Symbolic-Numeric Methods for Improving Structural Analysis of DAEs

Transcription

1 Symbolic-Numeric Methods or Improving Structural Analysis o DAEs Guangning Tan, Nedialko S. Nedialkov, and John D. Pryce Abstract Systems o dierential-algebraic equations (DAEs) are generated routinely by simulation and modeling environments, such as MapleSim and those based on the Modelica language. Beore a simulation starts and a numerical method is applied, some kind o structural analysis is perormed to determine which equations to be dierentiated, and how many times. Both Pantelides s algorithm and Pryce s Σ-method are equivalent in the sense that, i one method succeeds in inding the correct index and producing a nonsingular Jacobian or a numerical solution procedure, then the other does also. Such a success occurs on many problems o interest, but these structural analysis methods can ail on simple solvable DAEs and give incorrect structural inormation including the index. This article investigates the Σ- method s ailures, and presents two symbolic-numeric conversion methods or ixing these ailures. These methods convert a DAE on which the Σ-method ails to a DAE on which this structural analysis may succeed. 1 Introduction We consider DAEs o the general orm i (t, the x and derivatives o them) = 0, i = 1,...,n, (1) G. Tan School o Computational Science and Engineering, McMaster University, 1280 Main St. W., L8S 4L8, Hamilton, ON, Canada, tang4@mcmaster.ca N. S. Nedialkov Department o Computing and Sotware, McMaster University, 1280 Main St. W., L8S 4L8, Hamilton, ON, Canada, nedialk@mcmaster.ca J. D. Pryce School o Mathematics, Cardi University, Senghennydd Road, Cardi CF24 4AG, Wales, UK., pryced1@cardi.ac.uk 1

2 2 Guangning Tan, Nedialko S. Nedialkov, and John D. Pryce where the x (t), = 1,...,n are state variables that are unctions o an independent variable t, usually regarded as time. Pryce s structural analysis (SA), the Σ-method [9], determines or a DAE (1) its structural index, the number o degrees o reedom (DOF), variables and derivatives that need initial values, and constraints o this DAE. These SA results can help decide how to apply an index reduction algorithm [4], perorm a regularization process [12], or design a solution scheme or a Taylor series method [1, 2, 7]. The Σ-method is equivalent to Pantelides s algorithm [8]: they both produce the same structural index [9, Theorem 5.8]. This index is an upper bound or the dierentiation index, and oten they are the same [9]. The Σ-method provably succeeds on many problems o practical interest, producing a nonsingular System Jacobian. However, this SA method can ail hence Pantelides s algorithm can ail as well on some simple solvable DAEs, producing an identically singular System Jacobian. We investigate this SA s ailures and present two symbolic-numeric conversion methods or ixing these ailures. Ater each conversion, the value o the signature matrix is guaranteed to decrease, provided some conditions are satisied. We conecture that such a decrease should result in a better problem ormulation o a DAE, so that the SA may succeed and hence produce a nonsingular System Jacobian. Sect. 2 summarizes the Σ-method. Sect. 3 describes this SA s ailure. Sect. 4 presents our two symbolic-numeric conversion methods, each o which is illustrated with an example therein. Sect. 5 gives conclusions and indicates the uture work. 2 Summary o the Σ-method This SA method [9] constructs or a DAE (1) an n n signature matrix Σ, whose (i, ) entry σ i is either an integer 0, order o the highest derivative to which variable x occurs in equation i, or i x does not occur in i. A highest-value transversal (HVT) o Σ is a set T o n positions (i, ) with one entry in each row and each column, such that the sum o these entries is the largest possible. This sum is the value o Σ, written Val(Σ). Assume henceorth that Val(Σ) is inite, or equivalently, the DAE is structurally well posed. Then we ind 2n equation and variable osets c 1,...,c n and d 1,...,d n, respectively, which are integers satisying c i 0 or all i; d c i σ i or all i, with equality on a HVT. (2) Osets satisying (2) are called valid, but they are never unique; there exists an elementwise smallest solution o (2) termed the canonical osets. We associate with some valid osets an n n System Jacobian J, deined as J i = { i / x (σ i ) i d c i = σ i, and 0 otherwise. (3)

3 Symbolic-Numeric Methods or Improving Structural Analysis o DAEs 3 I one J is nonsingular and hence all J s are at a consistent point as deined in [5], then we say that the Σ-method succeeds and there exists (locally) a unique solution through this point [9]. In the success case, the SA uses the canonical osets to determine the structural index and the number o DOF: ν S = max i c i + DOF = Val(Σ) = { 1 i some d = 0, and 0 otherwise, (i, ) T σ i = d c i. i Example 1. We show 1 the above concepts or the simple pendulum (PEND), a DAE o dierentiation index 3. 0 = 1 = x + xλ 0 = 2 = y + yλ g 0 = 3 = x 2 + y 2 L 2 x y λ c i [ ] d x y λ [ ] 1 1 x 2 1 y 2x 2y 3 The state variables are x,y, and λ; G is gravity and L > 0 is the length o the pendulum. There are two HVTs o Σ, marked with and, respectively. A blank in Σ denotes, and a blank in J denotes 0. Since det(j) = 2(x 2 +y 2 ) = 2L 2 0, the System Jacobian is nonsingular, and the SA succeeds. The structural index is hence ν S = min i c i +1 = 2+1 = 3 (because d 3 = 0), which equals the dierentiation index. The number o DOF is Val(Σ) = d i c i = 4 2 = 2. 3 Structural Analysis s Failure We say that the Σ-method ails, i a DAE (1) has a inite Val(Σ) and an identically singular System Jacobian J. In the ailure case, the SA reports Val(Σ) as an apparent DOF but not a meaningul one. In this article, we only ocus on the case where such an identically singular J is not structurally singular that is, there exists a HVT T o Σ such that J i is generically nonzero or all (i, ) T. For a detailed description and discussion o SA s ailure, we reer the reader to [13, 4]. Example 2. We illustrate a ailure case with the ollowing DAE 2 in [3, p. 23]. 1 When we give a DAE or example, we present with it its signature matrix Σ, osets c i and d, and the associated System Jacobian J. 2 In the original ormulation, the driving unctions are 1, 2. We change them to g 1,g 2.

4 4 Guangning Tan, Nedialko S. Nedialkov, and John D. Pryce 0 = 1 = x +ty g 1 (t) 0 = 2 = x +ty g 2 (t) x y c [ ] i d 1 1 [ x y ] 1 1 t 2 1 t The SA ails since det(j) = 0. Now J is identically singular but not structurally singular: on a HVT marked by, each o J 11 = 1 and J 22 = t is generically nonzero. One simple ix is to replace 1 by 0 = 1 = = y + g 1 (t) g 2(t), which results in an algebraic problem; c. [4, Example 5]. 0 = 1 = y + g 1 (t) g 2(t) 0 = 2 = x +ty g 2 (t) x y c [ ] i d 0 0 [ x y ] t Now that det(j) = 1 0, the SA succeeds. Notice Val(Σ) = 0 < 1 = Val(Σ). Another simple ix is to introduce a new variable z = x +ty and eliminate x in 1 and 2 ; note z = x +ty + y. 0 = 1 = y + z g 1 (t) 0 = 2 = z g 2 (t) y z c [ ] i d 0 1 [ y z ] For this resulting DAE, det(j) = 1 0, and the SA succeeds. Ater solving or y and z, we can obtain x = z ty. This ix gives Val(Σ) = 0 < 1 = Val(Σ) also. We shall note that each o the above ixes actually uses each o the methods described below. 4 Conversion Methods We present two conversion methods or systematically ixing SA s ailures. The irst method is based on replacing an existing equation by a linear combination o some equations and derivatives o them. We call this method the linear combination (LC) method and describe it in Sect The second method is based on substituting newly introduced variables or some expressions and enlarging the system. We call this method the expression substitution (ES) method and describe it in Sect We only present the main eatures o these methods; the reader is reerred to [13] or more details and especially the proos o Theorems 1 and 2 below. Given a DAE (1), we assume henceorth that Val(Σ) is inite and that a System Jacobian J is identically singular but not structurally singular.

5 Symbolic-Numeric Methods or Improving Structural Analysis o DAEs Linear Combination Method Let u be a nonzero n-vector unction in the cokernel o J, that is, u coker(j) and J T u = 0. Let u i denote the ith component o u. Here, we consider J and u as unctions o t, the x s and derivatives o them. For such a unction ω, ω 0 means that ω is nonzero or all t in some time interval I. For our convenience, we let { order o the highest derivative to which x σ (x,u) = occurs in u; or i x does not occur in u. Denote I = {i u i 0}, c = min i I c i, and L = { i I c i = c }. (4) The LC method is based on the ollowing theorem. Theorem 1. I and we replace some l with l L by σ (x,u) < d c or all = 1,...,n, (5) = u i (c i c) i (6) i I (denoted as l ), then Val(Σ) < Val(Σ), where Σ is the signature matrix o the resulting DAE. We call (5) the condition or applying the LC method. The ollowing example illustrates this method. Example 3. Consider 0 = 1 = x 1 + x 3 0 = 3 =x 1 x 2 + g 1 (t) 0 = 2 = x 2 + x 4 0 = 4 =x 1 x 4 + x 2 x 3 + x 1 + x 2 + g 2 (t), where g 1 and g 2 are given driving unctions. x 1 x 2 x 3 x 4 c i d x 1 x 2 x 3 x x 2 x 1 4 x 2 x 1 A shaded entry σ i in Σ denotes a position (i, ) where d c i > σ i 0 and hence J i = 0 by (3). The SA ails since det(j) = 0, where J is identically (but not structurally) singular.

6 6 Guangning Tan, Nedialko S. Nedialkov, and John D. Pryce We use u = ( x 2,x 1,1, 1 ) T coker(j). Then (4) becomes I = { i u i 0 } = { 1,2,3,4 }, c = min c i = 0, i I and L = { i I c i = c = 0 } = { 1,2,4 }. The condition (5) holds since σ (x 1,u) = 0 < 1 0 = d 1 c, σ (x 2,u) = 0 < 1 0 = d 2 c, σ (x 3,u) = < 0 0 = d 3 c, σ (x 4,u) = < 0 0 = d 4 c. Choosing l = 4 L = { 1,2,4 } or example, we replace 4 by 4 = u i (c i c) i = x x = x 1 x 2 + g 1(t) g 2 (t). i I The resulting DAE is ( 1, 2, 3, 4 ) = 0. x 1 x 2 x 3 x 4 c i d x 1 x 2 x 3 x x 2 x Now Val(Σ) = 0 < 1 = Val(Σ). The SA succeeds at all points where det(j) = x 2 x 1 0. From (4) and (6), we can recover the replaced equation l by ( ) l = l i I\{l} u i (c /ul i c) i. Provided u l 0, it is not diicult to show that the original DAE and the resulting one have the same solution (i there exists one); see also [13, 5.3]. I the resulting DAE still has an identically (but not structurally) singular System Jacobian, we can iterate the LC method, provided the condition (5) is satisied. Since each conversion reduces the value o the signature matrix by at least one, the number o iterations does not exceed the original Val(Σ). 4.2 Expression Substitution Method Let v be a nonzero n-vector unction in the kernel o J, that is, v ker(j) and Jv = 0. Denote

7 Symbolic-Numeric Methods or Improving Structural Analysis o DAEs 7 { v 0 }, s = J, M = { i d c i = σ i or some J }, and c = max i M c i. (7) We choose an l J, and introduce s 1 new variables y = x (d c) v x (d l c) v l or all J \ { l }. (8) l In each i with i M, we substitute ( y + v ) x (d (c ci ) l c) v l l or every x (σ i ) with σ i = d c i and J \ { l }. (9) Denote by i the equations that result rom these substitutions, and write i = i or i / M. By (8), we append to these i s the equations 0 = g = y + x (d c) v x (d l c) v l or all J \ { l } (10) l that prescribe the substitutions. Hence the resulting enlarged system consists o ( ) equations 1,..., n = 0 and g = 0 or all J \ { l } in variables x 1,...,x n and y or all J \ { l }. Critical to the ES method is the ollowing theorem. Theorem 2. Let J, s, M, and c be as deined in (7). Assume σ (x,v) { d c 1 i J d c otherwise, and d c 0 or all J. (11) I we introduce s 1 new variables deined in (8), perorm substitutions in i or all i M by (9), and append the equations g in (10), then Val(Σ) < Val(Σ), where Σ is the signature matrix o the resulting DAE. We call the (11) the conditions or applying the ES method. Example 4. We illustrate the ES method with the artiicially constructed DAE below. 0 = 1 = x 1 + e x 1 x 2x 2 + h 1 (t) 0 = 2 = x 1 + x 2 x 2 + x h 2 (t) x 1 x 2 c [ ] i d 1 2 x 1 x 2 [ ] 1 α αx x 2

8 8 Guangning Tan, Nedialko S. Nedialkov, and John D. Pryce Here h 1 and h 2 are given driving unctions, and α = e x 1 x 2x 2. Obviously det(j) = 0 and the SA ails. Suppose we choose v = (x 2, 1) T ker(j). Then (7) becomes { 1,2 }, s = J = 2, M = { 1,2 }, and c = max i M c i = 1. We can apply the ES method as conditions (11) read σ (x 1,v) = 1 = 1 1 1= d 1 c 1, d 1 c = 1 1 = 0 0, σ (x 2,v) = 0 0 = 2 1 1= d 2 c 1, d 2 c = 2 1 = 1 0. For example, we choose l = 2 J. Now J \ { l } = { 1 }. Using (8) and (10), we introduce a new variable y 1 = x (d 1 c) 1 v 1 v 2 x (d 2 c) 2 = x (1 1) 1 x 2 1 x(2 1) 2 = x 1 + x 2 x 2, and append an equation 0 = g 1 = y 1 + x 1 + x 2 x 2. Then we substitute (y 1 x 2 x 2 ) or x 1 in 1 to obtain 1, and substitute y 1 x 2 x 2 or x 1 in 2 to obtain 2. The resulting DAE and its SA results are shown below. 0 = 1 = x 1 + e y 1 +x h 1 (t) 0 = 2 = y 1 + x h 2 (t) 0 = g 1 = y 1 + x 1 + x 2 x 2 x 1 x 2 y 1 c i g d x 1 x 2 y 1 [ ] 1 1 2x 2 β β 2 2x 2 1 g 1 1 x 2 Here β = e y 1 +x 2 2. Now Val(Σ) = 1 < 2 = Val(Σ). The SA succeeds at all points where det(j) = 2β(x 2 + x 2 ) x 2 0. From the steps o applying the ES method, we can undo the expression substitutions to recover the original DAE. Similar to the LC method, the ES method also guarantees that, provided v l 0, the original DAE and the resulting one have the same solution (i there is one); this is shown in [13, 6.3]. From our experience, we usually attempt the LC method irst, and when its condition (5) is violated, we try the ES method. Also, it is desirable to choose an l L [resp. l J] in the LC [resp. ES] method, such that an u l [resp. v l ] never becomes zero. For example, it can be a nonzero constant, x , or 2 + cosx 2. Such a choice o l guarantees that the resulting DAE is equivalent to the original one that is, they always have the same solution, i there exists one. We show below that the LC method does not work on the DAE in Example 4 because the condition (5) is not satisied. Choose u = ( 1,α ) T coker(j), where α = e x 1 x 2x 2. Then (4) becomes I = { i u i 0 } = { 1,2 }, c = min i I c i = 0, and L = { i I c i = c } = { 1 }. Since x 1 and x 2 occur o order 1 and 2 in u, respectively, we have

9 Symbolic-Numeric Methods or Improving Structural Analysis o DAEs 9 σ (x 1,u) = = d 1 c and σ (x 2,u) = = d 2 c. Hence (5) is not satiied. Choosing l = 1 I or example and replacing 1 by 1 = u u 2 2 = x 1 + h 1 (t) + α ( 1 + x 1 + x 2 x 2 + (x 2) 2 + 2x 2 x 2 + h 2(t) ) results in a DAE ( 1, 2 ) = 0. x 1 x 2 c [ ] i d 1 2 x 1 x 2 [ ] 1 γα γαx x 2 Here γ = x 1 + x 2x 2 + (x 2 )2 + 2x 2 x 2 + h 2 (t). The SA ails still, since J is identically (but not structurally) singular. Now Val(Σ) = Val(Σ) = 2. 5 Conclusions and Future Work We proposed two symbolic-numeric conversion methods or improving the Σ- method. They convert a DAE o inite Val(Σ) and an identically (but not structurally) singular System Jacobian to a DAE that may have a nonsingular System Jacobian. A conversion guarantees that both DAEs have (at least locally) the same solution i there exists one. The conditions or applying these methods can be checked automatically, and the main result o a conversion is Val(Σ) < Val(Σ), where Σ is the signature matrix o the resulting DAE. In our experiments, the linear combination (LC) method and the expression substitution (ES) method succeed in ixing many solvable DAEs on which the SA ails. We believe that these methods can help make the Σ-method more reliable. We also conecture that reducing the value o the signature matrix tends to give a better ormulation o a DAE in the SA s perspective and then a nonsingular System Jacobian. An implementation o the conversion methods requires (a) computing a symbolic orm o a System Jacobian J, (b) inding a vector in coker(j) or ker(j), (c) checking the conditions, and (d) generating equations o the resulting DAE. These symbolic computations may seem expensive. Fortunately, we can combine block triangularization techniques [10, 11] with the conversion methods. When J is identically singular, we can locate the diagonal block o which the Jacobian is singular, and perorm a conversion on this block. For DAEs that have a nontrivial block triangular orm (BTF), that is, having more than one diagonal blocks, the symbolic computations can be reduced signiicantly. For problems that do not have very complicated ormulas, the aorementioned symbolic computations can be handled reasonably well by a symbolic tool, such as MATLAB s Symbolic Math Toolbox [14]. We combine this toolbox with our structural analysis sotware DAESA [6, 11], and build a prototype code that applies the

10 10 Guangning Tan, Nedialko S. Nedialkov, and John D. Pryce conversion methods automatically. We aim to produce a solid implementation o these methods and incorporate this eature in a uture version o DAESA. Acknowledgements The authors wish to acknowledge with thanks the undings or supporting their research: GT supported in part by the Ontario Research Fund (ORF, Canada), NSN supported in part by the Natural Sciences and Engineering Research Council o Canada (NSERC), and JDP supported in part by the Leverhulme Trust (UK). GT wishes to thank R. McKenzie or prooreading the drat o this article and providing valuable suggestions. Reerences 1. Barrio, R.: Perormance o the Taylor series method or ODEs/DAEs. Appl. Math. Comp. 163, (2005) 2. Barrio, R.: Sensitivity analysis o ODES/DAES using the Taylor series method. SIAM J. Sci. Comput. 27, (2006) 3. Brenan, K., Campbell, S., Petzold, L.: Numerical Solution o Initial-Value Problems in Dierential-Algebraic Equations, second edn. SIAM, Philadelphia (1996) 4. Mattsson, S.E., Söderlind, G.: Index reduction in dierential-algebraic equations using dummy derivatives. SIAM J. Sci. Comput. 14(3), (1993) 5. Nedialkov, N.S., Pryce, J.D.: Solving dierential-algebraic equations by Taylor series (I): Computing Taylor coeicients. BIT Numerical Mathematics 45, (2005) 6. Nedialkov, N.S., Pryce, J.D., Tan, G.: Algorithm 948: DAESA: A Matlab tool or structural analysis o dierential-algebraic equations: Sotware. ACM Trans. Math. Sotw. 41(2), 12:1 12:14 (2015). DOI / URL 7. Nedialkov, N.S., Tan, G., Pryce, J.D.: Exploiting ine block triangularization and quasilinearity in dierential-algebraic equation systems. arxiv: (2014). 18 pages, download link: 8. Pantelides, C.C.: The consistent initialization o dierential-algebraic systems. SIAM. J. Sci. Stat. Comput. 9, (1988) 9. Pryce, J.D.: A simple structural analysis method or DAEs. BIT Numerical Mathematics 41(2), (2001) 10. Pryce, J.D., Nedialkov, N.S., Tan, G.: Graph theory, irreducibility, and structural analysis o dierential-algebraic equation systems. arxiv: (2014). 24 pages, download link: Pryce, J.D., Nedialkov, N.S., Tan, G.: DAESA: A Matlab tool or structural analysis o dierential-algebraic equations: Theory. ACM Trans. Math. Sotw. 41(2), 9:1 9:20 (2015). DOI / URL Scholz, L., Steinbrecher, A.: A combined structural-algebraic approach or the regularization o coupled systems o DAEs. Tech. Rep. 30, Reihe des Instituts ür Mathematik Technische Universität Berlin, Berlin, Germany (2013) 13. Tan, G., Nedialkov, N.S., Pryce, J.D.: Symbolic-numeric methods or improving structural analysis o dierential-algebraic equation systems. arxiv: [cs.sc] (2015). 84 pages. Available at The MathWorks, Inc.: Matlab Symbolic Math Toolbox (2015). URL