Snapshot location by error equilibration in proper orthogonal decomposition for linear and semilinear parabolic partial differential equations

Transcription

1 Numerical Analysis and Scientific Computing Preprint Seria Snapshot location by error equilibration in proper orthogonal decomposition for linear and semilinear parabolic partial differential equations R.H.W. Hoppe Z. Liu Preprint #22 Department of Mathematics University of Houston November 203

2 SNAPSHOT LOCATION BY ERROR EQUILIBRATION IN PROPER ORTHOGONAL DECOMPOSITION FOR LINEAR AND SEMILINEAR PARABOLIC PARTIAL DIFFERENTIAL EQUATIONS R.H.W. HOPPE AND Z. LIU Abstract. It is well-known that the performance of snapshot based POD and POD-DEIM for spatially semidiscretized parabolic PDEs depends on the proper selection of the snapshot locations. In this contribution, we present an approach that for a fixed number of snapshots selects the location based on error equilibration in the sense that the global discretization error is approximately the same in each associated subinterval. The global discretization error is assessed by a hierarchical-type a posteriori error estimator known from automatic time-stepping for systems of ODEs. We study the impact of this snapshot selection on error equilibration for the ROM and provide numerical examples that illustrate the performance of the suggested approach. Key words. POD, POD-DEIM, snapshot location, equilibration of the error, linear and semilinear parabolic PDEs AMS subject classifications.. Introduction. During the past decades, there has been significant progress in the development, analysis, and efficient implementation of Model Order Reduction MOR for the numerical simulation of dynamical systems systems of ordinary differential equations ODEs and partial differential equations PDEs. Among the most common techniques are Balanced Truncation Model Reduction BTMR, Proper Orthogonal Decomposition POD, and the Reduced Basis Method RBM. The common aim is to construct a Reduced Order Model ROM of a significantly reduced dimension which essentially captures the dynamics of the original Full Order Model FOM. This requires the specification of a ROM basis using data extracted from the FOM. In particular, one is interested both in a preservation of input-output relations and in a sufficiently accurate approximation of the state. For an overview of the stateof-the-art we refer to [, 3, 4, 3, 4, 24, 26, 27, 30, 34] and the references therein. POD can be applied to both linear and nonlinear parabolic PDEs and is usually based on snapshots of the solution at selected time instants snapshot locations. The snapshots can be chosen in function space POD-Galerkin or in Euclidean space for the spatially discretized PDEs. The basis for the ROM can be computed either by SVD of the matrix formed by the snapshots or by an Eigenvalue Decomposition EVD of associated covariance matrices. Estimates of the POD-Galerkin error in the state have been provided in [5, 20, 2]. As far as estimates of the POD state error in the ODE setting are concerned, we refer to the survey article [7] and the references therein. For nonlinear dynamical systems, POD suffers from the fact that the computational complexity for the evaluation of the nonlinear mapping still depends on the dimension N of the FOM. In [7], the authors have adopted the Empirical Interpolation Method EIM from [2] to the discrete case Discrete Empirical Interpolation Institute of Mathematics, University of Augsburg, D-8659, Augsburg, Germany, and Department of Mathematics, University of Houston, Houston, TX , USA rohop@math.uh.edu. The author has been supported by the DFG Priority Programs SPP 253 and SPP 506, by the NSF grants DMS , DMS-5658, and by the European Science Foundation within the Networking Programme OPTPDE. Department of Mathematics, University of Houston, Houston, TX , USA. The author has been supported by the NSF grant DMS-5658.

3 Method DEIM and suggested an approach where the nonlinear function is approximated by a function whose evaluation is independent on N and features a computational complexity being proportional to the number of reduced variables. This approach is now commonly referred to as POD-DEIM. Estimates of the POD-DEIM error in the state for spatially discretized nonlinear parabolic PDEs have been derived in [8]. It is well-known that the performance of POD depends on a proper selection of the snapshot locations. Given a fixed number of snapshot locations, in [22] the authors have addressed the optimal selection of a certain number of additional snapshot locations. They have formulated this problem as a nonlinear optimization problem and have provided an SQP Sequential Quadratic Programming-type algorithm based on the second order sufficient optimality conditions. In this paper, we will focus on a different approach based on an equilibration of the POD or POD-DEIM error in the state. To be more precise, the snapshot locations are chosen in such a way that the state error is approximately the same in each subinterval of the resulting partition of the global time interval. The evaluation of the state error is realized by an efficient and reliable hierarchical a posteriori error estimator known from automatic time-stepping for systems of ODEs, namely the Euclidean norm of the difference of the solutions obtained by the implicit Euler scheme and the implicit trapezoidal rule. It turns out that the equilibration of the error is inherited by the POD or POD-DEIM based ROM. In the nonlinear case, this can be explained by means of the affine covariant version of the Newton-Kantorovich convergence theorem [] for Newton s method which is used as an iterative scheme to compute the solutions of the nonlinear algebraic systems to be solved at each time step. The paper is organized as follows: Section 2 deals with initial-boundary value problems for linear and semilinear second order parabolic PDEs and their semidiscretization by the finite element method. The semidiscretized problems are considered as the Full Order Models FOMs. Section 3 gives a brief overview on snapshot based POD and POD-DEIM, whereas section 4 is devoted to snapshot selection by equilibration of the error. In the nonlinear case, implicit time stepping requires the solution of nonlinear algebraic systems which can be done by Newton s method. We perform a convergence analysis in an affine covariant framework with emphasis on the relationship between the Newton s method for the FOM and the POD and POD-DEIM based ROMs subsections 4. and 4.2. In subsection 4.3, we provide conditions which guarantee that the error equilibration for the FOM is inherited by the respective ROMs. Finally, section 5 contains a detailed documentation of numerical results illustrating the performance of the suggested approach. 2. Linear and semilinear Parabolic PDEs and their semidiscretization in space. We use standard notation from Lebesgue and Sobolev space theory [3] and consider semilinear parabolic PDEs that can be written as abstract evolution equations according to y t Ayt + ft, yt = 0, t 0, T ], 2.a y0 = y 0, 2.b in L 2 Ω, where y t := dyt/dt and A is a linear second order elliptic differential operator with DA = H 2 Ω H0 Ω, Ω being a bounded polyhedral domain in R d, d N, with boundary Γ := Ω. We further assume T > 0, y 0 L 2 Ω, and f C[0, T ], L 2 Ω with ft, 0 = 0, t [0, T ], satisfying a Lipschitz condition in the 2

4 second argument, i.e., ft, y ft, y 2 L2 Ω L f y y 2 L2 Ω, y, y 2 L 2 Ω, t [0, T ], 2.2 for some constant L f > 0. In particular, we assume A to be of the form Ay := d i,j= x j a ij y x i d i= where a ij L Ω, i, j d, such that for some α > 0 d a ij xξ i ξ j α ξ 2, ξ R d, f.a.a. x Ω, i,j= b i y x i cy, 2.3 b i L Ω, i d, and c L Ω such that cx 0 f.a.a. x Ω. Moreover, we assume the coefficient functions to be such that A ωi is dissipative for some ω > 0 with DA RI ha, 0 < h < /ω. Then, the solution operator of 2.a,2.b is a nonlinear semigroup [8, 9]. In case f does not depend on y, 2.a,2.b represents a linear evolution equation whose solution operator is a strongly continuous linear semigroup. Although snapshots may be considered in function space as in Galerkin-POD, the practical computation of snapshots is done with respect to a semidiscretization of the semilinear or linear evolution equation 2.a,2.b in space, e.g., by the finite element method [6, 0]. The finite element method is based on the variational formulation of 2.a,2.b. We define function spaces W 0, T := H 0, T, H Ω L 2 0, T, H0 Ω, W 0, T := {y W 0, T ft, y L 2 Ω, t 0, T }, 2.4a 2.4b and note that W 0, T C[0, T ], L 2 Ω cf., e.g., [29]. The variational formulation of 2.a,2.b amounts to the computation of y W 0, T such that for almost all t 0, T and v V := H 0 Ω it holds y t, v H Ω,H 0 Ω + ay, v + fy, v L2 Ω = g, v L2 Ω, y0, v L 2 Ω = y 0, v L 2 Ω, 2.5a 2.5b where the bilinear form a, : H 0 Ω H 0 Ω is given by av, w := d i,j= Ω a ij y x i y x j dx + d i= Ω b i y v dx + x i Ω cyv dx. 2.6 A function y W 0, T satisfying 2.5a,2.5b is called a weak solution of 2.a, 2.b. For the existence of a weak solution we refer to [6] and [23]. In particular, under the previous assumptions on the operator A and the nonlinear mapping f, the existence of a weak solution is guaranteed. For the finite element approximation of 2.5a,2.5b let {T h Ω} be a shape regular family of geometrically conforming simplicial triangulations of Ω. We denote by N h Ω 3

5 the set of nodal points of T h Ω in Ω and set N := cardn h Ω. For T T h Ω we refer to h T as the diameter of T and set h := max{h T T T h Ω}. We further refer to V h V = H 0 Ω the finite element space of continuous, piecewise linear finite elements with respect to T h Ω, i.e., denoting by P T the set of polynomials of degree on T, we have V h := {v h C Ω v h T P T, T T h Ω, v h Γ = 0}. 2.7 We note that dim V h = N and V h = span{φ h,, φn h }, where φi h, i N, stands for the nodal basis function associated with the nodal point a i N h Ω. Then, the finite element approximation of 2.5a,2.5b reads: Find y h C [0, T ], V h such that for all t [0, T ] and v h V h it holds y ht, v h L 2 Ω + ay h, v h + fy h, v h L 2 Ω = g, v h L 2 Ω, 2.8a y h 0, v h L2 Ω = y 0, v h L2 Ω. 2.8b For a priori estimates of the discretization error y y h in term of the mesh width h we refer to [32]. We may identify the finite element function y h, t with a vector yt := y h, t,, y h,n t T R N such that y h,i t := y h a h,i, t, a h,i N h Ω, i N. We denote by M R N N and A R N N the mass and the stiffness matrix M ij := φ i h, φj h L 2 Ω, i, j N, A ij := aφ j h, φi h, i, j N. We further refer to y 0 R N and g R N as the vectors with components y i := y 0, φ i h L 2 Ω,, g i := g, φ i h L 2 Ω, i N, and to ft, y : R N R N, t [0, T ], as the nonlinear map ft, y = f h, t, y h,, f h,n t, y h T, N f h,i t, y h := ft, y h,j φ j h, φi h L 2 Ω, i N. j= Then, the finite element approximation 2.8a, 2.8a can be equivalently written as the following initial-value problem for a system of nonlinear first order ODEs My t + Ayt + ft, yt = 0, 0 t T, 2.9a My0 = y b Since f is continuous and satisfies a Lipschitz condition in the second argument, due to the Theorem of Picard-Lindelöf the initial-value problem 2.9a,2.9a admits a unique solution. In the sequel, we consider 2.9a,2.9a as the Full Order Model FOM for which we will describe the application of snapshot based POD and POD-DEIM in the following section 3. 4

6 3. Snapshot Based POD and POD-DEIM. 3.. Snapshot Based POD. The method of snapshots for systems of ODEs assumes a certain number of snapshot times t j, 0 j M, and snapshots y j R N of the solution of 2.9a,2.9b. The aim is to find a set {ψ i } n i=, n N, of orthonormal vectors ψ i R N, i n, such that min Jψ,, ψ n := {ψ i } n i= M y j j= n y j, ψ i ψ i 2, 3.a subject to ψ i, ψ j = δ ij, i, j n. 3.b The following well-known result establishes the necessary and sufficient optimality conditions for 3.a,3.b. Theorem 3.. Let K := [y y M ] R N M. The necessary and sufficient optimality conditions for the minimization problem 3.a,3.b amount to the solution of the symmetric N N eigenvalue problem i= KK T ψ i = λ i ψ i, i n. 3.2 Proof. We refer to [33]. The orthonormal vectors ψ i, i n, can be computed by a Singular Value Decomposition SVD for K R N M which requires the computation of orthogonal matrices V l = [u u N ] R N N, V r = [v v M ] R M M, 3.3a 3.3b and a diagonal matrix D = diagσ,, σ l R l l, 3.3c with σ 2 i in decreasing order such that Vl T D 0 KV r = =: Σ R N M, 3.4a 0 0 Kv i = σ i u i, K T u i = σ i v i, KK T u i = σ 2 i u i, i l. 3.4b The orthonormal POD basis {ψ i } n i=, n l, and the associated eigenvalues λ i of KK T are then given by ψ i = u i, λ i = σ 2 i > 0, i n. 3.5 The SVD for K R N n implies the following error estimate. Theorem 3.2. Let ψ i, i n, as in 3.5 and λ i = σ 2 i, i l, with σ i from 3.3c. Then it holds Jψ,, ψ n = M y j j= 5 n y j, ψ i ψ i 2 i= l i=n+ λ i. 3.6

7 Proof. We refer to [33]. Setting V := [ψ ψ n ] R N n, the POD based ROM takes the form ˆMŷ t + Âŷt + ˆft, ŷt = 0, 0 t T, 3.7a ˆMŷ0 = ŷ 0, 3.7b where ŷ 0 := V T y 0, the matrices ˆM, Â Rn n are given by ˆM := V T MV, Â := V T AV, 3.8 and the nonlinear function ˆft, ŷt reads as follows ˆft, ŷt := V T ft, Vŷt, 0 t T. 3.9 Remark 3.. Denoting by I n and I N the n n and N N unit matrix, we have V T V = I n. Moreover, for x R N, the quantity τ POD x := I N VV T x represents the optimal POD norm error which satisfies M I N VV T y m 2 m= N m=m+ λ m. 3.0 Remark 3.2. We note that ˆf inherits the Lipschitz continuity in the second argument from f such that the reduced order initial-value problem 3.7a,3.7b is uniquely solvable. Remark 3.3. A commonly used criterion cf., e.g., [25] for the determination of the dimension n of the ROM 3.7a,3.7b is to select some Θ 0, and to choose n as the smallest integer for which n l λ i / λ i Θ. 3. i= For a discussion of this criterion and some more refined criteria we refer to [5]. i= 3.2. POD-DEIM. POD-DEIM relies on two orthonormal matrices. The first one is the matrix V R N n obtained from the SVD of the snapshot matrix, whereas the second one is the matrix W R N L, L N, from the SVD of the nonlinear snapshot matrix FF T, where F := ft, y ft L, y L. The DEIM is applied to the nonlinear function by using interpolation projection onto the columns of W. To this end, one selects L rows p i, i L, of W and solves the linear algebraic system P T W ct = P T ft, Vŷ, P := e p e pl R N L, 3.2 where e i stands for the i-th unit vector in R N. This results in the approximation f := WP T W P T f 3.3 6

8 of the nonlinear mapping f. We note that the evaluation of ft, Vŷ only requires OL components of Vŷ. As has been shown in [7], the DEIM error estimate is Here f f P T W I N WW T f C D I N WW T f. 3.4 C D := P T W + 2N L, 3.5 and I N WW T f stands for the optimal POD-DEIM norm error which satisfies L I N WW T ft l, y l 2 l= N l=l+ s l, 3.6 where s l, l N, are the eigenvalues of the nonlinear snapshot matrix FF T decreasing order. We define ˆf : [0, T ] R n R n according to in The POD-DEIM based ROM reads ˆft, ˆx := V T ft, Vˆx, ˆx R n. 3.7 ˆMŷ t + Âŷt + ˆft, ŷt = 0, 0 t T, 3.8a ˆMŷ0 = ŷ 0, 3.8b 4. Snapshot selection by Equilibration of the Error. Prescribing a fixed number M + of snapshots, we want to determine the snapshot locations 0 =: t 0 < t < < t M := T in such a way that the global discretization error yt m y m is equidistributed, i.e., yt m+ y m+ yt m y m for all m M. Since the exact solution y is not known, we substitute yt m y m by an efficient and reliable a posteriori error estimator η m and aim at η m+ η m, m M. Such a posteriori error estimators are well-known from automatic time-stepping for systems of ODEs cf., e.g., [2]. Here, we consider a hierarchical-type estimator based on the difference between the solution by the implicit Euler scheme M + τ m Ay m + τ m ft m, y m = My m, m M, 4.a and the implicit trapezoidal rule My 0 = y 0, M + τ m 2 Aym T + τ m 2 ft m, y m T = 4.b M τ m 2 Aym τ m 2 ft m, y m, m M, 4.2a MyT 0 = y 0, 4.2b which is known to be convergent of order 2 provided y C 3 [0, T ]. Hence, for sufficiently smooth y and sufficiently small τ the so-called saturation assumption yt m y m T θ yt m y m, m M, 0 θ <, 4.3 7

9 can be expected to hold true which infers efficiency and reliability of the estimator η m := y m yt m in the sense that + θ η m yt m y m θ η m. 4.4 In the sequel, we use a slightly modified estimator η m. Rearranging terms in 4.2a, we get We replace y m T M + τ m AyT m + τ m ft m, yt m = 4.5 My m + τ m 2 Aym T y m τ m ft m, y m ft m, yt m. 2 on the right-hand side of 4.5 with ym and thus obtain M + τ m AyT m + τ m ft m, yt m = 4.6 My m + τ m 2 Aym y m τ m ft m, y m ft m, y m. 2 Setting e m := y m y m T, it follows from 4. and 4.5 that em can be computed as the solution of where gt m is given by M + τ m Ae m τ m ft m, y m e m = gt m, m M, 4.7a Me 0 = 0. gt m := M + τ m 2 Aym y m + τ m Ay m + τ m ft m, y m ft m, y m b For error equilibration, we set η m := yt m y m, η av := M M m=0 η m, and determine {t,, t M } as the solution of the minimization problem subject to the constraints min {t,,t M } 2 M η m η av 2, 4.8a m= 0 =: t 0 < t < < t M < t M := T. 4.8b The minimization problem 4.8a,4.8b is solved by SQP Sequential Quadratic Programming with BFGS updates [5]. 4.. Newton s Method for the POD Based ROM. Given snapshot locations 0 =: t 0 < t < < t M := T, snapshots y m, 0 m M, can be computed by applying the implicit Euler scheme 4.a,4.b to the FOM 2.9a,2.9b. The solution of 4.a requires the computation of a zero x of the nonlinear map F : R N R N given by Fx := M + τ m Ax + τ m ft m, x My m, x R N

10 We observe that F inherits Lipschitz continuity from f in the sense that Fy Fy 2 L F y y 2, y i R N, i The Lipschitz constant L F reads L F := M + τ m A + τ m L f, 4. where L f is from 2.2. We further note that M + τ m A is invertible. Moreover, if f is continuously differentiable in its second argument and τ m is small enough, it follows from the Banach perturbation lemma cf., e.g., [28] that F is continuously differentiable with regular Jacobian F x := M + τ m A + τ m f y t m, x, x R N. 4.2 Hence, we may solve Fx = 0 by Newton s method F x k x k = Fx k, 4.3a x k+ = x k + x k, k 0, 4.3b using x 0 := y m as an initial iterate. Since Newton s method is affine covariant, i.e., invariant with respect to transformations in the range space, the convergence analysis should be provided in an affine covariant setting. An affine covariant version of the Newton-Kantorovich convergence theorem has been provided in []. Theorem 4.. Let F : D R N R N be continuously differentiable on D with an invertible Jacobian F x 0 for some initial guess x 0 D. Assume further that the following conditions hold true: F x 0 Fx 0 α m, 4.4 F x 0 F y F y 2 γ m y x, y i D, i 2, 4.5 h m := α m γ m < 2, 4.6 Bx 0, ρ m D, ρ m := 2h m γ m. 4.7 Then, for the sequence {x k } N0 of Newton iterates there holds i F x is invertible for all Newton iterates x = x k, k N 0, ii The sequence {x k } N of Newton iterates is well defined with x k Bx 0, ρ m, k N 0, and x k x Bx 0, ρ m k, where Fx = 0, iii The convergence x k x k is quadratic. iv The solution x of Fx = 0 is unique in Bx 0, ρ m D Bx 0, ρ m, ρ m := + 2h m γ m. Proof. We refer to the proof of Theorem 2. in []. Remark 4.. The previous result tells us that Newton s method is locally quadratically convergent, i.e., the initial iterate x 0 has to be situated in a sufficiently small 9

11 neighborhood of a zero x of F, the so-called Kantorovich neighborhood. If the initial iterate x 0 does not satisfy the assumptions of Theorem 4., convergence of Newton s method can be achieved by appropriate globalization techniques. One of these techniques is the damped Newton method combined with a so-called monotonicity test for which we refer to []. The POD based ROM 3.7a,3.7b can be solved by the implicit Euler scheme as well giving rise to the solution of a nonlinear algebraic system ˆFˆx = 0 where ˆF : R n R n is given by ˆFˆx := ˆM + τ m ˆx + τ mˆftm, ˆx ˆMŷ m, ˆx R n. 4.8 The following result shows that ˆF inherits essential properties of F. Theorem 4.2. If F is continuously differentiable in D R N with Jacobian F x, x D, then ˆF is continuously differentiable in ˆD R n, V ˆD D, with Jacobian ˆF ˆx, ˆx ˆD, given by ˆF ˆx = V T F VˆxV. 4.9 Let the Jacobian F x, x D, be regular and assume that F satisfies the affine covariant Lipschitz condition F x F y F y 2 γ m y y 2, y i D, i 2, γ m > Let ˆF V x : Rn R n be given by ˆF Vx := V T F xv, x D. 4.2 Then, V T F x V is an approximate inverse of ˆF V x in the sense that V T F x V = ˆF Vx I n D n, 4.22 where D n := I n V T F xi N VV T F x V In particular, if I N VV T F x V satisfies for some 0 q <, we have Moreover, assume that ˆx ˆD satisfies I N VV T F x V q/ V T F x, 4.24 ˆF Vx q VT F x V x Vˆx < /C F, C F := γ m q VT F x V T F x Then, ˆF is regular in ˆx and it holds ˆF ˆx q VT F x 0 C F x Vˆx. 4.27

12 Proof. For the Jacobian ˆF ˆx, ˆx R n, we obtain ˆF ˆx = V T M τ m AV + τ m V T f y t m, VˆxV = V T F VˆxV If F x is regular, so is ˆF V x. Using VT V = I n we have ˆF VxV T F x V = V T F xvv T F x V = I n V T F xi N VV T F x V, which implies Under the assumption 4.24, I n D n is invertible, whence. F Vx = V T F x V I n D n The estimate 4.25 is a consequence of the Neumann lemma cf., e.g., [28]. Observing 4.20, it follows that ˆF Vx ˆF ˆx = V T F x F VˆxV 4.29 V T F x F x F x F Vˆx γ m V T F x x Vˆx. Under the assumption 4.26, the regularity of ˆF ˆx and the estimate 4.27 follow from the Banach perturbation lemma cf., e.g., [28]. Remark 4.2. The assumption 4.26 in Theorem 4.2 can be expected to hold true, if ˆx V T x is sufficiently small as can be seen easily from x Vˆx x VV T x + ˆx V T x = τ POD x + ˆx V T x. We are now in a position to show that the convergence properties of Newton s method for the FOM are inherited by the POD based ROM. Theorem 4.3. Assume that for F : D R N R N the assumptions of Theorem 4. hold true and that 4.24 holds true for x = x 0. Then, for ˆF : ˆD R n R n it holds ˆF ˆx 0 ˆFˆx 0 ˆα m, 4.30a ˆF ˆx 0 ˆF ŷ ˆF ˆx ˆγ m ŷ ˆx, ˆx, ŷ ˆD. 4.30b Here, the constants ˆα m, ˆγ m are given by ˆα m := 4.30c q VT F x 0 V T F x 0 C F x 0 Vˆx 0 α m + ε m, ˆγ m := γ m q VT F x 0 V T F x 0 C F x 0 Vˆx d where ε m in 4.30c reads as follows ε m := F x 0 V T M y m Vŷ m + V T M + τ m A + τ m L f V T x 0 Vˆx e

13 Moreover, assume that ĥ m := ˆα m ˆγ m < 2, ˆBˆx 0, ˆρ m ˆD, ˆρ m := 2ĥm ˆγ m. 4.3 Then, the sequence {ˆx k } N0 of Newton iterates ˆF ˆx k ˆx k = ˆFˆx k, ˆx k+ = ˆx k + ˆx k is well defined, stays in ˆBˆx 0, ˆρ m, and converges quadratically to some ˆx ˆBˆx 0, ˆρ m with ˆFˆx = 0. The solution ˆx is unique in ˆBˆx 0, ˆρ m ˆD ˆBˆx 0, ˆρ m, ˆρ m := + 2ĥm/ˆγ m. Proof. In view of 4.8 we have ˆF ˆx 0 ˆFˆx 0 = 4.32 ˆF ˆx 0 V T M + τ m AVˆx 0 + τ m ft m, Vˆx 0 MVŷ m = ˆF ˆx 0 V T M + τ m Ax 0 + τ m ft m, x 0 My m + My m MVŷ m M + τ m Ax 0 Vˆx 0 + τ m ft m, x 0 ft m, Vˆx 0. Under the assumption 4.26 for x = x 0 and ˆx = ˆx 0, by 4.4 and 4.27 the first term on the right-hand side of 4.32 can be estimated from above as follows: ˆF ˆx 0 V T M + τ m Ax 0 + τ m ft m, x 0 My m + My m Vŷ m = ˆF ˆx 0 V T F x 0 F x 0 Fx 0 + My m Vŷ m ˆF ˆx 0 α m V T F x 0 + V T M y m Vŷ m q VT F x 0 V T F x 0 C F x 0 Vˆx 0 α m + V T F x 0 V T M y m Vŷ m. For the second term on the right-hand side it follows from 4.4, 4.25, and 4.27 that ˆF ˆx 0 V T M + τ m Ax 0 Vˆx 0 + τ m ft m, x 0 ft m, Vˆx 0 q VT F x 0 V T F x 0 C F x 0 Vˆx 0 V T M + τ m A + τ m L f V T x 0 Vˆx 0. The assertion 4.30a can be deduced by using the preceding two estimates in On the other hand, observing 4.5, 4.25, and 4.27, we obtain ˆF ˆx 0 ˆF ŷ ˆF ŷ 2 = ˆF ˆx 0 V T F Vŷ F Vŷ 2 V γ m ˆF ˆx 0 V T F x 0 ŷ ŷ 2 γ m q VT F x 0 V T F x 0 C F x 0 Vˆx 0 ŷ ŷ 2, which is The rest of the assertions is now a consequence of Theorem 4.. 2

14 4.2. Newton s Method for the POD-DEIM Based ROM. Throughout this subsection and in the sequel, we assume that the function f : [0, T ] R N R N is continuously differentiable in its second argument with invertible Jacobian f y t, x 0, t [0, T ], for some x 0 D R N and that f y satisfies the affine invariant Lipschitz condition f y t, x 0 f y t, y f y t, y 2 γ f y y 2, y i D, i 2, 4.33 for some γ f > 0. If we solve 2.9a,2.9b with f instead of f by the implicit Euler scheme, at each time step we have to solve a nonlinear algebraic system Fx = 0, where the nonlinear map F : R N R N is given by Fx := M + τ m Ax + τ m ftm, x My m On the other hand, if we solve the POD-DEIM based ROM 3.8a,3.8b by the implicit Euler scheme, we have to solve a nonlinear system with the nonlinear map ˆF : R n R n ˆFˆx := ˆM + τ m ˆx + τ m V T ftm, Vˆx ˆMŷ m As in the previous subsection, we are interested in the properties of ˆF. Theorem 4.4. Under the assumptions of Theorem 4.2 the POD-DEIM nonlinear map ˆF is continuously differentiable in ˆD R n, V ˆD D, with the Jacobian ˆF ˆx, ˆx ˆD, given by ˆF ˆx = V T M + τ m A + τ m fy t m, Vˆx V If ˆx ˆD satisfies x Vˆx /C F, 4.37 C F := q τ m γ f C D V T F x V T I N WW T f y t m, x, then ˆF is regular in ˆx and it holds ˆF ˆx q VT F x C F x Vˆx Proof. The representation 4.36 of the Jacobian follows directly from the definition of ˆF by Recalling 4.35 and 2.2, we further obtain ˆF ˆx F Vx = V T F Vˆx F x V 4.39 τ m V T WP T W P T f y t m, xf y t m, x f y t m, Vˆx f y t m, x τ m γ f C D V T I N WW T f y t m, x x Vˆx. Under the assumption 4.37 the estimate 4.38 follows again from the Banach perturbation lemma. 3

15 Theorem 4.5. Suppose that F : D R N R N satisfies the assumptions of Theorem 4. and that 4.24 holds true for x = x 0. Then, for ˆF : ˆD R n R n we have ˆF ˆx 0 ˆFˆx 0 ˆα m, 4.40a ˆF ˆx 0 ˆF ŷ ˆF ˆx ˆγ m ŷ ˆx, ˆx, ŷ ˆD. 4.40b Here, the constants ˆα m, ˆγ m are given by ˆα m := 4.40c q VT F x 0 V T F x 0 C F x 0 Vˆx 0 α m + ε m, ˆγ m := 4.40d q VT F x 0 V T F x 0 C F x 0 Vˆx 0 γ m + ε 2 m. where ε i m, i 2, read as follows ε m := V T F x 0 V T M + τ m A + τ m C D L f V T x 0 Vˆx e τ m C D V T I N WW T ft m, x 0 + V T M y m Vŷ m, ε 2 m := τ m γ f C D V T F x 0 V T I N WW T f y t m, x f Moreover, assume that ĥ m := ˆα m ˆγ m < 2, ˆBˆx 0, ˆρ m ˆD, ˆρ m := 2ĥm ˆγ m. 4.4 Then, the sequence {ˆx k } N0 of Newton iterates ˆF ˆx k ˆx k = ˆFˆx k, ˆx k+ = ˆx k + ˆx k is well defined, stays in ˆBˆx 0, ˆρ m, and converges quadratically to some ˆx ˆBˆx 0, ˆρ m with ˆFˆx = 0. The solution ˆx is unique in ˆBˆx 0, ˆρ m ˆD ˆBˆx 0, ˆρ m, ˆρ m := + 2ĥm/ˆγ m. Proof. Due to the definition of ˆF by 4.35 we have ˆF ˆx 0 ˆFˆx 0 = 4.42 ˆF ˆx 0 V T M + τ m AVˆx 0 + τ m WP T W P T ft m, Vˆx 0 MVŷ m = ˆF ˆx 0 V T M + τ m Ax 0 + τ m ft m, x 0 My m + My m Vŷ m V T M + τ m Ax 0 Vˆx 0 + τ m ft m, x 0 WP T W P T ft m, x 0 τ m V T WP T W P T ft m, x 0 ft m, Vˆx 0 = ˆF ˆx 0 V T F x 0 F x 0 Fx 0 + My m Vŷ m V T M + τ m Ax 0 Vˆx 0 + τ m ft m, x 0 WP T W P T ft m, x 0 τ m V T WP T W P T ft m, x 0 ft m, Vˆx 0. 4

16 Under the assumption 4.26 for x = x 0 and ˆx = ˆx 0, by 4.4 and 4.27 the first term on the right-hand side of 4.42 can be estimated from above as follows: ˆF ˆx 0 V T F x 0 F x 0 Fx 0 + My m Vŷ m q VT F x 0 C F x 0 Vˆx 0 α m V T F x 0 + V T M y m Vŷ m. Using 4.27 and 3.4, for the second term we obtain ˆF ˆx 0 V T M + τ m Ax 0 Vˆx 0 + τ m ft m, x 0 WP T W P T ft m, x 0 q VT F x 0 C F x 0 Vˆx 0 V T M + τ m A x 0 Vˆx 0 + τ m C D V T I N WW T ft m, x 0. Finally, observing 2.2, 4.27, and 3.5, the third can be estimated from above according to ˆF ˆx 0 V T τ m V T WP T W P T ft m, x 0 ft m, Vˆx 0 q τ m C D L f V T F x 0 V T C F x 0 Vˆx 0 Using the preceding three estimates in 4.42, it follows that x 0 Vˆx 0. ˆF ˆx 0 ˆFˆx q VT F x 0 C F x 0 Vˆx 0 α m V T F x 0 + V T M + τ m A + τ m C D L f V T x 0 Vˆx 0 + V T M y m Vŷ m + τ m C D V T I N WW T ft m, x 0, which readily gives 4.40a. In order to prove 4.40b, observing 4.35 we find ˆF ˆx 0 ˆF ŷ ˆF ŷ 2 = 4.44 ˆF ˆx 0 V T M + τ m AVŷ ŷ 2 + τ m V T WP T W P T f y t m, Vŷ f y t m, Vŷ V = ˆF ˆx 0 V T M + τ m AVŷ ŷ 2 + τ m V T f y t m, Vŷ τ m f y t m, Vŷ V T I N WP T W P T f y t m, Vŷ f y t m, Vŷ V = f y t m, Vŷ 2 V ˆF ˆx 0 V T F Vŷ F Vŷ 2 V τ m V T I N WP T W P T f y t m, Vŷ f y t m, Vŷ 2 V. Using 4.5 and 4.38, for the first term on the right-hand side of 4.44 we obtain 5

17 the estimate ˆF ˆx 0 V T F Vŷ F Vŷ 2 V = ˆF ˆx 0 V T F x 0 F x 0 F Vŷ F Vŷ 2 V γ m q VT F x 0 V T F x 0 C F x 0 Vˆx 0 ŷ ŷ 2. Taking advantage of 4.33, 4.38, and 3.4, the second term can be estimated from above according to τ m ˆF ˆx 0 V T I N WP T W P T f y t m, Vŷ f y t m, Vŷ 2 V τ m q C D V T F x 0 V T C F x 0 Vˆx 0 I N WW T f y t m, x 0 f y t m, x 0 f y t m, Vŷ f y t m, Vŷ 2 τ m γ f C D V T F x 0 V T C F x 0 Vˆx 0 I N WW T f y t m, x 0 ŷ ŷ 2. Then, 4.40b is a consequence of 4.44 and the preceding two estimates. The rest of the assertions follow from Theorem Error Equilibration for the POD and POD-DEIM Based ROMs. In this subsection, we show that the error equilibration for the FOM is inherited by the POD and POD-DEIM based ROMs. The proof relies on the affine covariant version of the Newton-Kantorovich theorem 4. applied to the respective ROMs and on the assessment of the POD and POD-DEIM error from [33] and [8]. We first study the POD based ROM. Recalling the equations 4.7a,4.7b satisfied by e m := y m y m T, we note that e0 = 0, whereas e m, m M, is a zero of the nonlinear map F : R N R N given by with Jacobian Fx := M + τ m Ax τ m ft m, y m x gt m 4.45 F x := M + τ m A + τ m f y t m, y m x We solve Fx = 0 by Newton s method with initial iterate x 0 = 0. The POD based ROM requires the computation of ê m, m M, as a zero of the nonlinear map ˆF : R n R n given by with Jacobian ˆFˆx := ˆM + τ m ˆx τ m V T ft m, y m Vˆx V T gt m 4.47 ˆF ˆx := ˆM + τ m  + τ m V T f y t m, y m VˆxV = V T F VˆxV We compute ê m by Newton s method with initial iterate ˆx 0 = 0. If we assume I N VV T F 0 V q/ V T F

18 for some 0 q < and take F 0 = V T F 0V into account, as in Theorem 4.2 we find that F 0 is regular and satisfies F 0 q VT F Then, the affine covariant Newton Kantorovich theorem applied to the POD based ROM reads as follows: Theorem 4.6. Let F : D R N R N fulfill the assumptions of Theorem 4. for x 0 = 0, i.e., F 0 F0 α, 4.5a F 0 F y F y 2 γ y y 2, y i D, i 2, 4.5b h := α γ < 2, 4.5c B0, ρ D, ρ := 2h. 4.5d γ Moreover, suppose that 4.49 is satisfied. Then, for ˆF : ˆD R n R n it holds ˆF 0 ˆF0 ˆα, 4.52a ˆF 0 ˆF ŷ ˆF ŷ 2 ˆγ ŷ ŷ 2, ŷ i ˆD, i 2, 4.52b where the constants ˆα, ˆγ are given by ˆα := α q VT F 0 V T F 0, 4.52c ˆγ := γ q VT F 0 V T F d Moreover, assume that ĥ := ˆα ˆγ < 2, 2ĥ ˆB0, ˆρ ˆD, ˆρ := ˆγ Then, the sequence {ˆx k } N0 of Newton iterates is well defined, stays in ˆB0, ˆρ, and converges quadratically to ê m ˆB0, ˆρ with ˆFê m = 0. The solution ê m is unique in ˆB0, ˆρ ˆD ˆB0, ˆρ, ˆρ := + 2ĥ/ˆγ. Proof. Using 4.50, we find as well as ˆF 0 ˆF0 ˆF 0 V T F 0F 0 F0V α q VT F 0 V T F 0 ˆF 0 ˆF ŷ ˆF ˆx ˆF 0 V T F 0F 0 FVŷ FVˆxV γ m q VT F 0 V T F 0 ŷ ˆx. 7

19 The assertions follow from Theorem 4.. On the other hand, we know from [33] that there exists a constant C > 0 such that e m Vê m = y m yt m Vŷ m VŷT m 4.54 N y m Vŷ m + yt m VŷT m C τ POD, τ POD := λ l /2. l=n+ Setting ˆη m := ê m, m M, and combining the results of Theorem 4.6 and 4.54, we can prove error equilibration of the POD based ROM. Theorem 4.7. In addition to the assumptions of Theorem 4.6 suppose that δ m := C τ POD /η m, m M. Then, for η m = η ε m, ε m, and Λ := maxˆρ, ρ it holds Λ ε m δ m Proof. We have ˆη m ˆη m Λ δ m ε m e m Vê m e m Vê m e m ê m = η m ˆη m, and hence, in view of 4.54 it follows that ˆη m η m e m Vê m η m C τ POD = η m C τ POD /η m Theorem 4.6 tells us Consequently, combining 4.56 and 4.57 we find ˆη m ˆρ, m M ˆη m η m C τ POD/η m η m C τ POD /η m ˆη m ˆρ maxˆρ, ρ and ˆη m η m C τ POD/η m ˆη m ˆρ η m C τ POD /η m. maxˆρ, ρ This leads to η m C τ POD /η m maxˆρ, ρ which readily results in ˆη m maxˆρ, ρ ˆη m η m C τ POD /η m, A similar result can be derived for the POD-DEIM based ROM. In this case, we have ˆFˆx := ˆM τ m ˆx τ m V T ftm, y m Vˆx V T gt m, with f from 3.3. The Jacobian reads ˆF ˆx = ˆM τ m  + τ m V T fy t m, y m VˆxV. 8

20 Theorem 4.8. Suppose that F : D R N R N satisfies the assumptions of Theorem 4.6 and that 4.49 is satisfied. Then, for ˆF : ˆD R n R n we have ˆF 0 ˆF0 ˆαm, 4.58a ˆF 0 ˆF ŷ ˆF ŷ 2 ˆγ m ŷ ŷ 2, ŷ i ˆD, i b The constants ˆα m, ˆγ m read ˆα m := ˆα + κ m, ˆγ m := ˆγ + κ 2 m, 4.58c 4.58d where ˆα, ˆγ, and κ i m, i 2, are given by κ m ˆα := α q VT F 0 V T F 0, 4.58e ˆγ := γ q VT F 0 V T F 0, 4.58f := τ m C D qˆα VT F 0 V T I N WW T ft m, y m, 4.58g κ 2 m := γ f C D qˆγ VT F 0 V T I N WW T f y t m, y m. 4.58h Moreover, assume that ĥ m := ˆα m ˆγ m < 2, ˆB0, ˆρm ˆD, ˆρ m := 2ĥm ˆγ m Then, the sequence {ˆx k } N0 of Newton iterates is well defined, stays in ˆB0, ˆρ m, and converges quadratically to ê m ˆB0, ˆρ m with ˆFê m = 0. The solution ê m is unique in ˆB0, ˆρ m ˆD ˆB0, ˆρ m, ˆρ m := + 2ĥm/ˆγ m. Proof. Under assumption 4.49 the Jacobian ˆF 0 is regular and we obtain ˆF 0 ˆF0 = τ m ˆF 0 V T I N WP T W P T ft m, y m V τ m ˆF 0 V T ft m, y m V = ˆF 0 V T F 0F 0 F0V + τ m ˆF 0 V T I N WP T W P T ft m, y m V For the first term on the right-hand side we get ˆF 0 V T F 0F 0 F0V q VT F 0 V T F 0, whereas the second term can be estimated from above as follows τ m ˆF 0 V T I N WP T W P T ft m, y m V τ m C D q VT F 0 V T F 0 V T I N WW T ft m, y m. 9

21 Then, 4.58a is a consequence of the preceding two estimates. 4.58b we note that ˆF 0 F ŷ F ŷ 2 = In order to prove τ m ˆF 0 V T F 0F 0 f y t m, y m Vŷ f y t m, y m Vŷ 2 V τ m ˆF 0 V T I N WP T W P T f y t m, y m Vŷ f y t m, y m Vŷ 2 V = τ m ˆF 0 V T F 0F 0 F Vŷ F Vŷ 2 V τ m ˆF 0 V T I N WP T W P T f y t m, y m f y t m, y m f y t m, y m Vŷ f y t m, y m Vŷ 2 V. The first term on the right-hand side can be estimated from above according to τ m ˆF 0 V T F 0F 0 F Vŷ F Vŷ 2 V γ q τ m V T F 0 V T F 0 ŷ ŷ 2. For the second term we obtain τ m ˆF 0 V T I N WP T W P T f y t m, y m f y t m, y m f y t m, y m Vŷ f y t m, y m Vŷ 2 V C D γ f q τ m V T F 0 V T ŷ ŷ 2. The preceding two estimates result in 4.58b. The rest of the assertions follow from Theorem 4.. Remark 4.3. Let ˆρ := 2ˆαˆγ/ˆγ for ˆαˆγ < /2 with ˆα, ˆγ as in 4.58e, 4.58f and let ˆρ m, m M, be given by 4.59 Then, with κ m as in 4.58g there holds ˆρ + κ m ˆρ m 2 ˆρ + κ m We know from [8] that there exists a constant C > 0 such that e m Vê m = y m yt m Vŷ m VŷT m 4.6 N N y m Vŷ m + yt m VŷT m C τ DEIM, τ DEIM := λ l + s l /2. l=n+ l=l+ Again, we set ˆη m := ê m, m M, and combine the results of Theorem 4.8 and 4.6 to prove error equilibration of the POD-DEIM based ROM. Theorem 4.9. In addition to the assumptions of Theorem 4.8 suppose that δ m := C τ DEIM /η m, m M. Then, for η m = η ε m, ε m, and Λ := maxˆρ, ρ it holds Λ ε m δ m 2 + κ m ˆη m Λ ˆη m 2 + κ m δ m ε m Proof. In much the same way as in the proof of Theorem 4.7 we derive η m C τ DEIM /η m maxˆρ, ρ which gives 4.62 by taking 4.60 into account. ˆη m maxˆρ, ρ ˆη m η m C τ DEIM /η m, 20

22 5. Numerical results. As an example for a semilinear parabolic initial-boundary value problem we consider y t y + y2 = f in Q, 5.a y = 0 on Σ, 5.b y, 0 = 0 in Ω, 5.c where Q := Ω 0,, Ω := 0,, Σ := Γ 0,, Γ := Ω, and the right-hand side f is given such that yx, t = x 3 x 3 t 2 t 2 arctan60x t /2 5.2 is the exact solution of 5.a-5.c cf. Figure 5.. Exact solution.5 x yt, x x t Fig. 5.. Exact solution of the semilinear parabolic PDE. As the nonlinear FOM we have chosen a finite element approximation in space by continuous, piecewise linear finite elements with respect to a uniform partition of the computational domain Ω with mesh size h = /28. We have further chosen 42 snapshot locations in the time interval 0, by the solution of the minimization problem 4.8 of section 4. We have used the implicit Euler finite element solutions to compute both the POD and the POD-DEIM singular values and basis functions. The singular values of the POD snapshot matrix and the POD-DEIM snapshot matrix are displayed in Figure 5.3 and Figure 5.4. Table contains the averages of the error estimators η m and ˆη m and its standard deviations for the FOM and the POD and POD-DEIM ROMs of different dimension N. The results reflect the theoretically derived estimates of subsection 4.3 2

23 Equilibration Time instance t 2 t 40 Plot of 40 time instances for error equilibration ith time instance Fig Snapshot locations t m, 0 m 4, by error equilibration for the FOM. 0 0 Singular values for POD snapshot matrix 0 5 σi 0 0 σ σ 2 σ σ ith singular value Fig Singular values of the POD snapshot matrix. REFERENCES [] A. C. Antoulas; Approximation of Large-Scale Systems, SIAM, Philadelphia, [2] M. Barrault, Y. Maday, N. C. Nguyen, and A. T. Patera; An empirical interpolation method: application to efficient reduced-basis discretization of partial differential equations, Comptes Rendus Mathematique 339, , [3] P. Benner, M. Hinze, and E.J.W. ter Maten eds.; Model Reduction for Circuit Simulation, Springer, Berlin-Heidelberg-New York, 20. [4] P. Benner, V. Mehrmann, and D. C. Sorensen eds.; Dimension Reduction of Large- Scale Systems, Lecture Notes in Computational Science and Engineering, Vol. 45, Springer, Berlin-Heidelberg-New York, [5] P.T. Boggs and J.W. Tolle; Sequential quadratic programming, Acta Numerica 4, 5, 995. [6] S.C. Brenner and L. Ridgway Scott, The Mathematical Theory of Finite Element Methods. 3rd Edition, Springer, New York, 2008 [7] S. Chaturantabut and D.C. Sorensen; Discrete empirical interpolation for nonlinear model reduction, SIAM J. Sci. Comput. 32, , 200. [8] S. Chaturantabut and D.C. Sorensen; A state space error estimate fpr POD-DEIM nonlinear model reduction, Technical Report TR0-32, CAAM, Rice University, Houston, 200. [9] Z. Chen and J. Feng; An adaptive finite element algorithm with reliable and efficient 22

24 0 0 Singular values for DEIM non-linear snapshot matrix 0 5 σi σ σ 2 σ σ ith singular value Fig Singular values of the POD-DEIM snapshot matrix. Table 5. Average of the error estimators η m and ˆη m and its standard deviations for the FOM and the POD and POD-DEIM ROMs of different dimension N FOM N POD POD-DEIM 3.707E 4 ± 5.025E E 4 ±.064E E 4 ±.094E E 4 ± 6.752E E 4 ± 6.906E E 4 ± 5.964E E 4 ±.90E E 4 ± 6.080E E 4 ± 6.080E 7 error control for linear parabolic problems. Math. Comp. 73, 67-93, [0] P.G. Ciarlet; The Finite Element Method for Elliptic Problems, SIAM, Philadelphia, [] P. Deuflhard; Newton Methods for Nonlinear Problems. Affine Invariance and Adaptive Algorithms, Springer, Berlin-Heidleberg-New York, [2] P. Deuflhard anf F. Bornemann; Scientific Computing with Ordinary Differential Equations, Springer, Berlin-Heidelberg-New York, [3] M. A. Grepl, Y. Maday, N. C. Nguyen, and A. T. Patera; Efficient reduced-basis treatment of nonaffine and nonlinear partial differential equations, Mathematical Modelling and Numerical Analysis 4, , [4] S. Gugercin and A.C. Antoulas; A survey of model reduction by balanced truncation and some new results. Int. J. Control 77, , [5] T. Henri and J.-P. Yvon; Convergence estimates of POD-Galerkin methods for parabolic problems. In: System Modeling and Optimization. IFIP Int. Fed. for Inf. Processing. Vol. 66, pages , Springer, Boston, [6] D. Henry; Geometric Theory of Semilinear Parabolic Equations, Springer, Berlin- Heidelberg-New York, 98. [7] C. Homescu, L. R. Petzold, and R. Serban; Error estimation for reduced-order models of dynamical systems, SIAM Review 49, , [8] T. Kato; Nonlinear semigroups and evolution equations, J. Math. Soc. Japan 9, , 967. [9] N. Kenmochi and S. Oharu; Difference approximation of nonlinear evolution equations and semigroups of nonlinear operators, Publ. RIMS, Kyoto Univ. 0, , 974. [20] K. Kunisch and S. Volkwein; Galerkin proper orthogonal decomposition methods for parabolic problems. Numer. Math. 90, 7 48, 200. [2] K. Kunisch and S. Volkwein; Galerkin proper orthogonal decomposition methods for a general equation in fluid dynamics. SIAM J. Numer. Anal. 40, ,

25 [22] K. Kunisch and S. Volkwein; Optimal snapshot location for computing POD basis functions, ESAIM: Mathematical Modelling and Numerical Analysis 44, , 200. [23] O.A. Ladyzenskaja, V.A. Solonnikov, and N.N. Ural ceva; Linear and Quasilinear Equations of Parabolic Type, AMS, Providence, 968 [24] H.V. Ly and H.T. Tran; Modelling and control of physical processes using proper orthogonal decomposition. Math. and Comput. Modelling 33, , 200. [25] B. C. Moore; Principal component analysis in linear systems: controllability, observability, and model reduction, IEEE Trans. Automat. Control 26, 7 32, 98. [26] B. R. Noack, M. Morzynski, and G. Tadmor eds.; Reduced-Order Modelling for Flow Control. Springer, Berlin-Heidelberg-New York, 20. [27] G. Obinata and B.D.O. Anderson; Model Reduction for Control System Design, Springer, Berlin-Heidelberg-New York, 200. [28] J.M. Ortega and W.C. Rheinboldt; Iterative Solution of Nonlinear Equations in Several Variables, SIAM, Philadelphia, [29] M. Renardy and R.C. Rogers, An Introduction to Partial Differential Equations, Springer, Berlin-Heidelberg-New York, 993. [30] W.H. Schilders, H.A. van der Vorst, and J. Rommes eds.; Model Order Reduction: Theory, Research Aspects, and Applications, Springer, Berlin-Heidelberg- New York, [3] L. Tartar; Introduction to Sobolev Spaces and Interpolation Theory, Springer, Berlin- Heidelberg-New York, [32] V. Thomée; Galerkin Finite Element Methods for Parabolic Problems, Springer, Berlin-Heidelberg-New York, 997. [33] S. Volkwein; Model Reduction using Proper Orthogonal Decomposition, Lecture Notes, Institute of Mathematics and Scientific Computing, University of Graz, 2008 see [34] K. Zhou, J. C. Doyle, and K. Glover; Robust and Optimal Control, Prentice Hall, Englewood Cliffs,