Improvement of preconditioned bi-Lanczos-type algorithms with residual norm minimization for the stable solution of systems of linear equations

In this paper, improved algorithms are proposed for preconditioned bi-Lanczos-type methods with residual norm minimization for the stable solution of systems of linear equations. In particular, preconditioned algorithms pertaining to the bi-conjugate gradient stabilized method (BiCGStab) and the generalized product-type method based on the BiCG (GPBiCG) have been improved. These algorithms are more stable compared to conventional alternatives. Further, a stopping criterion changeover is proposed for use with these improved algorithms. This results in higher accuracy (lower true relative error) compared to the case where no changeover is done. Numerical results confirm the improvements with respect to the preconditioned BiCGStab, the preconditioned GPBiCG, and stopping criterion changeover. These improvements could potentially be applied to other preconditioned algorithms based on bi-Lanczos-type methods.

Augmented Tikhonov Regularization Method for Dynamic Load Identification

We introduce the augmented Tikhonov regularization method motivated by Bayesian principle to improve the load identification accuracy in seriously ill-posed problems. Firstly, the Green kernel function of a structural dynamic response is established; then, the unknown external loads are identified. In order to reduce the identification error, the augmented Tikhonov regularization method is combined with the Green kernel function. It should be also noted that we propose a novel algorithm to determine the initial values of the regularization parameters. The initial value is selected by finding a local minimum value of the slope of the residual norm. To verify the effectiveness and the accuracy of the proposed method, three experiments are performed, and then the proposed algorithm is used to reproduce the experimental results numerically. Numerical comparisons with the standard Tikhonov regularization method show the advantages of the proposed method. Furthermore, the presented results show clear advantages when dealing with ill-posedness of the problem.

Solving the Nonlinear Heat Equilibrium Problems Using the Local Multiquadric Radial Basis Function Collocation Method

In this article, the nonlinear heat equilibrium problems are solved by the local multiquadric (MQ) radial basis function (RBF) collocation method. The system of nonlinear algebraic equations is solved by iteration based on the residual norm-based algorithm, in which the direction of evolution is determined by a linear equation. In addition, the role of the collocation point and source point is clearly defined such that in our proposed method the field value of any interested point can be expressed. Six numerical examples are shown to check the performance of the proposed method. As the number of supporting points (mp) increases, the accuracy of numerical solution increases. Among all examples, mp = 50 can perform well. In addition, the selection of shape parameter, c, affects the accuracy. However, as c < 2 the maximum relative absolute error percentage is less than 1%.

Preconditioned Subspace Descent Method for Nonlinear Systems of Equations

Nonlinear least squares iterative solver is considered for real-valued sufficiently smooth functions. The algorithm is based on successive solution of orthogonal projections of the linearized equation on a sequence of appropriately chosen low-dimensional subspaces. The bases of the latter are constructed using only the first-order derivatives of the function. The technique based on the concept of the limiting stepsize along normalized direction (developed earlier by the author) is used to guarantee the monotone decrease of the nonlinear residual norm. Under rather mild conditions, the convergence to zero is proved for the gradient and residual norms. The results of numerical testing are presented, including not only small-sized standard test problems, but also larger and harder examples, such as algebraic problems associated with canonical decomposition of dense and sparse 3D tensors as well as finite-difference discretizations of 2D nonlinear boundary problems for 2nd order partial differential equations.

Strain Energy Decomposition Influence in the Phase Field Crack Modelling at the Microstructural Level of Heterogeneous Materials

The prediction of a crack initiation and propagation occurring on the microstructural level of heterogeneous materials can be a very demanding problem. According to the results of recent investigations, the emerging phase field approach to fracture has a strong potential in modelling the complex crack behaviour in a simple manner. In this study, recently developed phase field staggered solution scheme with the residual norm stopping criterion has been employed for the fracture analysis of heterogeneous microstructure exhibiting complex crack phenomena. The microstructural geometries based on the metallographic images of the nodular cast iron and the material properties of an academic brittle material have been used in numerical simulations where the graphite nodules have been considered as porosities. Two commonly used energy decomposition models, the spectral decomposition and the spherical-deviatoric split, and their effects on the results of the phase field modelling are investigated. Numerical results show that the proposed algorithm recovers the complicated crack path driven by the complex microstructural topology.

Magnetic induction pneumography: a planar coil system for continuous monitoring of lung function via contactless measurements

Continuous monitoring of lung function is of particular interest to the mechanically ventilated patients during critical care. Recent studies have shown that magnetic induction measurements with single coils provide signals which are correlated with the lung dynamics and this idea is extended here by using a 5 by 5 planar coil matrix for data acquisition in order to image the regional thoracic conductivity changes. The coil matrix can easily be mounted onto the patient bed, and thus, the problems faced in methods that use contacting sensors can readily be eliminated and the patient comfort can be improved. In the proposed technique, the data are acquired by successively exciting each coil in order to induce an eddy-current density within the dorsal tissues and measuring the corresponding response magnetic field strength by the remaining coils. The recorded set of data is then used to reconstruct the internal conductivity distribution by means of algorithms that minimize the residual norm between the estimated and measured data. To investigate the feasibility of the technique, the sensitivity maps and the point spread functions at different locations and depths were studied. To simulate a realistic scenario, a chest model was generated by segmenting the tissue boundaries from NMR images. The reconstructions of the ventilation distribution and the development of an edematous lung injury were presented. The imaging artifacts caused by either the incorrect positioning of the patient or the expansion of the chest wall due to breathing were illustrated by simulations.

Numerical techniques for behavior of incompressible flow in steady two-dimensional motion due to a linearly stretching of porous sheet based on radial basis functions

In this paper the boundary layer flow of a micro-polar fluid due to a linearly stretching sheet which is a non-linear system two-point boundary value problem (BVP) on semi-infinite interval has been considered. This the sheets are included the suction and injection. We solve this problem by two different collecation approaches and compare their results with solution of other methods. The proposed approaches are equipped by the direct (DRBF) and indirect radial basis functions (IRBF). Direct approach (DRBF) is based on a differential process and indirect approach (IRBF) is based on an integration process. These methods reduce solution of the problem to solution of a system of algebraic equations. Numerical results and residual norm show that the IRBF performs better than the common DRBF, and has an acceptable accuracy and high rate of convergence of IRBF process.

A unified treatment for Tikhonov regularization using a general stabilizing operator

While dealing with the problem of solving an ill-posed operator equation Tx = y, where T : X → Y is a bounded linear operator between Hilbert spaces X and Y, one looks for a stable method for approximating [Formula: see text], a least-residual norm solution which minimizes a seminorm x ↦ ‖Lx‖, where L : D(L) ⊆ X → X is a (possibly unbounded) closed densely defined operator in X. If the operators T and L satisfy a completion condition ‖Tx‖2 + ‖Lx‖2 ≥ γ‖x‖2 for all x ∈ D(L*L) for some constant γ > 0, then Tikhonov regularization is one of the simple and widely used of such procedures in which the regularized solution is obtained by solving a well-posed equation [Formula: see text] where yδ is a noisy data and α > 0 is the regularization parameter to be chosen appropriately. We prescribe a condition on (T, L) which unifies the analysis for ordinary Tikhonov regularization, that is, L = I, and also the case of L = Bs with B being a strictly positive closed densely defined unbounded operator which generates a Hilbert scale {Xt}t>0. Under the new framework, we provide estimates for the best possible worst error and order optimal error estimates for the regularized solutions under certain general source condition which incorporates in its fold many existing results as special cases, by choosing regularization parameter using a Morozov-type discrepancy principle.