Solutions and stability for p-Laplacian differential problems with mixed type fractional derivatives

In this note, the main emphasis is to study two kinds of high-order fractional p-Laplacian differential equations with mixed derivatives, which include Caputo type and Riemann–Liouville type fractional derivative. Based on fixed point theorems on the cone, the existence-uniqueness of positive solutions for equations and two iterative schemes to uniformly approximate the unique solutions are discussed theoretically. What’s more, the sufficient conditions for stability of the equations are given. Some exact examples are further provided to verify the analytical results at the end of the article.

Two Nonmonotonic Self-Adaptive Strongly Convergent Projection-Type Methods for Solving Pseudomonotone Variational Inequalities

The primary objective of this study is to introduce two novel extragradient-type iterative schemes for solving variational inequality problems in a real Hilbert space. The proposed iterative schemes extend the well-known subgradient extragradient method and are used to solve variational inequalities involving the pseudomonotone operator in real Hilbert spaces. The proposed iterative methods have the primary advantage of using a simple mathematical formula for step size rule based on operator information rather than the Lipschitz constant or another line search method. Strong convergence results for the suggested iterative algorithms are well-established for mild conditions, such as Lipschitz continuity and mapping monotonicity. Finally, we present many numerical experiments that show the effectiveness and superiority of iterative methods.

A review of continuum damage and plasticity in concrete: Part II – Numerical framework

In this part II, companion article, we present the numerical review of continuum damage mechanics and plasticity in the context of finite element. The numerical advancements in local, nonlocal, and rate-dependent models are presented. The numerical algorithms, type of elements utilized in numerical analysis, the commercial software’s or in-house codes used for the analysis, iterative schemes, explicit or implicit approaches to solving finite element equations, and degree of continuity of element are discussed in this part. Lastly, some open issues in concrete damage modeling and future research needed are also discussed.

Convergence Criteria of Three Step Schemes for Solving Equations

We develop a unified convergence analysis of three-step iterative schemes for solving nonlinear Banach space valued equations. The local convergence order has been shown before to be five on the finite dimensional Euclidean space assuming Taylor expansions and the existence of the sixth derivative not on these schemes. So, the usage of them is restricted six or higher differentiable mappings. But in our paper only the first Frèchet derivative is utilized to show convergence. Consequently, the scheme is expanded. Numerical applications are also given to test convergence.

Convergence and Stability of New Approximation Algorithms for Certain Contractive-Type Mappings

We present new fixed points algorithms called multistep H-iterative scheme and multistep SH-iterative scheme. Under certain contractive-type condition, convergence and stability results were established without any imposition of the ’sum conditions’, which to a large extent make some existing iterative schemes so far studied by other authors in this direction practically inefficient. Our results complement and improve some recent results in literature.

Analysis of Subgradient Extragradient Iterative Schemes for Variational Inequalities

In this paper, we investigate the monotone variational inequality in Hilbert spaces. Based on Censor’s subgradient extragradient method, we propose two modified subgradient extragradient algorithms with self-adaptive and inertial techniques for finding the solution of the monotone variational inequality in real Hilbert spaces. Strong convergence analysis of the proposed algorithms have been obtained under some mild conditions.

On highly efficient derivative-free family of numerical methods for solving polynomial equation simultaneously

A highly efficient new three-step derivative-free family of numerical iterative schemes for estimating all roots of polynomial equations is presented. Convergence analysis proved that the proposed simultaneous iterative method possesses 12th-order convergence locally. Numerical examples and computational cost are given to demonstrate the capability of the method presented.

Design, Convergence and Stability of a Fourth-Order Class of Iterative Methods for Solving Nonlinear Vectorial Problems

A new parametric family of iterative schemes for solving nonlinear systems is presented. Fourth-order convergence is demonstrated and its stability is analyzed as a function of the parameter values. This study allows us to detect the most stable elements of the class, to find the fractals in the boundary of the basins of attraction and to reject those with chaotic behavior. Some numerical tests show the performance of the new methods, confirm the theoretical results and allow to compare the proposed schemes with other known ones.

Multi GPU parallelization of maximum likelihood expectation maximization method for digital rock tomography data

Digital rock is an emerging area of rock physics, which involves scanning reservoir rocks using X-ray micro computed tomography (XCT) scanners and using it for various petrophysical computations and evaluations. The acquired micro CT projections are used to reconstruct the X-ray attenuation maps of the rock. The image reconstruction problem can be solved by utilization of analytical (such as Feldkamp–Davis–Kress (FDK) algorithm) or iterative methods. Analytical schemes are typically computationally more efficient and hence preferred for large datasets such as digital rocks. Iterative schemes like maximum likelihood expectation maximization (MLEM) are known to generate accurate image representation over analytical scheme in limited data (and/or noisy) situations, however iterative schemes are computationally expensive. In this work, we have parallelized the forward and inverse operators used in the MLEM algorithm on multiple graphics processing units (multi-GPU) platforms. The multi-GPU implementation involves dividing the rock volumes and detector geometry into smaller modules (along with overlap regions). Each of the module was passed onto different GPU to enable computation of forward and inverse operations. We observed an acceleration of $$\sim 30$$ ∼ 30 times using our multi-GPU approach compared to the multi-core CPU implementation. Further multi-GPU based MLEM obtained superior reconstruction compared to traditional FDK algorithm.