Extended Krylov subspace methods for large-scale differential Sylvester matrix equations

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Extended Krylov subspace methods for large-scale differential Sylvester matrix equations

Abdeslem Hafid Bentbib, El Mostafa Sadek, Lakhlifa Sadek, Hamad Alaoui Talibi

To cite this version:

Abdeslem Hafid Bentbib, El Mostafa Sadek, Lakhlifa Sadek, Hamad Alaoui Talibi. Extended Krylov

subspace methods for large-scale differential Sylvester matrix equations. 2018. �hal-01964512�

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(will be inserted by the editor)

Extended Krylov subspace methods for large-scale differential Sylvester matrix equations

Abdeslem Hafid Bentbib · El Mostafa Sadek · Lakhlifa Sadek · Hamad Talibi Alaoui

Received: date / Accepted: date

Abstract In this paper, we present a new numerical methods for solving large-scale differential Sylvester matrix equations with low rank right hand sides. These differential matrix equations appear in many applications such as robust control problems, model reduction problems and others. We present two approaches based on extended global Arnoldi process the problem. The first one is based on the expression of the exact solution and a global extended Krylov method for the computation of the exponential of a matrix. The second one is based on low-rank approximate solutions and extended global Arnoldi algorithm. We give some theoretical results and report some numerical experiments to show the effectiveness of the proposed methods.

Keywords Extended global Krylov subspaces, Matrix exponential, low-rank approximation, Differential Sylvester equations.

Mathematics Subject Classification (2010) 65F·15A

Abdeslem Hafid Bentbib

Facult´e des Sciences et Techniques-Gueliz,

Laboratoire de Math´ematiques Appliqu´ees et Informatique, Morocco. E-mail: [email protected]

El Mostafa Sadek (corresponding author) Laboratory of Engineering Sciences for Energy, National School of Applied Sciences of El Jadida, University Chouaib Doukkali, Morocco.

E-mail: [email protected] Lakhlifa Sadek

Facult´e des Sciences d’El Jadida,

Morocco. E-mail: [email protected] Hamad Talibi Alaoui

Facult´e des Sciences d’El Jadida, Morocco. E-mail: talibi [email protected]

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1 Introduction

In this paper, we consider the differential Sylvester matrix equation (DSE in short) of the form X˙(t) =AX(t) +X(t)B+EF^T, t∈[t0,Tf]

X(t0) =X0 (1)

whereA∈R^n×n,B∈R^p×p, andE∈R^n×s,F∈R^p×s, are full rank matrices, withsn,p.

The initial condition is given in a factored form asX0=Z0Z˜₀^Tand the matricesAandBare assumed to be large and sparse.

Differential Sylvester matrix equations play a fundamental role in many problems in control, filter design theory, model reduction problems, differential equations and robust control problems; see, e.g.,[1, 10, 16] and the references therein.

The exact solution of the differential Sylvester matrix equation (1) is given by the following result.

Theorem 1 [1] The unique solution of the differential Sylvester equations (1) is defined by X(t) =e^(t−t⁰^)AX0e^(t−t⁰^)B+

Zt t₀

e^(t−τ)AEF^Te^(t−τ)Bdτ. (2)

For small or medium-sized differential Sylvester matrix equations, there are several methods to solve this equation, for example Backward Differentiation Formula (BDF) and Rosenbrock method [9, 25].

During the last years, several projection methods based on Krylov subspaces have been proposed for solving large matrix equation, see, e.g.,[2, 5, 6, 15–17, 28]. The main idea em- ployed in these methods is to use a extended Krylov subspace and then apply the Galerkin- type orthogonality condition. In [16], Jbilou and Hached are presented two approaches to solving the large differential Sylvester matrix equation, by using the block extended Krylov subspaces. We use in this work, the extended global Krylov subspaces for solving large differential Sylvester matrix equations.

The rest of the paper is organized as follows. In the next section, we give a new expression of the unique solutionX(t)of the differential Sylvester matrix equation (1). This expression based on the global Arnoldi process. In section 3, we recall the extended global Arnoldi algorithm with some of its properties. In Section 4, we define EGA-exp method by using the matrix exponential approximation by extended global Krylov subspaces and quadrature method for computing numerical solution of large-scale differential Sylvester matrix equations. In Section 5, we present a low-rank solution method for solving the problem (1) by using projections onto extended global Krylov subspacesKm^g(A,E)andKm^g(B^T,F). Fi- nally, Section 6 is devoted to numerical experiments.

Throughout the paper, we use the following notations, The Frobenius inner product of the matricesXandY is defined byhX,Yi_F=tr(X^TY), wheretr(Z)denotes the trace of a square matrixZ. The associated norm is the Frobenius norm denoted bykk_F. The Kronecker productA⊗B= [ai,jB]whereA= [ai,j]. This product satisfies the properties:(A⊗B)(C⊗ D) = (AC⊗BD)and(A⊗B)^T =A^T⊗B^T. We also using the matrix productdefined in [8]. The following proposition gives some properties satisfied by the this product.

Proposition 1 Let A,B,C∈R^n×ps, D∈R^n×n, L∈R^p×pandα∈R. Then we have, 1. (A+B)^TC=A^TC+B^TC.

2. A^T(B+C) =A^TB+A^TC.

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3. (αA)^TC=α(A^TC).

4. (A^TB)T=B^TA.

5. A^T(B(L⊗I_s)) = (A^TB)L.

A blockVm= [V1,V2, ...,Vm]is to beF-orthonormal matrix ifVm^TVm=Im. We also have the following result

Lemma 1 ([21])LetVm= [V1,V2, ...,Vm]is be an n×ms F-orthonormal matrix, let Z and Y be a matrices ofR^m×sandR^ms×q. Then we have

kVm(Z⊗I_s)k_F=kZk_F and kVmYk_F≤ kYk_F.

2 Expression of the exact solution of the differential Sylvester equation

In this section we will give a new expression of the unique solutionX(t)of the differential Sylvester matrix equation (1). This expression based on the global Arnoldi process andPA

be the minimal polynomial ofAforEof degreeqandP_BT be the minimal polynomial ofB^T forFof degreeq⁰.

The modified global Arnoldi process constructs anF-orthonormal basisV1,V2,· · ·,Vm

of the matrix Krylov subspace

Km(A,V) =span

V,AV,A²V, ...,A^m−1V . The modified global Arnoldi process is described as follows:

Algorithm 1The modified global Arnoldi process(MGA)

Inputs:A∈R^n×n,V∈R^n×sandman integer.

1. SetV₁=_kV^V_k

F; 2. Forj=1, ...,m 3. Ve=AV_j; 4. Fori=1, ...,j 5. hi,j=hVi,VeiF; 6. Ve=Ve−h_i,_jV_i; 7. end for(i).

8. Computehj+1,j=keVkF; 9. V_j+1=Ve/hj+1,j; 10. end for(j).

LetVm= [V1,V2, ...,Vm]andHe_mÂis the(m+1)×mupper Hessenberg matrix whose entries hi,j are defined by Algorithm 1 and H_mÂ is the m×m matrix obtained fromHe_mÂ by deleting its last row. We have the following relation

AVm=Vm(H_m^A⊗Is) +hm+1,mVm+1(e^T_m⊗Is), wheree^T_m= [0,0,· · ·,0,1].

The following result shows that the solutionXof (1) can be expressed in terms of the global Arnoldi basis.

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Theorem 2 Let q and q⁰be the degrees of the minimal polynomials of A for E and B^T for F respectively. LetVq= [V1,V2, ...,Vq]andWq⁰ =

W1,W2, ...,W_q0

are the F-orthonormal matrices constructed by applying simultaneously q and q⁰steps of the global Arnoldi algorithm to the pairs(A,E)and(B^T,F)respectively. Then the unique solution X of (1) can be expressed as:

X(t) =Vq(Y_qq⁰(t)⊗Is)Wq^T⁰. (3) where Y_qq0(t)is the solution of the low-order differential Sylvester equation:

Y˙_m(t)−H_q^AY_m(t)−Y_m(t) H_q^B0

T

−Ee_qFe_q^T0 =0.

whereEe_q=kEk_Fe^(q)₁ andFe_q0=kFk_Fe^(q₁⁰⁾.

Proof LetZbe the matrix defined byVq(Y_qq0(t)⊗Is)W_q^T0. Then we have,

Z(t)−˙ AZ(t)−Z(t)B−EF^T =Vq(Z(t)˙ ⊗I_s)W_q^T⁰ −AVq(Z(t)⊗I_s)W_q^T⁰ −Vq(Z(t)⊗I_s)W_q^T⁰B−EF^T.

=Vq

Y˙_qq0(t)−H_q^AY_qq0(t)−Y_qq0(t)

H_q^B0

T

−Ee_qFe_q^T0

⊗I_s

W_q^T⁰. WhenY_qq0the solution of the lower order differential Sylvester equation ˙Y_qq0(t)−T^q,AY_qq0(t)−

Y_qq⁰(t) H_q^B0

T

−EeqFe_q^T0 =0, we get

Z(t)−˙ AZ(t)−Z(t)B−EF^T=0.

Therefore, using the fact that the solution of (1) is unique, it follows thatX(t) =Vq(Y_qq⁰(t)⊗

Is)W_q^T⁰.

3 The extended global Arnoldi process

In this section, we recall the extended global Krylov subspace and extended global Arnoldi process. LetV be a matrix of dimensionn×s. Then the extended global Krylov subspace associated to(A,V)is given by

Km^g(A,V) =span

V,A⁻¹V,AV,A⁻²V,A²V, ...,A^m−1V,A^−mV (4) The extended global Arnoldi process constructs anF-orthonormal basis{V1,V2, ...,Vm} of the extended global Krylov subspaceKm^g(A,V)([17]). The algorithm is summarized as follows

Algorithm 2The extended global Arnoldi process(EGA)

Inputs:A∈R^n×n,V∈R^n×sandman integer.

1. Compute the globalQR[V,A⁻¹V], i.e ,[V,A⁻¹V] =V₁(Ω⊗I_s);

2. SetV0= [ ];

3. Forj=1, ...,m

4. SetV_j⁽¹⁾=V_j(:,1 :s);V_j⁽²⁾=V_j(:,s+1 : 2s);

5. Vj= [V^j−1,Vj];U= [AV⁽¹⁾_j ,A⁻¹V_j⁽²⁾];

6. Fori=1, ...,j 7. Hi,j=V_i^TU;

8. U=U−Vi(Hi,j⊗Is);

9. end for(i).

10. Compute the global QR decomposition ofU, i.e.,U=V_j+1(Hj+1,j⊗I_s);

11. end for(j).

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LetV^m= [V1,V2, . . . ,Vm]withVi∈R^n×2sand 2m×2mupper block Hessenberg matrix Tm,A=V^Tm(AVm).

We have the following relation

AVm=Vm+1(Tm,A⊗Is)

=Vm(Tm,A⊗I_s) +Vm+1(T_m+1,m^A E_m^T⊗I_s),

whereT^m,A=V^Tm+1(AV^m), andE_m^T= [0_2×2(m−1),I2]is the matrix formed with the last 2 columns of the 2m×2midentity matrixI2m.

4 Using the matrix exponential approximation and quadrature method

In this section, we show the approximation of the solution of the large differential Sylvester matrix equations (1), based on matrix exponential approximation with extended global Krylov subspace and quadrature method.

That the exact solution to (1) is given by X(t) =e^(t−t⁰^)AX0e^(t−t⁰^)B+

Zt t₀

e^(t−τ)AEF^Te^(t−τ)Bdτ. (5)

For obtain on approximate solution of (1), we use the approximate ofe^(t−τ)AEande^(t−τ)B^TF bay extended global Krylov subspaces, and finally, we find the approximate solution ofX(t) by the quadrature method.

In what follows, we consider projections onto extended global Krylov subspaces. Let Vm= [V1, ...,Vm]andWm= [W1, ...,Wm]be the F-orthonormal matrices whose columns form an F-orthonormal basis of the subspaceKm^g(A,E)andKm^g(B^T,F), respectively.

Following [29, 26, 27], an approximation toZ_A=e^(t−τ)AEcan be obtained as Z_m,A=Vm

e^(t−τ)^T^m,AVmE⊗I_s

, whereT^m,A=V^Tm(AV^m).

In the same way, an approximation toZ_B=e^(t−τ)B^TF, is given by Zm,B=Wm(e^(t−τ)^T^m,BWmF⊗I_s), whereT^m,B=W^Tm(B^TW^m).

Then, you have

e^(t−τ)AEF^Te^(t−τ)B≈Z_m,A(t)Z_m,B^T (t). (6) Assuming thatX(0) =0, therefore the approximation solution of the differential Sylvester equation (1) as follows

X_m(t) =Vm(Xm(t)⊗I_s)W^Tm, (7) where

X^m(t) = Zt

t₀X^m,A(τ)X^Tm,B(τ)dτ, (8)

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with

Xm,A(τ) =e^(t−τ)^T^m,AV^TmE X^m,B(τ) =e^(t−τ)^T^m,BW^TmF

The next result shows that the 2m×2mmatrix functionX^m(t)is solution of a low-order differential Sylvester matrix equation.

Theorem 3 Let X_m(t)be the matrix function defined by (8), then itXm(t)satisfies the following low-order differential Sylvester matrix equation

X˙m(t) =Tm,AXm(t) +Xm(t)T^Tm,B+Ee_mFe_m^T, t∈[t0,T_f], (9) whereEem=V^TmE andFem=W^TmF.

Proof The proof can be easily derived from the expression (8) and the result of theorem 1.

The residual associated to the approximationX_m(t)is defended bayR_m(t) =X˙_m(t)− AXm(t)−Xm(t)B−EF^T. Next, we give a result that allows us the computation of the norm of the residual.

Theorem 4 Let Xm(t) =V^m(X^m(t)⊗Is)W^Tmbe the approximation obtained at step m by the extended global Arnoldi method. Then the residual R_m(t)satisfies the relation

kR_m(t)k²_F≤ kT_m+1,m^A Xm(t)k_F²+kT_m+1,m^B Xm(t)k²_F, (10) whereX^m(t)is the2×2m matrix corresponding to the last2rows ofX^m(t).

Proof We have

Rm(t) =X˙m(t)−AXm(t)−Xm(t)B−EF^T, where Xm(t) =Vm(Xm(t)⊗Is)W^Tm. Therefore

Rm(t) =V^m(X˙^m(t)⊗Is)W^Tm−AV^m(X^m(t)⊗Is)W^Tm−V^m(X^m(t)⊗Is)W^TmB−EF^T. We use the following properties











AVm=V^m+1(Tb^m,A⊗I_s), W^TmB= (bTm,B^T ⊗I_s)W^Tm+1, V^m=V^m+1

I_2sm 02s,2s

, W^Tm=

I_2sm02s,2s

W^Tm+1,

and





 Tb^m,A=

Tm,A

T_m+1,m^A E_m^T

, Tb^Tm,B=

T^Tm,BE_m(T_m+1,m^B )^T ,

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we obtained R_m(t) =Vm+1

X˙^m(t)⊗Is 0

0 0

W^Tm+1−Vm+1

T^m,AXm(t)⊗I_s 0 T_m+1,m^A E_m^TXm(t)⊗I_s0

W^Tm+1

−V^m+1

X^m(t)T^Tm,B⊗IsX^m(t)Em(T_m+1,m^B )^T⊗Is

0 0

W^Tm+1−V^m+1

V^TmEF^TWm0

0 0

W^Tm+1

=V^m+1 "

X˙^m(t)−T^m,AX^m(t)−X^m(t)T^Tm,B−EemFe^T −X^m(t)Em(T_m+1,m^B )^T

−T_m+1,m^A E_m^TXm(t) 0

#

⊗Is

W^Tm+1

SinceXm(t)exact solution of the equation

X˙m(t) =Tm,AXm(t) +Xm(t)T^Tm,B+Ee_mFe^T. Then

Rm(t) =Vm+1

0 −X^m(t)Em(T_m+1,m^B )^T

−T_m+1,m^A E_m^TXm(t) 0

⊗Is

W^Tm+1.

If you noteXm(t) =E_m^TXm(t), we have

kR_m(t)k²_F≤ kT_m+1,m^A Xm(t)k²_F+kT_m+1,m^B Xm(t)k²_F.

The following result shows that the approximationX_m(t)is an exact solution of a per- turbed differential Sylvester equation.

Theorem 5 Let X_m(t)be the approximate solution given by (7). Then we have

X˙m(t) = (A−Fm,A)Xm(t) +Xm(t)(B−Fm,B) +EF^T, (11) where

Fm,A=Vm+1(T_m+1,m^A E_m^T⊗Is)(V^TmV^m)⁻¹V^Tm, Fm,B=W^Tm(WmW^Tm)⁻¹[Em(T_m+1,m^B )^T⊗I_s]W_m+1^T . Proof By multiplying the (9) left byV^mand right byW^Tm, we have

Vm(X˙m(t)⊗Is)W^Tm=Vm(Tm,AXm(t)⊗Is)W^Tm+Vm(Xm(t)T^Tm,B⊗Is)W^Tm+Vm(EemFe_m^T⊗Is)W^Tm

=V^m(T^m,A⊗Is)(X^m(t)⊗Is)W^Tm+V^m(X^m(t)⊗Is)(T^Tm,B⊗Is)W^Tm

+V^m(Eem⊗Is)(Fe_m^T⊗Is)W^Tm, and like

AVm=Vm(Tm,A⊗I_s) +Vm+1(T_m+1,m^A E_m^T⊗I_s), W^TmB= (T^Tm,B⊗Is)Wm^T+ (Em(T_m+1,m^B )^T⊗Is)W_m+1^T , so

X˙_m(t) = [AVm−V_m+1(T_m+1,m^A E_m^T⊗I_s)](Xm(t)⊗I_s)W^Tm

+Vm(Xm(t)⊗I_s)[BW^Tm−(Em(T_m+1,m^B )^T⊗I_s)W_m+1^T ] +EF^T

= [A−Vm+1(T_m+1,m^A E_m^T⊗Is)(V^TmV^m)⁻¹V^Tm]Xm(t)

+Xm(t)[B−W^Tm(WmW^Tm)⁻¹(Em(T_m+1,m^B )^T⊗I_s)W_m+1^T ] +EF^T, Then

X˙_m(t) = (A−F_m,A)Xm(t) +Xm(t)(B−F_m,B) +EF^T.

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The next result states that the error Em(t) =X(t)−Xm(t)satisfies also a differential Sylvester matrix equation.

Theorem 6 Let X(t)e the exact solution of (1) and let X_m(t)be the approximate solution obtained at step m. The errorEm(t) =X(t)−X_m(t)satisfies the following equation

E˙m(t) =AEm(t) +Em(t)B−Rm(t), (12) Proof We have

X(t) =˙ AX(t) +X(t)B+EF^T,

Rm(t) =X˙m(t)−AXm(t)−Xm(t)B−EF^T, so

E˙m(t) =X(t)˙ −X˙_m(t)

=AX(t) +X(t)B+EF^T−AXm(t)−Xm(t)B−EF^T−Rm(t)

=A(X(t)−X_m(t)) + (X(t)−X_m(t))B−R_m(t)

=AEm(t) +Em(t)B−R_m(t),

Notice that from theorem 6, the error Em(t)can be expressed in the integral form as follows

Em(t) =e^(t−t⁰^)AEm,0e^(t−t⁰^)B+ Zt

t₀

e^(t−τ)AR_m(τ)e^(t−τ)Bdτ, t∈[t0,T_f]. (13) whereEm,0=Em(t0).

Next, we give an upper bound for the norm of the error by using the 2-logarithmic norm defined byµ2(A) =¹₂λmax(A+A^T).

Theorem 7 Assume that the matrices A and B are such thatµ2(A) +µ2(B)6=0. Then at step m of the extended global Arnoldi process, we have the following upper bound for the norm of the errorEm(t),

kEm(t)k₂≤ kEm,0k₂e^(t−t⁰^)(µ²^(A)+µ²^(B))+αm

e^(t−t⁰^)(µ²^(A)+µ²^(B))−1

(µ2(A) +µ2(B)) , (14) where

αm= max

t∈[t₀,t]

(max{kT_m+1,m^A Xm(t)k₂,kT_m+1,m^B Xm(t)k₂}).

The matrixX^mis the2×2m matrix corresponding to the last2rows of Gm(t).

Proof We first point out thatke^tAk2≤e^µ²^(A)t.Using the expression (13) ofEm(t), we obtain the following relation

kEm(t)k₂=ke^(t−t⁰^)AEm,0e^(t−t⁰^)Bk₂+ Z t

t₀

ke^(t−τ)AR_m(τ)e^(t−τ)Bk₂dτ.

Therefore, using (13) and the fact thatke^(t−τ)Ak₂≤e^(t−τ)µ²^(A), we get kEm(t)k₂≤ kEm,0k₂e^(t−t⁰^)(µ²^(A)+µ²^(B))+max

t∈[t₀,t]

kR_m(t)k₂ Z t

t₀

e^(t−τ)µ²^(A)e^(t−τ)µ²^(B)dτ

≤ kEm,0k2e^(t−t⁰^)(µ²^(A)+µ²^(B))+max

t∈[t₀,t]

kR_m(t)k2e^t(µ²^(A)+µ²^(B)) Zt

t₀

e^τ(µ²^(A)+µ²^(B))dτ.

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Using the result of theorem 4, we obtain max

t∈[t₀,t]

kR_m(t)k₂=αmand then

kEm(t)k₂≤ kEm,0k₂e^(t−t⁰^)(µ²^(A)+µ²^(B))+αm

e^(t−t⁰^)(µ²^(A)+µ²^(B))−1 (µ2(A) +µ2(B)) . Next, we give another upper bound for the norm of the errorEm(t).

Theorem 8 Let X(t)be the exact solution to (1) and let X_m(t)be the approximate solution obtained at step m. Then we have

kEm(t)k₂≤ kFk₂e^tµ²^(B)Γ1,m(t) +kEe_mk₂e^tµ²^(A)Γ2,m(t), (15) where

(

Γ1,m(t) =^R_t^t

0e^{−τ µ}²^(B)kZ_A(τ)−Zm,A(τ)k₂dτ, Γ2,m(t) =^R_t^t

0e^{−τ µ}²^(A)kZ_B(τ)−Z_m,B(τ)k₂dτ.

Proof From the expressions ofX(t)andX_m(t), we have kEm(t)k2 =kX(t)−Xm(t)k2

=k Zt

t₀

(ZA(τ)Z^T_B(τ)−Zm,A(τ)Zm,B(τ)^T)dτk2

=k Zt

t₀

(ZA(τ)Z^T_B(τ)−Zm,A(τ)Z^T_B(τ) +Zm,A(τ)Z^T_B(τ)−Zm,A(τ)Z_m,B^T (τ))dτk₂

=k Zt

t₀

[(ZA(τ)−Z_m,A(τ))Z_B^T(τ) +Zm,A(τ)(ZB(τ)−Z_m,B(τ))^T]dτk₂

≤ Z t

t₀

kZ_B(τ)k₂k(Z_A(τ)−Zm,A(τ))k₂+kZm,A(τ)k₂k(Z_B(τ)−Zm,B(τ))k₂dτ, Now as

µ2(T^m,A) = 1

2λmax(T^m,A+T^Tm,A)

≤ 1

2λmax(A+A^T)

=µ2(A),

and since

kZ_B(τ)k₂≤e^(t−τ)µ²^(B)kFk₂,

kZm,A(τ)k2≤e^(t−τ)µ²⁽^T^m,A⁾kEemk2≤e^(t−τ)µ²^(A)kEemk2, we also have

kEm(t)k2≤ kFk2e^tµ²^(B) Zt

t₀

e^{−τ µ}²^(B)kZ_A(τ)−Zm,A(τ)k2dτ +kEemk2e^tµ²^(A)

Zt t₀

e^{−τ µ}²^(A)kZB(τ)−Zm,B(τ)k2dτ.

we get

kEm(t)k₂≤ kFk₂e^tµ²^(B)Γ1,m(t) +kEe_mk₂e^tµ²^(A)Γ2,m(t),

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One can use some known results [19, 27] to derive upper bounds forkZ_A(τ)−Z_m,A(τ)k₂, andkZB(τ)−Zm,B(τ)k2, when using the extended global Krylov subspaces.

Theorem 9 Let Z_m,A(τ)the approximate of Z_A(τ)by extended global Arnoldi method, we get the following upper bound for the exponential approximation error e_m,A(τ) =Z_A(τ)− Zm,A(τ):

kem,A(τ)k₂≤ kVm+1(T_m+1,m^A ⊗I_s)k₂ Z _τ

0

e^(u−τ)V(A)kLm,A(u)k₂du, (16) where

(

Lm,A(u) =E_m^Te^(t−u)^T^m,AEe_m⊗I_s, V(A) =λmax(^A+A₂^T).

Proof We have

ZA(τ) =e^(t−τ)AE,

Z_m,A(τ) =Vm(e^(t−τ)^T^m,AEe_m⊗I_s), Then

Z_A⁰(τ) =−Ae^(t−τ)AE=−AZ_A(τ), and

Z_m,A⁰ (τ) =−Vm(Tm,A⊗I_s)(e^(t−τ)^T^m,AEe_m⊗I_s)

=−[AVm−V_m+1(T_m+1,m^A E_m^T⊗I_s)](e^(t−τ)^T^m,AEe_m⊗I_s)

=−AZm,A(τ) +Vm+1(T_m+1,m^A ⊗Is)Lm,A(τ).

Therefore, the errorem,A(τ) =ZA(τ)−Zm,A(τ)is such that

e⁰_m,A(τ) =−Ae_m,A(τ)−V_m+1(T_m+1,m^A ⊗I_s)Lm,A(τ), which allows to give the following expression ofem,A:

em,A(τ) =− Z τ

0

e^(u−τ)AVm+1(T_m+1,m^A ⊗I_s)Lm,A(u)du. (17) Asτ−u>0, it follows that

ke^(u−τ)Ak₂≤e^(τ−u)µ²^(−A)=e^(u−τ)V^(A). Then, we get

kem,A(τ)k2≤ kT_m+1,m^A k2 Z τ

0

e^(u−τ)V^(A)kLm,A(u)k2du.

Notice that ifV(A)is not known butV(A)≥0 (which is the case for positive semidefinite matrices) then we get the upper bound

ke_m,A(τ)k₂≤ kV_m+1(T_m+1,m^A ⊗I_s)k₂ Z _τ

0

kL_m,A(u)k₂du.

To define a new upper bound for the norm of the global errorEm(t), we can use the upper bounds for the errorse_m,Aande_m,Bin the expression (15) stated in theorem8 to get kEm(t)k2 ≤ kFk2e^tµ²^(B)

Z t t₀

e^{−τ µ}²^(B)kem,A(τ)k2dτ+kEe_mk2e^tµ²^(A) Zt

t₀

e^{−τ µ}²^(A)kem,B(τ)k2dτ,

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and then we obtain

kEm(t)k2≤ kFk2e^tµ²^(B)kVm+1(T_m+1,m^A ⊗Is)k2 Z t

t₀

e^{−τ µ}²^(B)Sm,A(τ)dτ (18) +kEemk₂e^tµ²^(A)kW_m+1(T_m+1,m^B ⊗Is)k₂

Z t t₀

e^{−τ µ}²^(A)Sm,B(τ)k₂dτ, (19) where

S_m,A(τ) =^R₀^τe^(u−τ)V^(A)kL_m,A(u)k₂du, S_m,B(τ) =^R₀^τe^(u−τ)V^(B)kL_m,B(u)k₂du.

Sincemis generally very small (m<1000), the factorsL_m,AandL_m,Bcan be computed using theexpm function of Matlab, and we calculate the approximation of the integral ap- pearing in the right sides of (14) and (??) by quadrature formulas.

The approximate solution X_m(t)could be given as a product of two matrices of low rank. It is possible to decompose it asX_m=Z₁Z₂^T where the matrixZ₁andZ₂ are of low rank (lower than 2m). Consider the singular value decomposition of the 2m×2mmatrix

X^m(t) =Ge1ΣGe^T₂,

whereΣ is the diagonal matrix of the singular values of Xmsorted in decreasing order.

LetX1,l andX2,l be the 2m×lmatrices of the first l columns ofYe₁ andYe₂ respectively, corresponding to thel singular values of magnitude greater than some toleranced_tol. We obtain the truncated SVD

Xm(t)≈U_1,lΣlU_2,l^T, whereΣl=diag[σ1, . . . ,σl]. SettingZ1,m=V^m

U1,lΣ_l^1/2⊗Is

, andZ2,m=W^m

U2,lΣ_l^1/2⊗Is

it follows that

Xm≈Z1,mZ_2,m^T . (20)

This is very important for large problems when one doesn’t need to compute and store the approximationXmat each iteration. We notice that there are cheaper ways to compute the low rank representations of the approximate solution; see [2, 5, 4].

The algorithm of first approach for solving large differential Sylvester matrix equations (EGA-exp)is summarized as follows

Algorithm 3The extended global Arnoldi (EGA-exp) method for DSE’s

1. InputsX₀ann×smatrix,Aann×nmatrix, B anp×pmatrix,Eann×smatrix and F anp×smatrix.

2. Choose a tolerancetol>0 and an integerm_max. 3. Form=1 :m_max

(a) ComputeVmandTm,Aby Algorithm 1 applied to(A,E).

(b) ComputeWmandTm,Bby Algorithm 1 applied to(B^T,F).

(c) Set Ee_m = kEk_Fe₁^(2m), Fe_m = kFk_Fe^(2m)₁ and compute Xm,A(τ) = e^(t−^τ)T^m,AEe_m, Xm,B(τ) = e^(t−τ)T^m,BFe_mby using the Matlab function expm.

(d) Use a quadrature method to compute the integral (8) and get an approximation ofXm(t)for each t∈[t0,Tf].

(e) IfkR_m(t)k_F<tolstop.

(f) Using (20), the approximate solutionXm(t)is given byXm≈Z1,mZ_2,m^T . 4. End

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5 Low-rank approximate solutions by extended global Arnoldi algorithm

In this section, we will apply the extended global Arnoldi algorithm to obtain low-rank approximate solutions to the solutionX(t)of the large differential Sylvester matrix equations (1). This approach has been applied for Lyapunov, Stein or Riccati matrix equations, for more details see [2, 5, 6, 21].

Applying the extended global Arnoldi algorithm to the pairs (A,E)and(B^T,F), we get F-orthonormal bases{V₁,V2, ...,Vm} and{W₁,W2, ...,Wm}of extended global Krylov subspacesKm^g(A,E)andKm^g(B^T,F), respectively and we have

Tm,A=V^Tm(AVm) and Tm,B=W^Tm(B^TWm), where

V^m= [V1,V2, ...,Vm] and W^m= [W1,W2, ...,Wm].

We then consider approximate solutions of the large differential Sylvester matrix equations (1) that have the low-rank form

Xm(t) =V^m(Ym(t)⊗Is)W^Tm, (21) whereY_m(t)the solution of the reduced differential Sylvester matrix equation

Y˙m(t)−T^m,AYm(t)−Ym(t)T^Tm,B−EemFe_m^T=0. (22) withEe_m=kEk_Fe^(2m)₁ andFe_m=kFk_Fe^(2m)₁ .

Next, we give a result that allows us the computation of the norm of the residual.

Theorem 10 Let Xm(t) =V^m(Ym(t)⊗Is)W^Tmbe the approximation obtained at step m by the extended global Arnoldi algorithm. Then, the Frobenius norm of the residual R_m(t)associated to the approximation X_m(t)satisfies the relation

kR_m(t)k²_F ≤ kT_m+1,m^A Ym(t)k_F²+kT_m+1,m^B Ym(t)k²_F, (23) where Y_m(t)is the2×2m matrix corresponding to the last2rows of Y_m.

Proof The proof is similar to the theorem (4).

Note that, we have to solving the reduced order differential Sylvester matrix equation (22) by Backward Differentiation Method (BDF) or by the Rosenbrock method, see [9, 16, 25].

We summarize the steps of the second approach extended global Arnoldi whit PDF or Rosenbrock method in the following algorithm

Algorithm 4The extended global Arnoldi and PDF or Rosenbrock method for DSE’s (EGA)

1. InputX0, a tolerancetol>0, an integermmax. 2. Form=1 :m_max

(a) Apply the extended block Arnoldi algorithm to(A,E)and(B^T,F)to get theF−orthonormal matri- cesVmandWmand the upper block Hessenberg matricesTm,AandTm,B.

(b) Use the BDF or Rosenbrock method to solve the differential Sylvester equation (22).

(c) IfkR_m(t)k_F<tolstop.

(d) Compute the approximate solutionXm(t)by using (20).

3. End

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6 Numerical experiments

In this section, we present some numerical experiments of large and sparse differential Sylvester matrix equations. We give comparisons between the exponential approach (EGA- exp), low-rank approximate solutions by extended global Arnoldi algorithm via PDF method (EGA-PDF) and low-rank extended global Arnoldi method via Rosenbrock method (EGA- ROS). The algorithms are coded in Matlab2014. All the experiments were performed on a laptop with an Intel Core i5 processor and 4GB of RAM. In all of the examples, the coeffi- cients of the matricesEandFwere random values uniformly distributed on[0,1].

Example 1

In this first example, the matricesAandBare obtained from the centered finite difference discretization of the operators:

LA(u) =∆u+f1(x,y)∂u

∂x+,f2(x,y)∂u

∂y+f(x,y)u L_B(u) =∆u+g₁(x,y)∂u

∂x+g₂(x,y)∂u

∂y+g(x,y)u

on the unit square[0,1]×[0,1]with homogeneous Dirichlet boundary conditions. The number of inner grid points in each direction wasn₀andp₀for the operatorsL_AandL_B, respectively. The matricesAandBwere obtained from the discretization of the operatorLAandLB

with the dimensionsn=n²₀andp=p²₀, respectively. The discretization of the operatorLA(u) andL_B(u)yields matrices extracted from theLyapackpackage [23] using the command fdm 2d matrix and denoted as A = fdm(n0,’f 1(x,y)’,’f 2(x,y)’,’f(x,y)’). In this example, n=10,000 ands=4900, respectively, and are named asA=fdm(n0,f₁(x,y),f₂(x,y),f(x,y)) andB=fdm(s0,g1(x,y),g2(x,y),g(x,y))withf1(x,y) =−e^xy,f2(x,y) =−sin(xy),f(x,y) = y²,g₁(x,y) =−100e^x,g₂(x,y) =−12xyandg(x,y) =p

x²+y². For this experiment, we usedr=3. The time interval considered was[0,2]and the initial conditionX0=X(0)was X0=Z0Z₀^T, whereZ0=0n×2. The tolerance was set to 10⁻¹⁰for the stop test on the residual. For the EGA-BDF and Rosenbrock methods, we used a constant timesteph=0.1. The entries of the matricesEandF were random values uniformly distributed on the interval [0,1]and their rank were set tos=2, we chose a size of 2500×2500, 2500×2500 for the matricesAandB, respectively.

In Figure (1), we plotted the Frobenius norms of the residualskRm(Tf)kF at final timeTf

versus the number of extended global Arnoldi iterations for the EGA-exp, EGA-BDF and EGA-ROS methods.

In Figure (2),the matricesAandBare obtained from the discretisation of the operator L_A(u)andL_B(u)with dimensionsn=10000 ands=4900, respectively. we plotted the Frobenius norms of the residualskR_m(Tf)k_F at final timeTf versus the number of extended global Arnoldi iterations for the EGA-exp, EGA-BDF and EGA-ROS methods

Example 2

For the second set of experiments, we use the matricesadd32,pde2961, andthermalfrom the University of Florida Sparse Matrix Collection [12] and from the Harwell Boeing Col- lection (http://math.nist.gov/MatrixMarket). The tolerance was set to 10⁻⁷for the stop test

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0 5 10 15 20 25

−8

−6

−4

−2 0 2 4

m iterations log10(||R(m)||F)

EGA−exp EGA−BDF EGA−ROS

Fig. 1 Residual norm vs numbermof extended global Arnoldi iterations

0 5 10 15 20 25 30 35

−8

−6

−4

−2 0 2 4 6

on the residual. For the EGA-BDF and EGA-Ros methods, we used a constant timestep h=0.01,

In Figure (3),the matrices A=thermal and B=add32 with dimensions n=3456 and p=4960, respectively, ands=3. we plotted the Frobenius norms of the residuals kR_m(Tf)k_F at final timeT_f versus the number of extended global Arnoldi iterations for the EGA-exp, EGA-BDF and EGA-ROS methods

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0 2 4 6 8 10 12

−10

−8

−6

−4

−2 0 2 4 6 8

In Figure (4), we used the matricesA=pde2961 andB=fdm(s0,100e^x,p

2x²+y²,y²− x²)with dimensionsn=2961 ands=8100, respectively, andr=3. we plotted the Frobe- nius norms of the residualskR_m(Tf)k_Fat final timeTfversus the number of extended global Arnoldi iterations for the EGA-exp, EGA-BDF and EGA-ROS methods

1 2 3 4 5 6 7

−10

−8

−6

−4

−2 0 2 4 6