A new sufficiently descent algorithm for pseudomonotone nonlinear operator equations and signal reconstruction

Abstract

This paper presents a new sufficiently descent algorithm for system of nonlinear equations where the underlying operator is pseudomonotone. The conditions imposed on the proposed algorithm to achieve convergence are Lipschitz continuity and pseudomonotonicity which is weaker than monotonicity assumption forced upon many algorithms in this area found in the literature. Numerical experiments on selected test problems taken from the literature validate the efficiency of the new algorithm. Moreover, the new algorithm demonstrates superior performance in comparison with some existing algorithms. Furthermore, the proposed algorithm is applied to reconstruct some disturbed signals.

Appendix

1.1 List of test problems

We use the following system of nonlinear equation for the experiments in Section 3 where $Q(a)=(q_1(a),q_2(a),\ldots ,q_n(a))^T$ and $a=(a_1, a_2,\ldots , a_n)^T.$

Problem 5.1

[36]

$$\begin{aligned} q_1(a)= & {} e^{a_1}-1\\ q_i(a)= & {} e^{a_i}+a_{i-1}-1,\quad {}i=2,\ldots ,n. \end{aligned}$$

Problem 5.2

Logarithmic function [36]

$$\begin{aligned} q_i(a_i)=\log (a_i+1)-\frac{a_i}{n},\quad {}i=1,2,\ldots ,n. \end{aligned}$$

Problem 5.3

[37]

$$\begin{aligned} q_i(a) = 2a_i - \sin |a_i|,\quad {}i=1,2,\ldots ,n. \end{aligned}$$

Problem 5.4

[38]

$$\begin{aligned} q_i(a)=e^{a_i}-1, ~~ i=1,2,\ldots ,n. \end{aligned}$$

Problem 5.5

[39]

$$\begin{aligned} q_i(a)=a_i-\sin (\vert a_i-1\vert ), ~~ i=1,2,\ldots ,n. \end{aligned}$$

Problem 5.6

$$\begin{aligned} q_1(a)= & {} 2a_1-a_2+e^{a_1}-1,\\ q_i(a)= & {} -a_{i-1}+2a_i-a_{i+1}+e^{a_i}-1, ~~ i=2,\ldots ,n-1,\\ q_n(a)= & {} -a_{n-1}+2a_n+e^{a_n}-1. \end{aligned}$$

Problem 5.7

[40]

$$\begin{aligned} q_1(a)= & {} \frac{5}{2}a_1+a_2-1,\\ q_i(a)= & {} a_{i-1}+\frac{5}{2}a_i+a_{i+1}-1, ~~ i=2,\ldots ,n-1,\\ q_n(a)= & {} a_{n-1}+\frac{5}{2}a_n-1. \end{aligned}$$

Problem 5.8

$$\begin{aligned} q_i(a)=\frac{i}{n}e^{a_i}-1, ~~ i=1,2,\ldots ,n. \end{aligned}$$

Problem 5.9

Tridiagonal exponential problem [41]

$$\begin{aligned} q_1(a)= & {} a_1-\exp \left( \cos \left( \frac{a_1+a_2}{n+1}\right) \right) \\ q_i(a)= & {} a_i-\exp \left( \cos \left( \frac{a_{i-1}+a_i+a_{i+1}}{n+1}\right) \right) ,~~2\le i \le n-1,\\ q_n(a)= & {} a_n-\exp \left( \cos \left( \frac{a_{n-1}+a_n}{n+1}\right) \right) . \end{aligned}$$

Problem 5.10

$$\begin{aligned} q_1(a)= & {} a_1 + \sin (a_1) - 1\\ q_i(a)= & {} -a_{i-1}+2a_i + \sin (a_i)-1, ~i=2,\ldots ,n-1,\\ q_n(a)= & {} a_n + \sin (a_n) - 1. \end{aligned}$$

Problem 5.11

$$\begin{aligned} q_i(a)=\frac{i}{n+1}e^{a_i}-1, ~~ i=1,2,\ldots ,n. \end{aligned}$$

Problem 5.12

$$\begin{aligned} q_i(a)= & {} a_i-\frac{1}{100}a_{i+1}^3, ~~ i=1,2,\ldots ,n-1,\\ q_n(a)= & {} a_n-\frac{1}{100}a_{1}^3. \end{aligned}$$

Problem 5.13

Problem 76 in [41]

$$\begin{aligned} q_i(a)= & {} a_i-\frac{1}{10}a_{i+1}^2,\quad i=1,2,\ldots ,n-1,\\ q_n(a)= & {} a_n-\frac{1}{10}a_{1}^2. \end{aligned}$$

Problem 5.14

Nonsmooth IVP problem

$$\begin{aligned} Q(a)=\left( \begin{array}{ccc} 4 &{} -1 &{} \\ -1 &{} 4 &{} -1 \\ \ddots &{} \ddots &{} \ddots \\ \ddots &{} \ddots &{} -1\\ &{} -1 &{} 4 \\ \end{array}\right) a+(e^{ a_1}-1,\ldots ,e^{ a_n}-1)^T. \end{aligned}$$

Problem 5.15

$$\begin{aligned} Q(a)=\left( \begin{array}{ccc} \frac{5}{2} &{} -1 &{} \\ -1 &{} \frac{5}{2} &{} -1 \\ \ddots &{} \ddots &{} \ddots \\ \ddots &{} \ddots &{} -1\\ &{} -1 &{} \frac{5}{2} \\ \end{array}\right) a+(1,1,\ldots ,1)^T. \end{aligned}$$

Problem 5.16

$$\begin{aligned} Q(a)=\left( \begin{array}{ccc} 2 &{} -1 &{} \\ 0 &{} 2 &{} -1 \\ \ddots &{} \ddots &{} \ddots \\ \ddots &{} \ddots &{} -1\\ {} &{} -1 &{} 2 \\ \end{array}\right) a+(\sin a_{1}-1,...,\sin a_{n}-1)^T. \end{aligned}$$