Domain Decomposition and Model Reduction of Systems with Local Nonlinearities

Transcription

1 Domain Decomposition an Moel Reuction of Systems with Local Nonlinearities Kai Sun, Rolan Glowinsi, Matthias Heinenschloss, an Danny C. Sorensen Abstract The goal of this paper is to combine balance truncation moel reuction an omain ecomposition to erive reuce orer moels with guarantee error bouns for systems of iscretize partial ifferential equations (PDEs) with a spatially localize nonlinearities. Domain ecomposition techniques are use to ivie the problem into linear subproblems an small nonlinear subproblems. Balance truncation is applie to the linear subproblems with inputs an outputs etermine by the original in- an outputs as well as the interface conitions between the subproblems. The potential of this approach is emonstrate for a moel problem. Introuction Moel reuction sees to replace a large-scale system of ifferential equations by a system of substantially lower imension that has nearly the same response characteristics. This paper is concerne with moel reuction of systems of iscretize partial ifferential equations (PDEs) with spatially localize nonlinearities. In particular, we are intereste in constructing reuce orer moels for which the error between the input-to-output map of the original system an the input-to-output map of the reuce orer moel can be controlle. Balance truncation is a particular moel reuction technique ue to [6], which for linear time invariant systems leas to reuce orer moels which approximate the original input-to-output map with a user controlle error [, 6]. Although extensions of balance truncation to nonlinear systems have been propose, see, e.g., Kai Sun, Matthias Heinenschloss, Danny C. Sorensen Department of Computational an Applie Mathematics, MS-34, Rice University, 6 Main Street, Houston, Texas 77-89, USA. leinsun/heinen/sorensen@rice.eu Rolan Glowinsi University of Houston, Department of Mathematics, 6 P. G. Hoffman Hall, Houston, Texas , USA. rolan@math.uh.eu

2 Kai Sun, Rolan Glowinsi, Matthias Heinenschloss, an Danny C. Sorensen [9, 4], there are no bouns available for the error between the input-to-output map of the original system an that of the reuce orer moel. Proper Orthogonal Decomposition (POD) is often use for moel reuction of nonlinear systems. Error bouns are available for the error between the so-calle snapshots an the reuce orer moel, see, e.g., [, 3], but no bouns for the error between the input-tooutput map of original system an that of the reuce orer moel, unless the socalle snapshot set reflects all possible inputs. Our approach uses omain ecomposition techniques to ivie the problem into linear subproblems an small nonlinear subproblems. Balance truncation is applie only to the linear subproblems with inputs an outputs etermine by the original inan outputs as well as the interface conitions between the subproblems. We expect that this combination of omain ecomposition an balance truncation leas to a substantial reuction of the original problem if the nonlinearities are localize, i.e., the nonlinear subproblems are small relative to the other subomains, an if the interfaces between the subproblems are relatively small. To eep our presentation brief, we consier a moel problem which couples the D Burgers equation to two heat equations. This is motivate by problems in which one is primarily intereste in a nonlinear PDE which is pose on a subomain an which is couple to linear PDEs on surrouning, larger subomains. The linear PDE solution on the surrouning subomains nees to be compute accurately enough to provie acceptable bounary conitions for the nonlinear problem on the inner subomain. Such situations arise, e.g., in regional air quality moels. Our wor is also relate to [4], which is an example paper which iscusses the coupling of linear an nonlinear PDEs, but no imension reuction is applie. Domain ecomposition an POD moel reuction for flow problems with moving shocs are iscusse in []. POD moel reuction is applie on the subomains away from the shoc. The paper [8] iscusses a ifferent moel reuction technique for secon orer ynamical systems with localize nonlinearities. The papers [, ] an [] iscuss ifferent moel reuction an substructuring techniques for secon orer ynamical systems an moel reuction of interconnect systems respectively. The Moel Problem Let Ω = 3 = Ω, where Ω = (, ), Ω = (,) an Ω 3 = (,) an let T > be given. Our moel problem is given by y ρ t (x,t) µ y (x,t) = S (x,t), (x,t) Ω (,T ), (a) y (x,) = y (x), x Ω, =,3, (b) y (,t) =, y 3 (,t) = t (,T ), (c)

3 Domain Decomposition an Moel Reuction of Systems with Local Nonlinearities 3 y ρ t (x,t) µ y + y y (x,t) =, (x,t) Ω (,T ), () with the following interface conitions y (x,) = y (x), x Ω, (e) y (,t) = y (,t), y (,t) = y 3 (,t), t (,T ), (a) y µ (,t) = µ y (,t), µ y (,t) = µ y 3 3 (,t), t (,T ). (b) We assume that the forcing functions S, S 3 are given by S = n s i= b i (x)u i (t), =,3. (3) To obtain the wea form of () an (), we multiply the ifferential equations (a, ) by test functions v i H (Ω i ), i =,,3, respectively, integrate over Ω i, an apply integration by parts. Using the bounary conitions (c, h) this leas to ρ y v x + µ t Ω ρ y v x + µ t Ω y Ω Ω y v x µ y v x + v = S v x, =,3, (4a) Ω Ω y Ω y y v x µ v =. (4b) If v H (Ω ), =,3, satisfy v ( ) =, v 3 () =, then (c), (4a) imply y ( ) µ y 3 () µ 3 = y S v x + ρ y v x + µ Ω t Ω Ω y 3 y 3 v 3 x µ 3 Ω 3 Ω 3 = S 3 v 3 x ρ 3 Ω 3 t v x, v 3 x. (a) (b) If v H (Ω ) satisfies v ( ) = an v () =, then (4b) implies µ y ( ) y v = ρ y v x µ t Ω Ω x y Ω y v x. Finally, if v H (Ω ) satisfies v ( ) = an v () =, then (4b) implies µ y () y v = ρ y v x + µ t Ω Ω x + y Ω y v x. (c) () The ientities () are use to enforce the interface conitions (). We iscretize the ifferential equations in space using piecewise linear functions. We subivie Ω j, j =,,3, into subintervals. Let x i enote the subinterval enpoints an let v i be the piecewise linear basis function with v i (x i ) = an v i (x j ) = or all j i. We efine the following inex sets

4 4 Kai Sun, Rolan Glowinsi, Matthias Heinenschloss, an Danny C. Sorensen I I = {i : x i [, )}, I I = {i : x i (,)}, I I 3 = {i : x i (,]}, I Γ = {i : x i = }, I Γ 3 = {i : x i = }. Given y i for i I Γ IΓ 3, we compute functions y (t,x) = y i (t)v i (x) + y i (t)v i (x), =,3, (6a) i I I i I Γ y (t,x) = i I I y i (t)v i (x) + y i (t)v i (x) + y i (t)v i (x), i I Γ i I3 Γ where in (6a) we use I Γ = I Γ if = an IΓ = I3 Γ if = 3, as solutions of ρ y v i x + µ t Ω Ω y x v ix = S v i x,i I I, =,3, Ω ρ y v i x + µ t Ω y x v ix + y y v i x =,i I. I Ω Ω (6b) If we set y I = (y i) i I I, =,,3, y Γ j = (y i) i I Γ j, j {,3}, y Γ = (y Γ,yΓ 3 )T, an u = (u i ) i=,...,ns, =,3 (cf. (3)), the previous ientities can be written as M II M II t yi + A II y I + M IΓ t yi + A II y I + M IΓ M II 3 t yi 3 + A II 3 y I 3 + M IΓ 3 t yγ + A IΓ y Γ = B I u, (7a) t yγ + A IΓ y Γ + N I (y I,y Γ ) =, (7b) t yγ 3 + A IΓ 3 y Γ 3 = B I 3u 3. (7c) By construction, the functions y j, j =,,3, in (6) satisfy (a). To enforce (b) we insert the ientities (), (6) into (b). The resulting conitions can be written as M Γ I M Γ I 3 t yi + A Γ I y I + (M Γ Γ + M Γ Γ ) t yγ + (A Γ Γ + A Γ Γ )y Γ (8a) +M Γ I t yi + A Γ I y I + N Γ (y I,y Γ ) = B Γ u, (8b) t yi 3 + A Γ 3 I y I 3 + (M Γ 3 Γ + M Γ 3 Γ ) t yγ 3 + (A Γ 3 Γ + A Γ 3 Γ )y Γ 3 (8c) +M Γ 3 I t yi + A Γ 3 I y I + N Γ 3(y I,y Γ 3) = B Γ 3 u 3. (8) To summarize, our iscretization of () an () is given by (7) an (8). As outputs we are intereste in the solution of the PDE at the spatial locations ξ =,ξ =,ξ 3 =. Thus the output equations are y (t,ξ ) = i I I y i (t)v i (ξ ), =,,3, which can be written as z I (t) = CI jy I (t), where CI R II, =,,3.

5 Domain Decomposition an Moel Reuction of Systems with Local Nonlinearities 3 Balance Truncation Moel Reuction Given E R n n symmetric positive efinite, A R n n, B R n m, C R q n, an D R q m, we consier linear time invariant systems in state space form E t y(t) = A y(t) + Bu(t), t (,T ), y() = y, (9a) z(t) = C y(t) + Du(t), t (,T ). (9b) Projection methos for moel reuction generally prouce n r matrices V,W with r n an with W T E V = I r. One obtains a reuce form of equations (9) by setting y = V ŷ an projecting (imposing a Galerin conition) so that W T [E V ŷ(t) AV ŷ(t) Bu(t)] =, t (,T ). t This leas to a reuce system of orer r with matrices Ê = W T E V = I, A = W T AV, B = W T B, C = C V, an D = D. Balance reuction is a particular techniqe for constructing the projecting matrices V an W, see, e.g., [, 6]. One first solves the controllability an the observability Lyapunov equation A PE + E PA T + BB T = an A T QE + E QA + C T C T =, respectively. Uner the assumptions of stability, controllability an observability, the matrices P,Q are both symmetric an positive efinite. There exist methos to compute (approximations of) P = UU T an Q = LL T in factore form. In the large scale setting the factorization is typically a low ran approximation. See, e.g., [8, 7]. The balancing transformation is constructe by computing the singular value ecomposition U T E L = ZSY T an then setting W = UZ r, V = LY r, where S r = iag(σ,σ,...,σ r ) is the r r submatrix of S = S n. The singular values σ j are in ecreasing orer an r is selecte to be the smallest positive integer such that σ r+ < τσ where τ > is a prespecifie constant. The matrices Z r,y r consist of the corresponing leaing columns of Z,Y. It is well nown [6] that A must be stable an that for any given input u we have z ẑ L u L (σ r σ n ), () t yγ, AIΓ where ẑ is the output (response) of the reuce moel. Moel reuction techniques for infinite imensional systems are reviewe in, e.g., [3]. We want to apply balance truncation moel to the linear subsystems an 3 in (7) an (8). We nee to ientify the input-output relations for these subsystems in the context of the couple system to ensure that balancing techniques applie to these subsystems leas to a reuce moel for the couple system with error bouns. To ientify the appropriate input-output relations, we focus on subsystem. Examination of (7a,b) an (8a) shows that M IΓ yγ an BI u are the inputs into system an C Γ I are the outputs. Hence, if yi, MΓ I t yi + AΓ IyI

6 6 Kai Sun, Rolan Glowinsi, Matthias Heinenschloss, an Danny C. Sorensen then we nee to apply moel reuction to M II M IΓ = an M Γ I =, () t yi = A II y I A IΓ y Γ + B I u (a) z I = C I y I, z Γ = A Γ I y I. (b) The system () is exactly of the form (9) an we can apply balance truncation moel reuction to obtain M II t ŷi = ÂII ẑ I = ĈI ŷi, ŷi ÂIΓ y Γ + B I u (3a) ẑ Γ = ÂΓ I ŷi. (3b) Subsystem 3 can be reuce analogously. The reuce moel for the couple nonlinear system (7) an (8) is now obtaine by replacing the subsystem matrices for subsystems an 3 by their reuce matrices. Whether the balance truncation error boun () can be use to erive an error boun between the original couple problem (7) an (8) an its reuce moel is uner investigation. In our finite element iscretization we use mass lumping to obtain (). However other iscretizations, such as spectral elements or iscontinuous Galerin methos satisfy () irectly, see [, ]. 4 Numerical Results We subivie Ω j into equiistant subintervals of length h = /N, =,,3, an we use piecewise linear basis functions.the size of the system (7), (8) is 9(N + N 3 )+N +. The parameters in the PDE are ρ =, =,,3, an µ =., µ =., µ =.. For subsystem an 3 we compute low-ran approximate solutions of the controllability an observability Lyapunov equations using the metho escribe in [8]. We truncate such that σ r+ < τσ, where τ = 4. The sizes of the full an of the reuce orer moels for various iscretization parameters are shown in Table. The subsystems an 3 reuce substantially an the size of the subsystem limits the amount of reuction achieve overall. For example, for N = N 3 = the subsystems an 3 are each reuce in size from 8 to. The size of the couple system is reuce from 36 + N to 3 + N. Next, we compare the system output given forcing functions S = u (t), S 3 = u 3 (t) (cf., (3)) with u (t) = sin(3t)(.8t/t ), u 3(t) = sin(t)(.3 +.7t/T ) on (,T ) = (,). The full orer moel (7), (8) an the corresponing reuce orer moel are solve using the moifie θ-scheme [7, 9] with (macro) time step t = T /. Figure shows the outputs, i.e., the approximate solution of the PDE at ξ =,ξ =,ξ 3 =. The left plot in Figure shows the solution of the reuce orer iscretize PDE. The solution of the iscretize PDE is visually inistinguishable

7 Domain Decomposition an Moel Reuction of Systems with Local Nonlinearities 7 Table Dimension of the full an of the reuce orer moels for various iscretization parameters N, N, N3 an τ = 4. N = N3 N size of full orer moel size of reuce orer moel Fig. Outputs,, 3 of the full orer system corresponing to the iscretization N = N = N3 = are given by, an, respecitively. Outputs,, 3 of the reuce orer system are given by otte, ashe an soli lines, respectively. 4 t $!!$ t!! x t!! x Fig. Solution of the reuce orer iscretize PDE (left) an error between the solution of the iscretize PDE an the reuce orer system (right) for iscretization N = N = N3 =. from the solution of the reuce orer iscretize PDE, as inicate by the size of the error shown in the right plot in Figure. The error is larger in the right subomain because the PDE solution is positive an the avection term in () avects the solution to the right. Our numerical results inicate that the coupling of balance truncation reuction for linear time variant subsystems with spatially localize nonlinear moels leas to a couple reuce orer moel with an error in the input-to-output map that is comparable to the error ue to balance truncation moel reuction applie to the linear subsystems alone. The efficiency of the approach epens on the size of the interface an on the size of the localize nonlinearity. Investigations for higher imensional problems are unerway to explore the overall gains in efficiency.

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