A Hybrid Algorithm based on Iterative Regularization Methods for semi-explicit Integro-Differential-Algebraic Equations

Integro-Differential-Algebraic Equations arise in many in real world applications, in particular those related to the memory kernel identification problem in heat conduction, viscoelasticity, etc.  The main focus of this paper is to present a numerical method based on the iterative regularization algorithms. Owing to the ill-posed behavior of these equations, we are looking for iterative type methods including the regularization schemes in order to fixed the difficulties which may arise in their numerical solvability. Of particular interest would be on the Landweber iteration which is developed for its efficiency and fast performance to solve linear inverse equations as well as many ill-posed problems. Typically, the discretized form of these problems leads to a large, sparse and ill-conditioned linear system of equations. As the regularity of the solution is closely related to regularity of kernels, degree of smoothness and properties of the integral operator, we assume throughout the paper that all the functions are satisfied in the regular conditions.
The proposed

In this paper, a hybrid algorithm is developed in terms of the Landweber type iterative regularization and the Galerkin procedure.  The problems have been first discretized by a Galerkin type method with piecewise constant functions as basis functions. We then start from a suitable initial value and experimentally choose the regularization parameter. The proposed method terminates when the maximum number of iterations is reached or a stopping rule is satisfied. The strategy will be accomplished rather fast which leads to efficient and fast numerical algorithm.

Experimental

The validity and accuracy of the proposed algorithm are demonstrated through some illustrative examples. To clarify our results, we consider some test problems from the previously work and report the numerical results for different values of the number of nodes and iteration number.  We also reported the CPU time for each kernel evaluation.
It should be noted that, the iterative methods based on Landweber algorithm typically can cause a semi-convergence phenomenon, which means the error initially decreases while after some iterations begin to increase. This behavior depends on different factors such as level of the noise, the relaxation parameter and the starting point.  Under these crucial issues, the iterative algorithms may lead to fast or slow semi-convergence which indicates the key role of verification the semi-convergence.  Our experimental results show that the regularization parameter is important either in postponing the semi-convergence or decreasing the divergence. If it is chosen too large, it gives an over-smoothed solution which may lacks the desired solution, otherwise it yields a solution that is unnecessarily, and possibly severely contaminated by propagated error.

Here, the numerical solution of semi-explicit Integro-differential algebraic equations, using a hybrid algorithm based on iterative regularization method is presented. The efficiency and accuracy of the algorithm are experimentally discussed. The main advantages of the proposed method are the low computational complexity and its convenient numerical implementation. The intrinsic property of the method is stipulated by its ability to efficiently control the number of iterations by varying the regularization parameter and subdivisions.