A Fast Fully Discrete Mixed Finite Element Scheme for Fractional Viscoelastic Models of Wave Propagation

In this paper, probability theory is applied to the problem of estimating the decay rate constants in data that are known to contain sums of exponentials. In many instances the number of exponential components is unknown. One might naively believe that the correct strategy, to estimate the parameters, is to first determine the number of exponentials and then estimate the parameters from a model containing that number of exponentials. However, this is not what the rules of probability theory indicate should be done. Probability theory indicates that the best parameter estimates are obtained from the probability for the parameter of interest independent of the number of exponentials in the data. This probability density function is a weighted average. It is a sum over the probability for the parameter given the number of exponentials weighted by the probability that this number of exponentials is the correct value. When the number of exponentials in the data are well determined, this reduces to the problem of determining the number of exponentials and then estimating the parameters given that number of exponentials. However, when the data fail to strongly support a single value for the number of exponentials the result from probability can differ significantly from this intuitive procedure. In this paper a sketch of these calculations is presented, and numerical examples are used to illustrate the calculations.

Fractional calculus approach, providing novel models through the introduction of fractional-order calculus to optimization methods, is employed in machine learning algorithms. This scheme aims to attain optimized solutions by maximizing the accuracy of the model and minimizing the functions like the computational burden. Mathematical-informed frameworks are to be employed to enable reliable, accurate, and robust understanding of various complex biological processes that involve a variety of spatial and temporal scales. This complexity requires a holistic understanding of different biological processes through multi-stage integrative models that are capable of capturing the significant attributes on the related scales. Fractional-order differential and integral equations can provide the generalization of traditional integral and differential equations through the extension of the conceptions with respect to biological processes. In addition, algorithmic complexity (computational complexity), as a way of comparing the efficiency of an algorithm, can enable a better grasping and designing of efficient algorithms in computational biology as well as other related areas of science. It also enables the classification of the computational problems based on their algorithmic complexity, as defined according to the way the resources are required for the solution of the problem, including the execution time and scale with the problem size. Based on a novel mathematical informed framework and multi-staged integrative method concerning algorithmic complexity, this study aims at establishing a robust and accurate model reliant on the combination of fractional-order derivative and Artificial Neural Network (ANN) for the diagnostic and differentiability predictive purposes for the disease, (diabetes, as a metabolic disorder, in our case) which may display various and transient biological properties. Another aim of this study is benefitting from the concept of algorithmic complexity to obtain the fractional-order derivative with the least complexity in order that it would be possible to achieve the optimized solution. To this end, the following steps were applied and integrated. Firstly, the Caputo fractional-order derivative with three-parametric Mittag-Leffler function \((\alpha ,~\beta ,~\gamma )\) was applied to the diabetes dataset. Thus, new fractional models with varying degrees were established by ensuring data fitting through the fitting algorithm Mittag-Leffler function with three parameters \((\alpha ,~\beta ,~\gamma )\) based on heavy-tailed distributions. Following this application, the new dataset, named the mfc_diabetes, was obtained. Secondly, classical derivative (calculus) was applied to the diabetes dataset, which yielded the cd_diabetes dataset. Subsequently, the performance of the new dataset as obtained from the first step and of the dataset obtained from the second step as well as of the diabetes dataset was compared through the application of the feed forward back propagation (FFBP) algorithm, which is one of the ANN algorithms. Next, the fractional order derivative model which would be the most optimal for the disease was generated. Finally, algorithmic complexity was employed to attain the Caputo fractional-order derivative with the least complexity, or to achieve the optimized solution. This approach through the application of fractional-order calculus to optimization methods and the experimental results have revealed the advantage of maximizing the model’s accuracy and minimizing the cost functions like the computational costs, which points to the applicability of the method proposed in different domains characterized by complex, dynamic and transient components.KeywordsComputational complexityComplex systemsFractional calculus and complexityFractional-order derivativesCaputo fractional-order derivativeClassical derivativesMittag-Leffler functionsInteger-order derivativesComputational and nonlinear dynamicsMathematical biologyDynamic biological modelsData analysisData fittingUncertaintyNonlinearityNeural networksMultilayer perceptron algorithmData-driven fractional biological modeling

We propose a fast L2-1σ finite element method for solving the time fractional Keller–Segel equations with a Caputo fractional derivative of α∈(0,1). Firstly, the fast L2-1σ scheme on the graded mesh is used to discretize the time fractional derivative. This approach relies on the sum of exponentials (SOE) skill to speed up the convolution kernel. Thus, we overcome the computational cost caused by the nonlocality of fractional derivatives. Then, by combining finite element discretization in spatial direction, a fully implicit numerical scheme is derived. Subsequently, we establish the stability and an α-robust error analysis of the fully discrete scheme. Finally, we present some numerical examples to demonstrate the correctness of our theoretical results.

In this paper, we study the variable-order (VO) time-fractional diffusion equations. For a VO function \(\alpha (t)\in (0,1)\), we develop an exponential-sum-approximation (ESA) technique to approach the VO Caputo fractional derivative. The ESA technique keeps both the quadrature exponents and the number of exponentials in the summation unchanged at different time level. Approximating parameters are properly selected to achieve the efficient accuracy. Compared with the general direct method, the proposed method reduces the storage requirement from \({\mathcal {O}}(n)\) to \({\mathcal {O}}(\log ^2 n)\) and the computational cost from \({\mathcal {O}}(n^2)\) to \(\mathcal {O}(n\log ^2 n)\), respectively, with n being the number of the time levels. When this fast algorithm is exploited to construct a fast ESA scheme for the VO time-fractional diffusion equations, the computational complexity of the proposed scheme is only of \({\mathcal {O}}(mn\log ^2 n)\) with \({\mathcal {O}}(m\log ^2n)\) storage requirement, where m denotes the number of spatial grid points. Theoretically, the unconditional stability and error analysis of the fast ESA scheme are given. The effectiveness of the proposed algorithm is verified by numerical examples.

Long time [formula omitted]-stability of fast L2-1[formula omitted] method on general nonuniform meshes for subdiffusion equations

A fractional order control model for Diabetes and COVID-19 co-dynamics with Mittag-Leffler function

In this manuscript, a new approach to study the fractionalized Oldroyd-B fluid flow based on the fundamental symmetry is described by critically examining the Prabhakar fractional derivative near an infinitely vertical plate, wall slip condition on temperature along with Newtonian heating effects and constant concentration. The phenomenon has been described in forms of partial differential equations along with heat and mass transportation effect taken into account. The Prabhakar fractional operator which was recently introduced is used in this work together with generalized Fick’s and Fourier’s law. The fractional model is transfromed into a non-dimentional form by using some suitable quantities and the symmetry of fluid flow is analyzed. The non-dimensional developed fractional model for momentum, thermal and diffusion equations based on Prabhakar fractional operator has been solved analytically via Laplace transformation method and calculated solutions expressed in terms of Mittag-Leffler special functions. Graphical demonstrations are made to characterize the physical behavior of different parameters and significance of such system parameters over the momentum, concentration and energy profiles. Moreover, to validate our current results, some limiting models such as fractional and classical fluid models for Maxwell and Newtonian are recovered, in the presence of with/without slip boundary wall conditions. Further, it is observed from the graphs the velocity curves for classical fluid models are relatively higher than fractional fluid models. A comparative analysis between fractional and classical models depicts that the Prabhakar fractional model explains the memory effects more adequately.

In this paper we introduce an advanced supervised training method for neural networks. It is based on Jacobian rank deficiency and it is formulated, in some sense, in the spirit of the Gauss-Newton algorithm. The Levenberg-Marquardt algorithm, as a modified Gauss-Newton, has been used successfully in solving nonlinear least squares problems including neural-network training. It outperforms (in terms of training accuracy, convergence properties, overall training time, etc.) the basic backpropagation and its variations with variable learning rate significantly, however, with higher computation and memory complexities within each iteration. The new method developed in this paper is aiming at improving convergence properties, while reducing the memory and computation complexities in supervised training of neural networks. Extensive simulation results are provided to demonstrate the superior performance of the new algorithm over the Levenberg-Marquardt algorithm.

Mixed Finite Elements (FE) constitute an elegant remedy for the approximation of constrained boundary value problems, where the capability of the classical FE method is limited. This thesis comprises in a first step the mathematical analysis and numerical investigation of different mixed FE approaches in the case of linear elasticity. In a second step novel strategies for the extension of the considered formulations to the nonlinear hyperelastic framework are discussed. Within the main objective of reliable and efficient FE based approximations including large deformations, a focus of the proposed work is set on the construction of elements based on the Hellinger-Reissner variational framework. This family of elements is characterized by a direct discretization of the stresses as well as the displacements and a challenging extension to the large strain regime. The investigation of the efficiency and reliability of the proposed FE schemes is emphasized by a comparison to well established formulations using nontrivial numerical benchmarks. Additionally to the common constraint situations of incompressibility and thin-walled structures, the important case of inextensibility is regarded. This work results in a couple of novel FE discretizations, which are characterized by a notable gain in efficiency and robustness. In addition, further insights considering the reliability and stability of mixed Finite Elements in the hyperelastic framework are gained.

Hidden pools, hidden modes, and visible repeated eigenvalues in compartmental models

Chapter 9 - Computational fractional-order calculus and classical calculus AI for comparative differentiability prediction analyses of complex-systems-grounded paradigm

Exponential Sum-Fitting of Dwell-Time Distributions without Specifying Starting Parameters

Analysis of a fractional order model for HPV and CT co-infection

Unveiling the generalization of the derivative order with a novel application of the fractional order model to green soybean oil extraction

Mixed finite element methods (MFEM) form a general mathematical framework for the spatial discretisation of partial differential equations, mainly applied to elliptic equations of second order. They become increasingly important for the solution of nonlinear problems. In contrast to standard finite element schemes the mixed finite element discretisation of problems in divergence form, i.e. f +{\rm div} \, \sigma = 0 where \sigma = A (\nabla \, u) , \sigma \in L and u \in H , allows more flexibility in the design of the discrete approximation spaces contained in L and H , i.e. in the spaces for the direct variables and the Lagrange multipliers. The workshop focuses on new developments in the field of mixed and non-standard finite element methods. The main points are The workshop aimed at bridging the gap between the computational engineering community and applied mathematicians and in consequence to unify the scientific language and foster later collaboration. Nonlinear mixed schemes were of particular concern for problems in elasticity and plasticity, but electromagnetics and mathematically related topics were also included. Mixed finite element methods for elliptic problems are based on a variational description in saddle-point form. Side conditions such as divergence free velocity fields in incompressible fluid dynamics are usually treated in this framework. The appearance of ‘soft’ side conditions is typical for structural mechanics as is the case with nearly incompressible materials or plates and shells with small thickness parameters. We also mention materials which almost satisfy the Kirchhoff condition, i.e. problems with a high but finite shear stiffness. In such cases, which are by no means ‘soft’ from the mathematical point of view, mixed methods lead to a more robust discretisation. The arising stability conditions and computational techniques cannot be understood fully by intuitive mechanical principles; however, from the mathematician's point of view their reasoning is natural, clear and insightful. Mixed and non-standard finite element methods gain increasing prominence in the prevention of locking phenomena. We highlight a topic which is currently actively investigated: the development of stable and efficient plate and shell elements with regard to shear locking, which is more intricate than volume locking. Here it is important to understand how techniques based on heuristic ideas are consistent with more modern mathematical methods. Availability of fast solvers is decisive for the competitiveness of numerical techniques. For a variety of applications, multigrid methods are crucial for the efficiency of the implementation. Methods have been proposed which do not appear plausible if one wants to deduce the algorithms directly from the physical model. The advanced methods depend on rigorous error estimators in order to guarantee that the numerical solutions represent the exact solutions of the physical model.