Newton and Quasi-Newton Methods for the Non- - WIT Press · Newton and Quasi-Newton Methods for the Non-linear Richards Equation Claudia Fassino* and Gianmarco Manzini^ i D^arfzmenZo

Newton and Quasi-Newton Methods for the Non-

linear Richards Equation

Claudia Fassino* and Gianmarco Manzini^

i D^arfzmenZo dz Ma^ema^cG; 77

Vergata, Roma, Italy

Istituto di Analisi Numerica—CNR, Pavia, Italy

EMail: [email protected]

Abstract

In this work we investigate the effectiveness and efficiency of Newton, Picard

and quasi-Newton linearizations of the non-linear algebraic problem which is

originated by an RTo — PQ mixed-hybrid formulation of the 2 — D non-linear

Richards' equation. Numerical experiments are shown when the methods

are applied to a stationary and a time-dependent benchmark problem.

1 Introduction

The Richards equation, which is commonly used in modelizing ground-

water flow in partially saturated porous media, can contain some

nonlinearities due to pressure head dependencies in the general stor-age term and in the relative hydraulic conductivity. The RT^ — PQ

mixed-hybrid formulation, reported in Bergamaschi & Putti [5] and

in Fassino & Manzini [3], provides with the following non-linear al-gebraic problem

(1)

where we have introduced the vectors q, t/> and A, which contains the

unknowns associated to the approximate velocity, pressure head and

Transactions on Ecology and the Environment vol 17, © 1998 WIT Press, www.witpress.com, ISSN 1743-3541

586 Computer Methods in Water Resources XII

pressure head trace (lagrangian multipliers), considered at the time-

step t + A£. In eqn. (1) A(ifi) is the pressure head dependent mass

matrix, B is the divergence matrix, C is the lagrangian multipliers

matrix and D(if>} is the pressure head dependent diagonal matrix

arising from an implicit 1^-order backward approximation of the time

derivative of the pressure head. The right-hand-side terms gi, g2(VO?and ga take into account possible sources and Neumann boundary

condition values. In the time-dependent case, we adopted a two-stage

2^-order semi-implicit Runge-Kutta scheme. The non-linear system

in eqn. (1) is solved by an iterative algorithm. Once given an estimate

for A(if>) and D(?/)), the resulting linearized system is solved by a

standard static condensation technique, see Brezzi & Fortin [1], and

a GMRES linear algebra solver, see Kelley [2]. In this work, we assess

and compare the convergence rate performance and computational

efficiency of Newton, Picard and two quasi-Newton linearizations,

based on the fast implementation of the Broyden iterative algorithm,

as proposed in Kelley [2]. Full details are given in the extended

technical report by Fassino & Manzini [3].

The outline of the paper is as follows. In Section 2 we illustrate

the iterative algorithms, in Section 3 we present and compare their

performance and in Section 4 some final considerations are given.

2 The Iterative Solution Methods

The non-linear algebraic problem in eqn. (1) can be rewritten as

)z - 6(z) = 0, (2)

where the following compact notation x = (q, i/>, A)^ has been intro-

duced for the unknown arrays. The matrix M(x) and the vector b(x)

are respectively the matrix operator and the right-hand-side.

2.1 Newton and Picard Linearizations

The classical approach for solving the non-linear problem (2) con-sists in the locally quadratically-convergent Newton method, which

requires at each iteration the computation of the Jacobian matrixJ(x). Starting from a given initial point XQ, the Newton sequence of

the approximate solutions is built by updating the vector % with the

k*h Newton displacement s^ , see Algorithm 1.


Computer Methods in Water Resources XII 587

Algorithm 1 : Newton

1 - Choose XQ.

2 - Repeat from k — 0 until "convergence"

. solve Js = -F% for 6

The Jacobian matrix of F is formally given by

(3)

with the following definitions of derivatives

^ [A(i/,)q] and D i/,) =

The linearly convergent Picard method can be given as a modified

Newton scheme, in which the asymmetric Jacobian matrix J(x ) is

approximated by the symmetric matrix M(%&), and the contributions

of the derivatives of M(x) and b(x) with respect to x are neglected.

The Picard sequence of approximate solutions is thus built as

%k + «4 \ where the fc-th Picard displacement s is the solution of( Pi

the linear problem M(xk)s\ ' = -F(x/J. In order to improve the

convergence behavior, sj can be relaxed by a factor (]& G [0.1, 0.5].

2.2 Preconditioned Fast Broyden Methods

The superlinearly convergent fast Broyden method has been appliedto the following preconditioned form of eqn. (2)

)=0, (4)

where f(x) is an assigned function, see Fassino & Manzini [3].

Algorithm 2 : preconditioned fast Broyden

1 - Choose XQ, BQ = /. Compute d$ = -BQ^F(XQ).2 - Repeat from k = 0 until "convergence"



Each Broyden displacement dk, used to update the fc^ iterative

solution £&, is formally the solution of the linear problem B^d^ =

—F(xk), where B^ is the Broyden approximation of the Jacobian

matrix. In the fast version, dk is computed by the Sherman-Morrison

formula. The initial Broyden matrix B$ should be choosen as an

approximation of the initial Jacobian matrix Jp(x$). Since Jp(x] ~

M~ (f(x}}M(x] the algorithm can start with BQ = / if the function

f(x) is such that /(#o) = o, see Kelley [2].

Algorithm 2 is uniquely determined once the function f(x) is

specified. The simplest possible choice consists in the constant func-

tion f(x) = #o, and the resulting iteration scheme, referred in the

paper as the fast Broyden method, is equivalent to a direct applica-

tion of the Broyden linearization technique to the non-preconditioned

form of the non-linear problem given in eqn. (2), with the same start-

ing solution XQ and the initial matrix BQ — M(XQ). If we set f(x) = x,as proposed in Fassino & Manzini [3], the Broyden method is actually

applied to the non-linear problem F(x) = M~~ (x}F(x) — 0, where/\ t p\

the evaluation of F(xk) = —$k requires the calculation of the local

Picard displacement. This motivates the name Picard-Broyden given

to the resolution strategy through all the present work.

3 Numerical Experiments

The performance of the previously reported resolution strategies hasbeen tested on the stationary and the evolutionary 2-D model prob-

lems, originally presented in Paniconi & Putti [4], and referred under

the labels 2S and 2T. The storage term rj(i/j) and the non-linear rela-

tive conductivity kr(i/j) are modeled by some characteristic relations,whose functional forms are detailed in Paniconi & Putti [4] - see

eqn (9) and eqns (11-12) therein - and in Bergamaschi & Putti [5],

Fassino & Manzini [3].

3.1 The Stationary Model Problem

This model problem is concerned with the calculation of a steady

state flow through a square embankment. In this section we show the

performances in three cases, indicated resp. with 25a, 25*6, and 25c,

differing for the values of the parameters (3 and n, as described in Pan-

iconi & Putti [4], Bergamaschi & Putti [5], Fassino & Manzini [3], and



TABLE 1Parameters used in the model problems

/?n

7

2Sa

1

2

1

2Sb

1

4

1

2Sc J[ 2Ta

2

2

1

1

1

-1

2Tb [

3

4

-3

shown in Table 1. Figure 1 illustrates the convergence curves for the

Newton, the quasi-Newton and the Picard linearizations. In Table 2

we report the performance of the algorithms in all the benchmark

cases. All the computations are performed on a 50 x 50 regular trian-

gularized mesh and stopped after 200 iterations or when the iterative

residual error became less that 10~ Newton and quasi-Newton

methods generally show a local superlinear convergence behavior. In

order to achieve stronger robustness, the starting approximation of

the solution can be improved via some (relaxed) Picard steps. The

crucial parameters are the number of initial Picard steps and the

relaxing factor O. Several combinations of values for these two pa-

rameters were adopted to test the performance dependencies and in

Table 2 we report the best performance results. The two rows of

each entry give the results when the method is respectively applied

without and with some initial (relaxed) Picard iterations. In the "Pi-

card" column, the second row refers, instead, to the relaxed Picard

method. The CPU costs are given in seconds and the relaxing pa-

rameter for the initial Picard iterations takes the value Q = .25, if not

otherwise indicated. It is evident that the Newton method is always

the less expensive solution strategy, even if requiring some Picard it-erations to converge. The quasi-Newton methods are also effective in

terms of CPU cost and convergence rate. However, it is worth noting

that the fast Broyden algorithm always requires to be initialized by

some relaxed Picard iterations. The Picard-Broyden algorithm in-stead converges even if not initialized. Moreover, this latter method

is always less expensive than the fast Broyden one because it is better

preconditioned. The more expensive and less effective method is thePicard one.



TABLE 2Summary of Results for Test Case 2S.

2Sa

2Sb

2Sc

Newton

Niter CPU

10 154

5 +10 73

failed —

10 +8 124

failed —

5^+7 180

Picard

N^r CPU

91 1469

127 2024

174 2583

130 2004

180 2627

131 2004

Pic.-Broy.

N^ CPU

27 453

10 +25 609

37 576

10 +25 703

31 480

10 +31 659

F. Broyden

Niter CPl

failed

20 +24 73

failed

20 +33 82

failed

20 +29 77

() : unrelaxed Picard iters., () • relaxed Picard iters.

10"

o111"TO

'wo>DC

10"

icr

10'

10'50 100

Iterations150 2C

FIGURE 1 - Convergence curves for the solution of the station-

ary model problem, test case 2Sc on a 50 x 50 regular mesh;

N(Newton), PB(Picard-Broyden), FB(fast Broyden), rP(relaxed Pi-card), P (Picard).



3.2 The Evolutionary Model Problem

This model problem is concerned with the calculation of a transient

flow in an unsaturated slab. Two different computations, respec-

tively indicated with the labels 2Ta - 2Tb and whose parameters

are given in Table 1, have been considered, in accord with Pani-

coni & Putti [4]. The approximate solution is advanced in time by

a two-stage 2^-order semi-implicit Runge Kutta method, where the

adaptive timestep strategy described in Bergamaschi & Putti [5] has

been adopted. All the computations are performed on a 60 x 10 reg-

ular triangularized mesh up to the final time T = 5 days, when the

groundwater flow reaches an almost steady state configuration. Each

non-linear iterative process is initialized using the solution from the

previous time step, which for small At is a very good approxima-

tion of the final iterative solution, and stopped when the value of

the iterative residual error becomes less than 10~ . The results are

summarised in Table 3. For each case, we report the total time steps

to reach the final solution, the average, the smallest and the largest

timestep At, the average number of non-linear iterations per timestep and the total CPU costs in seconds.

The Newton and the Picard-Broyden algorithms achieve superlin-

ear convergence rate at a very early stage of the iterative process

and result to be the more effective solution strategies. The Picard

method suffers from stagnation at a number of timesteps, triggering

some backsteppings and timestep size reductions. The fast Broydenmethod, instead, seems to be the less robust of the four ones, oftenrequiring some Picard initializations.

4 Final Remarks

The Newton and quasi-Newton algorithms achieve convergence in asmaller number of iterations than the Picard one, due to their resp.

quadratic and superlinear convergence rates. Since the computa-

tional costs of a Newton, a quasi-Newton and a Picard iterative step

are almost equivalent, the Newton method results the best scheme inthe benchmark problems we considered. Nevertheless, in many situ-

ations the Picard-Broyden linearization has been shown to provide acomparable efficiency and robustness.



TABLE 3Summary of Results for Test Case 2T

2Ta

2Tb

iterative

method

Newton*

Picard

fast Broy.

Pic.-Broy.

Newton

Picard

fast Broy.**

Pic.-Broy.

^tirne

steps

14

20

15

14

14

58

42

28

aver. min. max.

At At At

0.36 0.10 0.89

0.25 0.10 0.74

0.33 0.10 0.89

0.36 0.10 0.89

0.36 0.10 0.89

0.08 0.01 0.46

0.12 0.005 0.57

0.18 0.05 0.53

aver. #it.

per step

4.8

18.0

14.3

14.1

4.8

21.8

19.8

21.2

CPU

(sees)

143

706

401

417

141

2273

1541

1238

= initialized by one Picard step, ** = initialized by 5 Picard steps)

Acknowledgements

The authors thank Dr M. Arioli, Dr L. Bergamaschi, Dr C. Paniconi

and Dr M. Putti for many fruitful discussions and suggestions.

References

[1] F. Brezzi and M. Fortin. Mixed and Hybrid Finite Element Meth-

ods. Springer Verlag, Berlin, 1991.

[2] C.T. Kelley. Numerical Methods for Unconstrained Optimization

and Nonlinear Equations. Prentice-Hall, Englewood Cliffs, N.J.,

1995.

[3] C. Fassino and G. Manzini. Non-linear iterative methods for the

Richards' equation. Technical Report IAN-CNR-1073, 1997.

[4] C. Paniconi and M. Putti. A comparison of Picard and Newtoniteration in the numerical solution of multidimensional variably

saturated flow problems. Water Resources Research, 30:3357-3374,1994.

[5] L. Bergamaschi and M. Putti. Mixed finite elements and Newton-

type linearizations for the solution of Richards' equation, submit-

ted to Int. J. Numer. Meth. Engng., 1997.


Documents

Newton and Quasi-Newton Methods for the Non- - WIT Press · Newton and Quasi-Newton Methods for the Non-linear Richards Equation Claudia Fassino* and Gianmarco Manzini^ i D^arfzmenZo