Abstract

<p>It is well known that the classical numerical algorithm of the steepest descent method (SDM) is effective for well-posed linear systems, but performs poorly for ill-posed ones. In this paper we propose accelerated and/or bidirectional modifications of SDM, namely the accelerated steepest descent method (ASDM), the bidirectional method (2DM), and the accelerated bidirectional method (A2DM). The starting point is a manifold defined in terms of a minimum functional and a fictitious time variable; nevertheless, in the end the fictitious time variable disappears so that we arrive at purely iterative algorithms. The proposed algorithms are justified by dynamics-theoretical and optimization interpretation. The accelerator plays a prominent role of switching from the situation of slow convergence to a new situation that the functional tends to decrease stepwise in an intermittent and ceaseless manner. Three examples of solving ill-posed systems are examined and comparisons are made with exact solutions and with the existing algorithms of the SDM, the Barzilai-Borwein method, and the random SDM, revealing that the new algorithms of ASDM and A2DM have better computational efficiency and accuracy even for the highly ill-posed systems.</p>

Highlights

  • 1.1 Ill-posed Problems and RemedyIn this paper we propose robust and implemented new methods to solve the system of linear algebraic equations Ax = b, (1)where the coefficient matrix A ∈ Rn×n is a given positive definite matrix, which may be ill-conditioned; the right-hand side vector b ∈ Rn is the input data, which may be corrupted by noise; and x ∈ Rn is the unknown vector to be sought for

  • In this paper we propose accelerated and/or bidirectional modifications of SDM, namely the accelerated steepest descent method (ASDM), the bidirectional method (2DM), and the accelerated bidirectional method (A2DM)

  • Three examples of solving ill-posed systems are examined and comparisons are made with exact solutions and with the existing algorithms of the SDM, the Barzilai-Borwein method, and the random SDM, revealing that the new algorithms of ASDM and accelerated 2DM (A2DM) have better computational efficiency and accuracy even for the highly ill-posed systems

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Summary

Ill-posed Problems and Remedy

Where the coefficient matrix A ∈ Rn×n is a given positive definite matrix, which may be ill-conditioned; the right-hand side vector b ∈ Rn is the input data, which may be corrupted by noise; and x ∈ Rn is the unknown vector to be sought for. We encounter the problem that the numerical solution to Eq (1) may deviate from the exact one to a great extent, when A is highly ill-conditioned and/or b is disturbed by noise. The solution of ill-posed linear algebraic equations is an important issue for many engineering and scientific problems. The linear algebraic system (1) with A having a large condition number usually suffers from the numerical instability problem that an arbitrary small perturbation on the right-hand side b may lead to a large perturbation to the solution x on the left-hand side. A famous example of highly ill-conditioned matrices is the Hilbert matrix. It arises naturally in finding an n-degree polynomial function p(x) = a0 + a1 x + .

The Steepest Descent Method
A New Stage—Invariant Manifold
Dynamics of Iterative Algorithms
How Does the Accelerator Work?
Bidirectional Methods
Further Study by Numerical Experiments
Example 1
Example 2
Example 3
Concluding Remarks

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