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1 Journal of ELECTRICAL ENGINEERING, VOL. 52, NO. /s, 2, 48{52 COMPARISON OF BROYDEN AND NEWTON METHODS FOR SOLVING NONLINEAR PARABOLIC EQUATIONS Ivan Cimrak If the time discretization of a nonlinear parabolic dierential equation is realized by Rothe's method then the problem of solving degenerate elliptic equations occurs. After performing a space discretization on each time level the problem can be transformed into solving large systems of nonlinear equations. In this paper Broyden's method and NewtonKantorovich method are used for solving such systems. The numerical implementations of both methods are compared from the point of view of time eectiveness. Several numerical experiments comparing the Broyden and NewtonKantorovich methods are studied. K e y w o r d s: Degenerate parabolic equation, slow diusion, quasinewton method, Broyden method, Newton Kantorovich method, Barenblatt solution. 2 Mathematics Subject Classication: 35K65, 65P5, 65M6 INTRODUCTION The purpose of this paper is to compare computational results of two methods for solving nonlinear degenerate parabolic equations. Computational eciency of Newton's method is compared with the Broyden quasi Newton method. These methods are used for solving a degenerate nonlinear elliptic problem in each time step level. We will be mainly concerned with the following nonlinear parabolic equation: t u (u) = ; in (; T ) ; () where is a smoothly bounded domain in R N. A solution to () is subject to the Dirichlet boundary condition and initial condition u = ; (2) u(x; ) = u (x) : (3) The initialboundary value problem (){(3) will be denoted by (P ) hereafter. We assume that the function satises the assumption (A) is C smooth function such that. Notice that in the case () = the problem (P ) is a degenerated parabolic equation. In general, a solution to (P ) need not be necessarily smooth and that is why we have to deal with weak solutions only. Denition. A function u is called a weak solution to the problem (P ) i (i) u 2 L 2 (I; H ()) \ L (I ); (ii) u satises the integral identity Z T Z (u(x; t) u (x))v t (x; t) r(u(x; t))rv(x; t) dx dt = ; for all v 2 L 2 (I; H ()) such that v t 2 L (I ) and v(; T ) =. It is worth to note that the problem (P ) has at most one weak solution (see [9]). In this paper we investigate numerical aspects of Broyden's and NewtonKantorovich methods. We implement both methods and compare their eciency and computational time needed for numerical results of the same precision. In [7] a proof of the existence of a solution as well as convergence of Rothe's method used for solving parabolic equations of the form () has been shown. In [6] authors suggested a linear approximation scheme for solving the problem ( P ) and they proved the convergence of approximative solutions to an exact solution. For the numerical implementation of this scheme we refer to [8]. The comparison of this scheme and Broyden's method has been studied in [2]. Recall that the Broyden method as a part of quasinewton methods is often used in optimization problems like, e.g., minimization of nonlinear functionals in nite dimensional spaces. In the context of nonlinear parabolic equations, the Broyden method has been successfully applied as a tool for minimizing penalty functionals in inverse problems. We refer to recent papers by Soria and Pegon [] and Yu [] for applications of Broyden's method used for solving nonlinear systems arising from optimization problems for nonlinear parabolic equations. However, inspection of the literature shows that this method has been rarely used for solving Department of Mathematical Analysis, Ghent University, Galglaan 2, 9 Ghent, Belgium, cimocage.rug.ac.be ISSN c 2 FEI STU
2 Journal of ELECTRICAL ENGINEERING VOL. 52, NO. /s 2 49 nonlinear systems of equations in innite dimension such as nonlinear elliptic problems. On the other hand, the Newton method and the socalled Jager Kacur method attracted a lot of attention with respect to numerical solution of degenerate parabolic equations and systems. In [4] Fuhrmann investigated a numerical solution to nonlinear parabolic equations like, e.g., Richard's equation or the equation describing the motion of a viscous compressible uid in a porous medium. Special attention was paid to problems with coecients varying in space. In his approach the solution scheme is based on an implicit time discretization combined with Newton's method. For the solution of linear problems, iterative methods are used. The author compares in [3] the Newton method with linear approximation scheme of Jager and Kacur introduced in [6]. TIME DISCRETIZATION As usual, time discretization of a nonlinear parabolic equation () is done by using standard Rothe's step functions. The time interval I = h; T i will be divided into n subintervals ht i ; t i i such that jt i t i j = = n. The time derivative u t is approximated by the backward Euler approximation ui ui. For each xed n we therefore have to solve n elliptic problems of the form u i u i (u i ) = ; (4) u i = : (5) known when computing the new time level v i. If we denote F (v) := v v i f (v) (6) then our aim is to nd a solution v to the following nonlinear equation: F (v) = ; F : R K! R K : (7) 2 SOLUTION IN TIME LEVEL It is dicult to nd an exact solution to the problem (7) in general. There are several numerical methods used in order to nd approximation of the exact solution. We will be concerned with iterative schemes of Newton and quasinewton types. Hereafter, we will denote all inner iterations of these schemes by an upper index. We will seek for a solution v to (7) as a limit v = lim k! v k, where fv k g k= is a suitable sequence of approximations of v constructed via either Newton's or quasinewton's (Broyden's) method. The NewtonKantorovich method This method is commonly used in solving nonlinear problems of the form (7). The socalled Newton's iterations are dened by v k+ = v k [F (v k )] F (v k ) : (8) By solving the above elliptic problems we obtain functions u ; u ; : : : ; u n. Using these functions one can construct Rothe's step function u n : u n (x; t) = u i (x) if t 2 ht i ; t i i : It was proved in [7] that the sequence fu n g converges to a weak solution to the problem (P ) in the norm of the space L 2 (I; H ). Now the main problem is how to nd functions u i. One can discretize the elliptic problem by using several methods, e.g., Finite Elements Method, Finite Dierences Method and others. In any case, the space of functions in which we look for u i is approximated by nite dimensional spaces equivalent to R K. After time and space discretization is performed, functions u i are approximated by vectors v i = (vi ; v2 i ; : : : ; vk i ) 2 RK and problem (P ) is reduced to the following problem v i v i f (v i ) = ; (R) with respect to the approximation of the Dirichlet boundary condition. The operator f approximates the Laplace operator. Of course, it depends on the way how the space discretization has been performed. It is important to emphasize that the vector v i is already We refer to a book [5] by Hutson for a proof of local quadratic convergence of Newton's iterations to a solution of equation (7). Let us recall that assumptions needed for the proof of local quadratic convergence require that the norm kf (v )k is suciently small. This can be guaranteed by taking v = v i and assuming that the time discretization step > is small enough. Broyden's method This method is based on the socalled Broyden's update formula for quasinewton iterations: v k+ = v k B k F (vk ) : (9) In the NK method matrices we have B k = F (v k ). In Broyden's method, B k, k = ; 2; : : :, represent only approximations of the Jacobi matrix F (v k ). If we denote s k = v k+ v k ; y k = F (v k+ ) F (v k ) ; then the Broyden update for a new approximation of the Jacobi matrix B k+ is given by B k+ = B k + y k B k s k ks k k 2 s k > :
3 5 I. Cimrak: COMPARISON OF BROYDEN AND NEWTON METHODS FOR SOLVING NONLINEAR PARABOLIC EQUATIONS t x.25.5 Fig.. The exact Barenblatt solution.8 Table. Computational time in the D problem for the spatial grid size h = :35. NK method Broyden's method :5 :66 : :25 :28 :225 :6 :89 :53 :25 :25 :745 Table 2. Computational time in the D problem for the time step = :25. h NK method Broyden's method :7 :68 :349 :466 :98 :78 :35 :28 :225 :28 :57 2:9 :233 :87 2:826 :2 :26 3:98 results based on this method. Otherwise, we have to pay for this precision with more operations needed to compute new iterates of the NK method compared to relatively cheap implementation requirements of the Broyden method. We have considered a simple example of a diusion equation with a known exact solution. Let us consider the following equation t u u m = ; x 2 R N ; t 2 (; T i ; () where m >. This equation can be used in modeling slow diusion problems. Denote z(x; t; a; ) = " (t + ) k a 2 Cjxj2 (t + ) 2k N # m N where [] + = max[; ], k =, C = k(m ) N(m )+2 2Nm and a; > are positive constants. Taking u(x; t) = z(x; t; a; ) one can easily verify that such a function u is a solution to () with the initial condition u(x; ) = z(x; ; a; ). This explicit solution is referred to as the Barenblatt solution to (). The support fx 2 R N j u(x; t) > g of the solution is a bounded subset of R N for every t. (See Fig..) We have tested this example in one and in two spatial dimensions. We focused on the eectiveness of the two methods. In every example we set parameters such as the number of NK or Broyden's iterations to gain the same precision of numerical solution. Then we measured the computational time needed for both methods. + ; Thanks to ShermanMorrison formula (see Allgower and Georg []) we can directly compute B k+ if B k is known. That is the main advantage in comparison to solving systems using NK method. No systems of linear equations with large matrices have to be solved. Such a formula can be derived by the following nice geometric motivation discussed in more detail in []. In this book one can also nd a proof of a local superlinear convergence of Broyden's iterates to the root of (7). The assumptions needed for the proof of a local superlinear convergence require closeness of the initial iterate v and the root v. Again, this requirement can be guaranteed by taking v = v i and assuming <. 3 COMPARISON OF BOTH METHODS In this section we compare the results of implementation of both the NewtonKantorovich and Broyden methods for solving one time step level problems in time discretization of the degenerate parabolic problem. Since theoretical results indicate a better rate of convergence for the NK method, we expected more precise numerical One spatial dimension The Laplace operator xx was approximated by nite dierences dened on a uniform spatial mesh with the grid size equals to h. Such an approximation of xx yields a simple three diagonal matrix F from (8). Therefore we can use a fast solver for a linear system of equations with a three diagonal matrix. The time needed to compute numerical solution is apparently shorter for NK scheme. (See Tables and 2). By using Broyden's scheme we need to compute several scalar products in each time step and this is why, in one spatial dimension, Broyden's method turned to be less eective than the NK method. Since the support of the Barenblatt solution is bounded, for any time t > we can take suciently large and we can assume zero Dirichlet boundary conditions. We performed experiments in two ways. First we investigated the dependence of the computational TIME for a xed grid size h > and varying time steps. Next we have xed x and we varied the grid size h >. In order to compare precisions of the two methods we used discrete approximations of continuous norms of the Banach spaces L 2 (), L and W 2 () spaces. Let
4 Journal of ELECTRICAL ENGINEERING VOL. 52, NO. /s 2 5 In Tab. 2 the tolerances were set as follows: : < 3 ku NK u g k L 2() < 4:689 ; Fig. 2. The initial prole Table 3. Computational time in the 2D problem for the spatial grid size h = :6. NK method Broyden's method :5 3:89 :55 :25 7:97 2:3 :6 9:8 3:45 :25 3:372 4:6 Table 4. Computational time in the 2D problem for the time step = :5. h NK method Broyden's method :6 3:89 :55 :52 4:433 :278 :45 5:54 :47 :39 5:773 :566 :33 6:59 :78 :28 7:32 :844 us dene the following discrete norms of a vector u = (u ; u 2 ; : : : ; u K ): kuk Lp = n kuk L = max K i= nx i= ju i j p p ; () (ju i j) ; (2) :22524 < 3 ku B u gk L2() < 4:5984 ; :388 < 2 ku NK u gk L() < 2:978 ; :378 < 2 ku B u g k L() < 2:967 : Two spatial dimensions The Laplace operator xx + yy was approximated by nite dierences dened on a uniform spatial mesh with a grid size h > in each direction. Such an approximation leads to a more complicated matrix F (see (8)). The matrix is only block diagonal and it was not possible to use a fast solver for this system. When the solver for full matrices is used, the computational time in 2D for NK method is about 2 times longer than for Broyden's method. Then, we used the standard package meschach for operations with sparse matrices. Some problems with indefinite matrices occur and the software had to be modied. The computational time needed to compute a numerical solution is rapidly growing for NK if we enlarge spatial discretization (see Tabs. 3 and 4). Broyden's scheme requires less operations and it seems to be more ecient compared with NK scheme in 2D. While solving problems in 3D, there are extremely large matrices and using of Broyden's scheme could be very eective. We used discrete L p norms from (2) and () to guarantee the same precision of the methods again. In the case which is shown in Tab. 3 the controlled difference between numerical solutions u NK ; u B and exact solution u g at grid points was estimated by 2:4362 < 2 ku NK u gk L2() < 2:443 ; 2:4344 < 2 ku B u g k L 2() < 2:4378 ; 9:3359 < 2 ku NK u g k L() < 9:3377 ; 9:339 < 2 ku B u gk L() < 9:337 : For the comparison of computational times shown in Tab., as a stopping criterion we assumed the following tolerances for various norms of dierences of numerical solutions u NK ; u B and the exact solution u g : 2:2866 < 3 ku NK u g k L 2() < 2: ; 2:568 < 3 ku B u g k L 2() < 2:986 ; :2984 < 2 ku NK u gk L() < :3448 ; :34 < 2 ku B u g k L() < :3433 : In the last case (see Tab. 4) we obtained the following estimates: :63896 < 2 ku NK u g k L 2() < :93294 ; :63697 < 2 ku B u g k L 2() < :9399 ; 7:3388 < 2 ku NK u gk L() < 8:86 ; 7:336 < 2 ku B u g k L() < 8:7947 :
5 52 I. Cimrak: COMPARISON OF BROYDEN AND NEWTON METHODS FOR SOLVING NONLINEAR PARABOLIC EQUATIONS t= s t=.33 s t=.66 s t=. s t=.33 s t=.66 s Fig. 3. The time evolution of the interface 4 OTHER NUMERICAL EXPERIMENTS We solve one more equation with unknown exact solution (taken from [8]). Let us consider a parabolic equation with homogeneous Dirichlet boundary conditions 2 t u x 2 (um )+ 2 y 2 (um ) +Cu p = ; x 2 ; t 2 (; T ); where = ( L; L) ( L; L) and p; C are positive constants. We consider an initial prole u(x; ) = u (x) (see Fig. 2) where u (x) = 8 < : ; i h x = ; y = ; (x2 +y 2 ) 2 ; xy 6= : (x 6 +y 6 ) 2 The parameters were chosen: m = 2 ; p = :5 ; c = 5 ; = :3 ; T = :2 ; L = :5 ; h = 4 : The evolution of the interface sptfu(; t)g is depicted in Fig. 3. Acknowledgment + [3] FUHRMANN, J. : Numerical Solution Schemes For Nonlinear Diusion Problems Based on Newton's Method, Proceedings of contributed papers and posters, ALGORITMY'97, Conference on Scientic Computing, West Tatra Mountains, September 25 (997) pp. 32{4. [4] FUHRMANN, J. : On Numerical Solution Methods for Nonlinear Parabolic Problems, Vieweg. Notes Numer. Fluid Mech. 59 (997), 7{8. [5] HUTSON, V. PYM, J. S. : Applications of Functional Analysis and Operator Theory, Academic Press, London, 98. [6] J AGER, W. KACUR, J. : Solution of Porous Medium Type Systems by Linear Approximation Schemes, Numer. Math. 6 (99), 47{427. [7] KACUR, J. : Method of Rothe in Evolution Equations, BSB Teubner Verlag, Leipzig, 985. [8] MIKULA, K. : Numerical Solution of Nonlinear Diusion with Finite Extinction Phenomenon, Acta Math. Univ. Comenianae 64 No. 2 (995), 73{84. [9] OTTO, F. : L Contraction and Uniqueness for Quasilinear EllipticParabolic Equations, Journal of Dierential Equations 3 (996), 2{38. [] SORIA, A. PEGON, P. : QuasiNewton Iterative Strategies Applied to the Heat Diusion Equation, Int. J. Numer. Methods Eng. 4 (99), 66{677. [] YU, W. : A QuasiNewton Method in InniteDimensional Spaces and its Application for Solving a Parabolic Inverse Problem, J. Comput. Math. 6 No. 4 (998), 35{38. Received 29 May 2 Revised October 2 The author is greatly indebted to D. Sevcovic for his constructive remarks. References [] ALLGOWER, E. L. GEORG, K. : Numerical Continuation Methods, Springer Verlag, Berlin, 99. [2] CIMR AK, I. SEVCOVIC, D. : Application of Broyden's Method for Solving Nonlinear Degenerate Parabolic Equations, submitted. Ivan Cimrak (Mgr) is a graduate student in mathematics at the Department of Mathematical Analysis of the Faculty of Engeneering, University of Ghent, Belgium. His supervisor is prof. Roger Van Keer. His research interests include electromagnetism, magnetic hysteresis, FEM. As a high school student he parcitipated in International Mathematical Olympiads in Toronto (995) and in Bombay (996) and he won bronze medals. In 2 he nished study of Numerical Analysis at the Faculty of Mathematics, Physics and Informatics, Comenius University, Bratislava, Slovakia.
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