linearization or discretization?

For nonlinear infinite dimensional equations, which to begin with: linearization or discretization? Laurence Grammont∗, Mario Ahues†, Filomena D. d’Almeida‡ Dedicated to the memory of Alain Largillier

Abstract To tackle a nonlinear equation in a functional space, two numerical processes are involved: discretization and linearization. In this paper we study the differences between applying them in one or in the other order. Linearize first and discretize the linear problem will be in the sequel called option (A). Discretize first and linearize the discrete problem will be called option (B). As linearization scheme, we consider the Newton method. It will be shown that, under certain assumptions on the discretization method, option (A) converges to the exact solution, contrarily to option (B) which converges to a finite dimensional solution. These assumptions are not satisfied by the classical Galerkin, Petrov-Galerkin and collocation methods, but they are fulfilled by the Kantorovich projection method. The problem to be solved is a nonlinear Fredholm equation of the second kind involving a compact operator. Numerical evidence is provided with a nonlinear integral equation. Keywords: Nonlinear equations, Newton-like methods, Kantorovich projection approximation, integral equations. AMS Classification: 65J15, 45G10, 35P05 ´ Universit´e de Lyon, Institut Camille Jordan, UMR 5208, 23 rue du Dr Paul Michelon, 42023 Saint-Etienne Cedex 2, France. [email protected] † ´ Universit´e de Lyon, Institut Camille Jordan, UMR 5208, 23 rue du Dr Paul Michelon, 42023 Saint-Etienne Cedex 2, France. [email protected] ‡ CMUP − Centro de Matem´ atica and Faculdade Engenharia da Universidade Porto, Rua Roberto Frias, 4200-465 Porto, Portugal. [email protected] ∗

1

1

Introduction

We consider a complex Banach space X and a nonlinear Fr´echet differentiable operator F : O ⊆ X → X defined on a nonempty open set O of X . The problem is set as (1)

Find ϕ ∈ O :

F (ϕ) = 0,

where 0 is the null vector of X . The exact Newton method in function spaces leads to the Newton sequence (ϕ(k) )k≥0 defined through the relation : (2)

F 0 (ϕ(k) )(ϕ(k+1) − ϕ(k) ) = −F (ϕ(k) ),

ϕ(0) ∈ O.

(For convergence results on Newton method, see the slide of the conference of C. Villani [15] or the book of I.K. Argyros [4]). Three options are to be considered: (NK) Solve (2) in exact arithmetic (i.e. with so-called analytical methods). (A) Discretize (2) and solve a finite-dimensional linear problem. (B) Discretize (1), apply Newton’s method to the discrete nonlinear problem, and solve the corresponding finite dimensional linear problem. The first option is often impossible to perform, so one has to choose between option (A) and option (B). The treatment of (1) depends on the kind of operator involved in the definition of F . If F involves compact operators as integral operators, projection and iterated projection methods or numerical quadrature rules are well known techniques to compute approximate solutions to such a nonlinear operator equation : in [12], the author explains how to build a sequence of approximate solutions ϕn of the operator equation ϕ = Kϕ, where K is a nonlinear operator in X and ϕn is the solution of an approximate operator equation ϕn = Kn ϕn . He analyses the error estimates for Kn = πn K where (πn )n≥1 is a sequence of linear projections onto finite dimensional subspaces. In [13], chap4, pp 244. and [14], the authors approach the solution with that of a perturbed Galerkin equation ϕn = πn Kϕn + Sn ϕn where Sn is some other nonlinear operator. We can also find in [6] a comparison between the Galerkin approxG imation, obtained as a solution of the equation ϕG n = πn K(ϕn ), and the iterated Galerkin S G approximation defined as ϕn := K(ϕn ). The authors prove, under appropriate assumptions, S that both kϕG n − ϕk and kϕn − ϕk tend to 0 as n → ∞ and they give corresponding error estimates. In [9], the authors propose accelerated projection and iterated projection methods. They consist in decomposing the equation into two components, a finite dimensional one and an infinite dimensional one. In all the above mentioned papers, one notices that methods start with a discretization procedure which leads to a nonlinear approximate equation in a finite dimensional linear space. Next, in these papers, the authors apply a numerical scheme to treat that nonlinear equation, such as Newton method or some of its many variants. This scheme

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involves at each step the resolution of an n×n updated linear system. This method corresponds to option (B). If the nonlinear operator F is sufficiently smooth, we suggest to proceed in the opposite sense: First, linearize the original equation in the infinite dimensional context, and then solve numerically the underlying sequence of linear equations using some discretization scheme such as a numerical quadrature rule, if the function F involves an integral operator, or a projection method. This method corresponds to option (A). In the domain of numerical PDEs, operators are not compact. Many papers have been published in the last decades on that issue (see [3], [8], [10] and [16]). In them the discretization procedure is a Galerkin or a Petrov-Galerkin method in which case option (A) and option (B) coincide ([16]). The notion of mesh independence is not linked to option (A) or (B) as it could seem at the first glance : “Mesh independence of Newton’s method means that Newton’s method applied to a family of finite dimensional discretizations of a Banach space nonlinear operator equation behaves essentially the same for all sufficiently fine discretizations” [10]. So in this domain, with this particular discretization, the linearization and discretization commute and our question has no interest. For Fredholm equations of the second kind involving a compact operator, there exist numerical methods which are more efficient than the Galerkin or Petrov-Galerkin method. The aim of our paper is to exhibit a class of discretization scheme for which linearization and discretization do not commute. Under certain conditions on the discretization scheme and on the nonlinear operator F , we prove that option (A) behaves better than option (B) for a same and fixed discretization level. The paper is organized as follows: In section 2, we present the main theoretical result which is that, under suitable assumptions on the operator and on the discretization process, if a Newton type method is applied first and the discretization process of order n is used at each step of the Newton like method, then the sequence of iterates converges to ϕ, the exact solution of (1), for any fixed integer n large enough. This means that one can attain any desired accuracy by employing a suitable discretization method of such a fixed order n. In section 3, we illustrate this result with an application to a nonlinear Fredholm equation of the second kind involving a compact operator K and a given function f : F (x) := x − K(x) − f. We discretize (2) with the so-called Kantorovich scheme (see [2], page 186) : (k+1) (k) (k) (I − πn K 0 (ϕ(k) − ϕ(k) n ))(ϕn n ) = −ϕn + K(ϕn ) + f,

where (πn )n≥1 is a sequence of linear bounded projections. We remark that πn acts only on K 0 , which makes a significant difference with respect to Galerkin, Petrov-Galerkin or collocation schemes in which πn acts on the whole equation : (k+1) (k) (k) πn [(I − K 0 (ϕ(k) − ϕ(k) n ))(ϕn n )] = πn [−ϕn + K(ϕn ) + f ].

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The scheme, which we propose in our paper, converges when K is compact (and hence K 0 too) and the sequence (πn )n≥1 is pointwise convergent to the identity operator I. Our work shows that options (A) and (B) are not equivalent and that (A) should be preferred to (B). Then we exhibit some numerical examples involving an integral operator, confirming in practice our theoretical results. In section 4, we give the discrete version of option (A). In section 5, we list some concluding remarks and provisory conclusions. In this paper, L(X ) denotes the real Banach algebra of all bounded linear operators from X into itself, Bρ (u) denotes the closed ball with center u ∈ X and radius ρ > 0 and F 0 (x) denotes the Fr´echet derivative of F at x.

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Option (A) vs Option (B)

2.1

Theoretical analysis of option (A)

The solution of problem (1) is characterized as the limit of the sequence (ϕ(k) )k≥0 defined through the relation (3)

F 0 (ϕ(k) )(ϕ(k+1) − ϕ(k) ) = −F (ϕ(k) ),

ϕ(0) ∈ O.

These iterations are functional linear equations. To solve them numerically, in most cases, we apply a discretization scheme. For example, if F is a Fr´echet differentiable nonlinear integral operator then, for each x ∈ O, F 0 (x) is a linear integral operator, and the discretization process could be a numerical quadrature such as the Nystr¨om scheme or a projection method such as Kantorovich scheme. If F 0 (ϕ(k) ) is invertible, equation (3) can be rewritten as ϕ(k+1) = ϕ(k) − F 0 (ϕ(k) )−1 F (ϕ(k) ). Let Σn : O → L(X ) be such that, in some discretized sense, for each x ∈ O, Σn (x) is an approximation to F 0 (x)−1 . Then corresponding discretized iterates satisfy:

(4)

ϕ(0) n ∈ O,

(k) (k) ϕ(k+1) = ϕ(k) n n − Σn (ϕn )F (ϕn ).

We give sufficient conditions on the approximate operator Σn to ensure the cited conditions. For this purpose, we will interpret equation (4) as a Newton-like method. Theorem 1 A priori convergence theorem Suppose that F , O, ϕ ∈ O, µ > 0, R > 0, ` > 0 and α ∈]0, 1] are such that: 1. F (ϕ) = 0, F 0 (ϕ) is invertible and kF 0 (ϕ)−1 k ≤ µ, 2. The closed ball BR (ϕ) is included in the open set O, and F 0 : O → L(X ) is (`, α)−H¨ older continuous on BR (ϕ),

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3. For n r := min R,

o 1 , (2µ`)1/α

there exists γn ∈]0, 1[ such that sup kI − Σn (x)F 0 (x)k ≤ γn , x∈Br (ϕ) (0)

4. The starting approximation ϕn is chosen in the closed ball Bρn (ϕ), where n  ρn := min r,

1 − γn 1/α o . 4`µ(1 + γn )

(k)

Then, for all k, ϕn ∈ Bρn (ϕ), and kϕ(k) n − ϕk ≤ ρn

 1 + γ k n

2

→0

as k → ∞.

Proof : We first prove that, for all x ∈ Br (ϕ), F 0 (x) is invertible and kF 0 (x)−1 k ≤ 2µ. In fact, for all x ∈ Br (ϕ), F 0 (x) = F 0 (ϕ) + F 0 (x) − F 0 (ϕ) = F 0 (ϕ)[I + F 0 (ϕ)−1 (F 0 (x) − F 0 (ϕ))]. Since

1 kF 0 (ϕ)−1 (F 0 (x) − F 0 (ϕ))k ≤ kF 0 (ϕ)−1 k kF 0 (x) − F 0 (ϕ)k ≤ µ`rα ≤ , 2 we conclude that F 0 (x) is invertible and that its inverse is uniformly bounded on Br (ϕ):  −1 F 0 (x)−1 = I + F 0 (ϕ)−1 (F 0 (x) − F 0 (ϕ)) F 0 (ϕ)−1 , so kF 0 (x)−1 k ≤ µ

∞ X

kF 0 (ϕ)−1 (F 0 (x) − F 0 (ϕ))kk ≤ 2µ.

k=0

Concerning Σn (x), we remark that, for all x ∈ Br (ϕ), Σn (x) = F 0 (x)−1 − (I − Σn (x)F 0 (x)))F 0 (x)−1 hence kΣn (x)k ≤ 2µ(1 + γn ). Since

  (k) (k) (k) ϕ(k+1) − ϕ = ϕ − ϕ − Σ (ϕ ) F (ϕ ) − F (ϕ) , n n n n n

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and F (ϕ(k) n ) − F (ϕ) =

Z

1

  F 0 (1 − t)ϕn(k) + tϕ (ϕ(k) n − ϕ) dt,

0

then ϕ(k+1) n

Z 1h i 0 (k) −ϕ= I − Σn (ϕ(k) )F ((1 − t)ϕ + tϕ) (ϕ(k) n n n − ϕ) dt. 0

Let

(k) F 0 (ϕn )

(k)

be added to and subtracted from F 0 ((1 − t)ϕn + tϕ). We get

ϕ(k+1) n

−ϕ =

Z 1h

i I − Σn (ϕn(k) )F 0 (ϕn(k) ) (ϕn(k) − ϕ) dt +

0

Z 1   0 (k) 0 (k) ) + Σn (ϕ(k) F ((1 − t)ϕ + tϕ) − F (ϕ ) (ϕ − ϕ(k) n n n n ) dt, 0

and (k) kϕ(k+1) − ϕk ≤ kI − Σn (ϕn(k) )F 0 (ϕ(k) n n )k kϕn − ϕk + Z 1 0 (k) (k) kF 0 ((1 − t)ϕ(k) +kΣn (ϕ(k) )k kϕ − ϕk n + tϕ) − F (ϕn )k dt. n n 0 (k)

(k)

(k)

Let ϕn ∈ Bρn (ϕ). Then kI − Σn (ϕn )F 0 (ϕn )k ≤ γn , and since Br (ϕ) is convex, for t ∈ [0, 1], (k) (1 − t)ϕn + tϕ ∈ Br (ϕ) and 0 (k) (k) α kF 0 ((1 − t)ϕ(k) n + tϕ) − F (ϕn )k ≤ `kϕn − ϕk .

Hence   (k) (k) α kϕ(k+1) − ϕk ≤ kϕ − ϕk γ + 2µ`(1 + γ )kϕ − ϕk n n n n n ≤

1 + γn (k) kϕn − ϕk. 2 (k+1)

Since 1 + γn < 2 the previous inequality implies that ϕn kϕ(k) n − ϕk ≤ ρn

 1 + γ k n

2

→0

∈ Bρn (ϕ) and that

as k → ∞.

The proof is complete. (k)

Remark 1 Notice that the sequence (ϕn )k≥0 tends to ϕ which does not depend on n. This means that we are not constrained by the value of n, provided the assumptions are satisfied. We (k) will see in section 3 that the numerical computation of ϕn may require to solve an n×n linear (k) system. That is, by solving k linear systems of order n, we get an appoximation ϕn of any desired accuracy, if k is large enough.

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In [11], option (A) is applied to a Fredholm equation of the second kind ϕ − K(ϕ) = f , involving an integral operator K, with the Nystr¨om method as the discretization process. In section 3, we propose to discretize with a projection method - the Kantorovich projection method - fulfilling the assumptions of the theorem. If the classical Galerkin method or collocation method, built upon the finite rank projection πn , (k) are applied to equation (3), then option (A) leads to a sequence (ϕn )k≥0 in Xn , the range of πn , satisfying (k+1) (k) (I − πn K 0 (ϕ(k) − ϕ(k) ϕ(0) ∈ O. n ))(ϕn n ) = −πn F (ϕ ), (k)

(k)

Then Σn (ϕn ) := (I−πn K 0 (ϕn ))−1 πn is not invertible so that the condition kI−Σn (x)F 0 (x)k ≤ γn will not be satisfied.

2.2

Option (B)

There is no possible general analysis of option (B). We can only state that, starting with some discretization scheme in an n-dimensional subspace of X applied to problem (1), we are led at some stage of computations to solve a nonlinear system of equations Find x(∞) ∈ Cn×1 : n

Fn (x(∞) n ) = 0.

This problem will be linearized and become a sequence of linear updated systems. For instance, under suitable conditions, the Newton method can be applied. With such a method, a sequence (k) (xn )k≥0 in Cn×1 is built with the recursion formula (k+1) (k) − x(k) F0n (x(k) n )(xn n ) = −Fn (xn ). (0)

Sufficient conditions on the starting point xn for the convergence of this sequence may be (k) found, for instance, in [1]. Next, the vector xn should allow us to compute a function ψn which will be called an n-order approximation of ϕ.

3 Kantorovich Projection Approximation for Equations of the Second Kind Let K : O → X be a Fr´echet-differentiable nonlinear compact operator and T := K 0 denote the Fr´echet d´erivative of K. Remark 2 We have chosen T to denote K 0 because the methods presented in this paper involve an approximation of K 0 built through an operator of finite rank n. A symbol such as Kn0 is ambiguous: it may denote both the derivative of some operator Kn or an n-order approximation of K 0 . The advantage of writing T for K 0 is that Tn will always denote some n-order approximation of T .

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The problem is Given f ∈ X , find ϕ ∈ O :

(5)

ϕ − K(ϕ) = f.

Problem (5) will be handled as Find ϕ ∈ O :

F (ϕ) = 0,

where, for all x ∈ O, F (x) := x − K(x) − f,

F 0 (x) = I − T (x).

and hence

Let πn be a projection onto an n-dimensional subspace of X spanned by an ordered basis en := [en,1 , . . . , en,n ] ∈ X 1×n . Let e∗n := [e∗n,1 , . . . , e∗n,n ] ∈ (X ∗ )1×n be an ordered basis of the annihilator of the null space of πn which is adjoint to [en,1 , . . . , en,n ]. Each e∗n,j is a bounded semi-linear functional and πn is characterized by πn x =

n X hx , e∗n,j ien,j ,

x ∈ X,

j=1

where hx , e∗n,j i := e∗n,j (x). Remark that, for all j ∈ [[1, n ]], πn∗ e∗n,j = e∗n,j .

πn en,j = en,j ,

We shall use the following notations which allows us to simplify the description of matrices and linear combinations: en x :=

n X

x(j)en,j for all x ∈ Cn×1 ,

j=1

h, e∗n

(i, j) := hhj , e∗n,i i for all h := [h1 , . . . , hp ] ∈ X 1×p , Lx := [Lx1 , . . . , Lxm ] for all x := [x1 , . . . , xm ] ∈ X 1×m , and all L : X → X .

For example, with such notations, πn x = en x, e∗n ,

3.1

x ∈ X.

Option (A)

With option (A), we apply the Kantorovich projection discretization to the linear operator equation issued from the Newton scheme: (6)

Find ϕ(k+1) ∈X : n

(k+1) ϕ(k+1) − πn T (ϕ(k) = gn(k) , n n )ϕn

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where (k) gn(k) := K(ϕn(k) ) − πn T (ϕ(k) n )ϕn + f.

We suppose that      

(7)

    

(i) (ii) (iii)

Equation (5) has a unique solution ϕ ∈ O, I − T (ϕ) is invertible, T : O → L(X ) is `-Lipschitz.

In the following, we prove that the assumptions of Theorem 1 are fulfilled. Let µ > 0 and R > 0 be such that k(I − T (ϕ))−1 k ≤ µ,

BR (ϕ) ⊂ O.

Concerning the constants of Theorem 1, fix α = 1,

n 1 o . r = min R, 2µ`

The discretization process is based upon the approximation Tn (x) := πn T (x),

x ∈ BR (ϕ).

As in Theorem 1, we can prove that, for all x ∈ Br (ϕ), I − T (x) is invertible and k(I − T (x))−1 k ≤ 2µ. Proposition 1 Suppose that (7) holds, and also that 1. For all x ∈ X , πn x → x as n → ∞. 2. The set W := {T (x)h : x ∈ BR (ϕ), h ∈ X , khk = 1} is relatively compact. Then lim

sup

n→∞ x∈B (ϕ) R

kTn (x) − T (x)k = 0.

Proof : W := {T (x)h : x ∈ BR (ϕ), h ∈ X , khk = 1} is relatively compact, πn tends to I pointwise and pointwise convergence is uniform on relatively compact sets.

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Proposition 2 Under the assumptions of Proposition 1, for n large enough, the approximate inverse defined by Σn (x) := (I − Tn (x))−1 ,

(8)

exists and is uniformly bounded for x ∈ Br (ϕ). Proof : Set δn :=

sup kTn (x) − T (x)k. x∈Br (ϕ)

Since δn → 0 as n → ∞, for all n large enough, δn
> n. If F involves an integral operator, FN can be built from F , replacing the integrals by numerical quadratures. Then a discrete version of Theorem 1 can be written replacing the operator F by FN . Let us define the sequence of discrete Newton iterates as (13)

(0)

ϕN,n ∈ ON ,

(k+1)

ϕN,n

(k)

(k)

(k)

= ϕN,n − ΣN,n (ϕN,n )FN (ϕN,n ).

Theorem 2 A priori convergence of the discrete version Suppose that FN , ON , ϕN ∈ O, µN > 0, RN > 0, `N > 0 and αN ∈]0, 1] are such that: 1. FN (ϕN ) = 0, FN0 (ϕN ) is invertible and kFN0 (ϕN )−1 k ≤ µN , 2. The closed ball BRN (ϕN ) is included in the open set ON , and FN0 : ON → L(X ) is (`N , αN )−H¨ older continuous on BRN (ϕN ), 3. For n rN := min RN ,

16

o 1 , (2µN `N )1/αN

there exists γN,n ∈]0, 1[ such that sup

kI − ΣN,n (x)FN0 (x)k ≤ γN,n ,

x∈BrN (ϕN ) (0)

4. The starting approximation ϕn is chosen in the closed ball BρN,n (ϕN ), where n  ρN,n := min rN ,

1/αN o 1 − γN,n . 4`N µN (1 + γN,n )

(k)

Then, for all k, ϕN,n ∈ BρN,n (ϕN ), and (k)

kϕN,n − ϕN k ≤ ρn

1 + γ

N,n

2

k

→0

as k → ∞.

Proof : Is is the same proof as for theorem 1, replacing F by FN . Remark 3 The discrete version of option A is not equivalent to the application of option B, (k) except in the case where n := N and ΣN,N (ϕN ) := (FN0 (ϕN ))−1 . The philosophy of the discrete version of option A is to have the solution ϕN of FN (x) = 0 at a cheaper cost. (k) The application of option B leads to linear systems of size N whose solutions ψN tend to ϕN whereas the discrete version of option A leads to the resolution of linear systems of size n 0, r > 0, ` > 0 and α ∈]0, 1] are such that: (a) F (ϕ) = 0, (0)

(b) kΣn (ξn )k ≤ µn , (c) The closed ball Br (ϕ) is included in the open set O, and F 0 : O → L(X ) is (`, α)−H¨ older continuous on Br (ϕ), (0)

(d) The starting approximation ξn and γn ∈]0, 1[ satisfy ξn(0) ∈ B%n (ϕ), where

kI − Σn (ξn(0) )F 0 (ξn(0) )k ≤ γn ,

n  1 − γ 1/α o n %n := min r, . 3`µn (k)

Then, for all k, ξn ∈ B%n (ϕ), and kξn(k) − ϕk ≤ (3µn `%αn + γn )k → 0

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as k → ∞.

Proof : The keys are the identity ξn(k+1) − ϕ = [I − Σn (ξn(0) )F 0 (ξn(0) )](ξn(k) − ϕ) + Σn (ξn(0) )[F 0 (ξn(0) ) − F 0 (ξn(k) )](ξn(k) − ϕ) Z 1 (0) −Σn (ξn ) [F 0 (ϕ + t(ξn(k) − ϕ)) − F 0 (ξn(k) )](ξn(k) − ϕ) dt, 0

and the fact that 3µn `%αn + γn < 1. 4. To conclude, let us remark that the classical Galerkin approximation to the equation (I − T (x))v = g built upon the projection πn does not enter the framework of this paper. Indeed, this approximation is defined as the solution of the zero projected residual: πn ((I − T (x))vn − g) = 0. This means that, for n large enough and provided I − πn T (x)πn is invertible, the Galerkin solution is given by vn = (I − πn T (x)πn )−1 πn g. In other words, in the context of this paper, the role of the operator Σn (x) is played by (I − πn T (x)πn )−1 πn which is certainly not invertible. Hence, hypotheses 3 of Theorem 1 and (d) of Theorem 3 will never be satisfied. Acknowledgements This research has been partially supported by the project IFCPAR-CEFIPRA 4101-1 and got financial support provided through CMUP by the European Regional Development Fund through the programme COMPETE and by the Portuguese Government through the FCT Funda¸c˜ ao para a Ciˆencia e Tecnologia under the project PEst-C/MAT/UI0144/2011. Part of this work ´ was done when the third author was on leave at the University of Saint Etienne as Visiting Professor during May 2011.

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