we also can transform (2.12) to an equation with a diagonal matrix. These transforma- tions allow easy computation of the regularization parameter α > 0 such that the solution (2.10) satisfies (2.12) by Newton’s method analogously as described above. Moreover, Algo- rithm 3.1 with D = µIncan be executed efficiently when A has the factorization (3.25). This is illustrated in the following section.

In document Monitorización hemodinámica intraoperatoria: Estudio comparativo entre la utilización en el mantenimiento anestésico del Eco-Doppler esofágico y la presión venosa central como guía para reposición de fluidos en pacientes sometidos a Cirugía Mayor (página 45-54)