A fast multiscale solver for modified Hammerstein equations

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Abstract

This paper presents a fast solver, called the multilevel augmentation method, for modified nonlinear Hammerstein equations. When we utilize the method to solve a large scale problem, most of components of the solution can be computed directly, and lower frequency components can be obtained by solving a fixed low-order algebraic nonlinear system. The advantage of using the algorithm to modified equations is that it leads to reduce the cost of numerical integrations greatly. The optimal error estimate of the method is established and the nearly linear computational complexity is proved. Finally, numerical examples are presented to confirm the theoretical results and illustrate the efficiency of the method.

Introduction

In this paper, we study fast multiscale methods for solving the modified Hammerstein integral equationu(t)=ψt,Ek(t,s)u(s)ds+f(t),tE,where ERd(d1) is a compact domain, f, k and ψ are given functions defined on E, E × E and E×R respectively, and u is the unknown to be determined.

The Eq. (1) is equivalent to the original Hammerstein equation (cf. [35])z(t)=Ek(t,s)ψ(s,z(s))ds+f(t),tE,where z and u satisfyz(t)=Ek(t,s)u(s)ds+f(t)andu(t)=ψ(t,z(t)).

Hammerstein equations are a class of typical nonlinear integral equations, which are also integral forms of some nonlinear elliptic boundary value problems and portray a lot of mathematical physics problems. A lot of substantial literatures have developed the existence, uniqueness and regularity of the solutions for (1), (2) and some relative nonlinear integral equations (see, for example, [2], [7], [20], [27], [29], [36]). In the past decade, many numerical methods for solving Hammerstein equation (2) have been developed (cf. [3], [4], [5], [14], [21], [22], [23], [24], [25], [26], [27], [28], [29], [30], [31], [32], [33], [34]). Usually, these methods lead to systems of nonlinear equations which should be solved by some kind of iteration methods. It is therefore a marked disadvantage of the numerical methods for (2) that a lot of integrals should be evaluated at each step of the iteration. If we adopt the Newton iteration method, then we have to recalculate Jacobian matrix for each iteration step and the entries of the Jacobian matrix have to be computed by numerical integration. When the size of matrix becomes large, the calculational cost would be staggering. To overcome the problem, [35] proposed the collocation method for solving the modified Hammerstein integral equation (1), which is equivalent to (2). An advantage of the method is that integrals of a similar nature need to be calculated once only. However, if the dimension of approximate space is large, the Jacobian matrix computation is still costly.

In recent years, the numerical methods for solving integral equations were rapidly developed, especially multiscale methods were presented to achieve a breakthrough (see [1], [6], [8], [9], [10], [11], [12], [13], [14], [15], [16], [17], [18], [37], [38]). One of the valuable methods makes use of the multiscale basis, which was originally constructed by Micchelli and Xu in [37] and later further developed by Chen et al. in [9]. The other one of valuable methods is the multilevel augmentation methods for solving the second kind integral equations developed by Chen et al. [13]. This algorithm makes the program for solving the discrete linear equations runs at several or several dozens times, and for the large scale problem it becomes more efficient. Chen et al. [14] applied the multilevel augmentation methods to Hammerstein equation (2) and proposed a nonlinear multilevel augmentation algorithm with the optimal convergence order. This algorithm is only required to solve nonlinear equations in a smaller fixed size, which makes the computational complexity substantially reduced. However, the algorithm has to calculate a lot of numerical integrals at each iteration step. In order to reduce and avoid overmany numerical integrals, in this paper, we apply the wavelet basis which was originally constructed in [9], [37] and the multilevel augmentation method to the modified Hammerstein equation (1), and extended the corresponding algorithm to the case that the nonlinear function ψ(t, v) has the continuious second derivative (see the condition (H3)). We will show that the algorithm has nearly linear computational complexity while preserving optimal order of convergence.

Throughout this paper, we assume that the functions f, k, ψ in Eq. (1) satisfy the following conditions:

  • (H1)

    f  C(E).

  • (H2)

    The kernel function k satisfiessuptEE|k(t,s)|ds<,andlimttk(t,·)-k(t,·)=0.

  • (H3)

    The function ψ(s, v) and it’s partial derivatives ψv(s,v),ψvv(s,v) are continuous on E×R. Furthermore, for any vC(E),ψ(·,v(·)),ψv(·,v(·))C(E).

This paper is organized in five sections. In Section 2, we present the multilevel augmentation method for the modified Hammerstein equation (1). We prove in Section 3 the existence and convergence of the proposed method and establish the optimal convergence estimates. In Section 4 we give the discrete form of the algorithm based on the collocation method and prove that it has a nearly linear computational complexity. Finally, in Section 5 we present several numerical examples to illustrate our theoretical results.

Section snippets

The multilevel augmentation method

In this section we present the multilevel augmentation method for solving the modified Hammerstein integral equation (1).

Let X be the Banach space L2(E) or L(E). Assume that K:XX is a linear operator defined byKu(t)Ek(t,s)u(s)ds,tE,and that Ψ:XX is a nonlinear operator defined by(Ψu)(t)ψ(t,u(t)),tE.By using (5), (6), (1) can be rewritten into the operator formu=Ψ(Ku+f).

Let {Xn:nN0} be a sequence of finite-dimensional subspaces of X satisfyingXnXn+1,andC(E)nN0Xn¯X,where N0{0,1,2,}

Existence and convergence analysis

In this section we discuss the existence of multilevel augmentation solutions and establish their optimal convergence estimates.

For this purpose, we assume that (1) has an isolated solution u  L(E). According to the assumption (H1) and (H2), we can define constant M0 byM0suptEE|k(t,s)u(s)|ds+|f(t)|.This allows us to construct a truncated function ψ˜C(E×R) satisfying (H3) and having properties that for some ϵ > 0, there exists a positive constant Cϵ such thatψ˜(t,v)=ψ(t,v),-M0-ϵvM0+ϵ,andsup

Discrete scheme and its computational complexity

In this section, we will describe a discrete form of the multilevel augmentation method based on multiscale collocation schemes, and then estimate the computational complexity of the algorithm for solving (8), which shows the advantages of the multilevel augmentation method.

To realize the multilevel augmentation method and analyze its computational complexity, we need a discrete form of Algorithm 2.2. We present it by giving appropriate multiscale basis functions and functionals (cf. [9], [12]

Numerical examples

We present two numerical examples in this section to confirm the theoretical estimates for the convergence order. The numerical algorithms is run on a notebook computer with Intel Core2 T5600 1.83 GHz CPU and 2 GB RAM, and the programs with single thread are compiled by Visual C++ 2005.

Example 5.1

Consider the following nonlinear integral equationu(t)=01sin(π(t+s))u(s)ds+f(t)2,t[0,1],where f(t)sin(πt)-43πcos(πt). And it has an isolated solution u(t) = sin2(πt).

Let Xn be the space of piecewise linear

Acknowledgement

Supported in part by the Natural Science Foundation of China under grants 10771224, 11071264, 10601070, 11071286, and the Science and Technology Section of SINOPEC. Supported in part by Guangdong Provincial Government of China through the ”Computational Science Innovative Research Team” program.

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