scholarly journals Numerical Inversion for the Multiple Fractional Orders in the Multiterm TFDE

2017 ◽  
Vol 2017 ◽  
pp. 1-7 ◽  
Author(s):  
Chunlong Sun ◽  
Gongsheng Li ◽  
Xianzheng Jia
Keyword(s):  
Numerical Stability ◽  
Noisy Data ◽  
Fractional Diffusion ◽  
Exact Order ◽  
Inversion Problem ◽  
Inversion Algorithm ◽  
Regularization Algorithm ◽  
Fractional Orders ◽  
Diffusion Behaviors ◽  
Key Parameter

The fractional order in a fractional diffusion model is a key parameter which characterizes the anomalous diffusion behaviors. This paper deals with an inverse problem of determining the multiple fractional orders in the multiterm time-fractional diffusion equation (TFDE for short) from numerics. The homotopy regularization algorithm is applied to solve the inversion problem using the finite data at one interior point in the space domain. The inversion fractional orders with random noisy data give good approximations to the exact order demonstrating the efficiency of the inversion algorithm and numerical stability of the inversion problem.

scholarly journals A Simultaneous Inversion Problem for the Variable-Order Time Fractional Differential Equation with Variable Coefficient

2019 ◽  
Vol 2019 ◽  
pp. 1-13 ◽  
Author(s):  
Shengnan Wang ◽  
Zhendong Wang ◽  
Gongsheng Li ◽  
Yingmei Wang
Keyword(s):  
Inverse Problem ◽  
Noisy Data ◽  
Fractional Diffusion ◽  
Basis Functions ◽  
Variable Order ◽  
Inversion Problem ◽  
Inversion Algorithm ◽  
Regularization Algorithm ◽  
Simultaneous Inversion ◽  
Fractional Differential

This paper deals with an inverse problem of simultaneously determining the space-dependent diffusion coefficient and the fractional order in the variable-order time fractional diffusion equation by the measurements at one interior point. Numerical solution to the forward problem is given by the finite difference scheme, and the homotopy regularization algorithm is applied to solve the inverse problem utilizing Legendre polynomials as the basis functions of the approximate space. The inversion solutions with noisy data which give good approximations to the exact solution demonstrate effectiveness of the inversion algorithm for the simultaneous inversion problem.

Numerical Inversion for the Initial Distribution in the Multi-Term Time-Fractional Diffusion Equation Using Final Observations

2017 ◽  
Vol 9 (6) ◽  
pp. 1525-1546 ◽  
Author(s):  
Chunlong Sun ◽  
Gongsheng Li ◽  
Xianzheng Jia
Keyword(s):  
Diffusion Equation ◽  
Initial Distribution ◽  
Fractional Diffusion ◽  
Fractional Diffusion Equation ◽  
Numerical Inversion ◽  
Inversion Problem ◽  
Inversion Algorithm ◽  
Regularization Algorithm ◽  
Lower Semi ◽  
Time Fractional Diffusion Equation

AbstractThis article deals with numerical inversion for the initial distribution in the multi-term time-fractional diffusion equation using final observations. The inversion problem is of instability, but it is uniquely solvable based on the solution's expression for the forward problem and estimation to the multivariate Mittag-Leffler function. From view point of optimality, solving the inversion problem is transformed to minimizing a cost functional, and existence of a minimum is proved by the weakly lower semi-continuity of the functional. Furthermore, the homotopy regularization algorithm is introduced based on the minimization problem to perform numerical inversions, and the inversion solutions with noisy data give good approximations to the exact initial distribution demonstrating the efficiency of the inversion algorithm.

scholarly journals The noise handling properties of the Talbot algorithm for numerically inverting the Laplace transform

2018 ◽  
Vol 13 ◽  
pp. 174830181879706 ◽  
Author(s):  
Colin L Defreitas ◽  
Steve J Kane
Keyword(s):  
Fourier Series ◽  
Laplace Transform ◽  
Noisy Data ◽  
Standard Test ◽  
Numerical Inversion ◽  
Test Functions ◽  
Inversion Algorithm ◽  
Inverse Transform ◽  
Exact Data ◽  
The Laplace Transform

This paper examines the noise handling properties of three of the most widely used algorithms for numerically inverting the Laplace transform. After examining the genesis of the algorithms, their error handling properties are evaluated through a series of standard test functions in which noise is added to the inverse transform. Comparisons are then made with the exact data. Our main finding is that the for “noisy data”, the Talbot inversion algorithm performs with greater accuracy when compared to the Fourier series and Stehfest numerical inversion schemes as they are outlined in this paper.

scholarly journals Analysis of fractional non-linear diffusion behaviors based on Adomian polynomials

2017 ◽  
Vol 21 (2) ◽  
pp. 813-817 ◽  
Author(s):  
Guo-Cheng Wu ◽  
Dumitru Baleanu ◽  
Wei-Hua Luo
Keyword(s):  
Taylor Expansion ◽  
Numerical Solutions ◽  
Fractional Diffusion ◽  
Diffusion Equations ◽  
Linear Diffusion ◽  
Adomian Polynomials ◽  
Fractional Diffusion Equations ◽  
Non Linear ◽  
Recurrence Formulae ◽  
Diffusion Behaviors

A time-fractional non-linear diffusion equation of two orders is considered to investigate strong non-linearity through porous media. An equivalent integral equation is established and Adomian polynomials are adopted to linearize non-linear terms. With the Taylor expansion of fractional order, recurrence formulae are proposed and novel numerical solutions are obtained to depict the diffusion behaviors more accurately. The result shows that the method is suitable for numerical simulation of the fractional diffusion equations of multi-orders.

scholarly journals Discrete fractional diffusion model with two memory terms

2015 ◽  
Vol 19 (4) ◽  
pp. 1177-1181
Author(s):  
Yan-Mei Qin ◽  
Hua Kong ◽  
Kai-Teng Wu ◽  
Xiao-Ming Zhu
Keyword(s):  
Numerical Simulation ◽  
Fractional Calculus ◽  
Diffusion Model ◽  
Discrete Time ◽  
Anomalous Diffusion ◽  
Fractional Diffusion ◽  
Fractional Difference ◽  
Linear Evolution ◽  
Time Domains ◽  
Diffusion Behaviors

Fractional calculus can always exactly describe anomalous diffusion. Recently the discrete fractional difference is becoming popular due to the depiction of non-linear evolution on discrete time domains. This paper proposes a diffusion model with two terms of discrete fractional order. The numerical simulation is given to reveal various diffusion behaviors.

scholarly journals 3D magnetic amplitude inversion in the presence of self-demagnetization and remanent magnetization

Geophysics ◽  
2019 ◽  
Vol 84 (5) ◽  
pp. J69-J82 ◽  
Author(s):  
Boxin Zuo ◽  
Xiangyun Hu ◽  
Yi Cai ◽  
Shuang Liu
Keyword(s):  
Remanent Magnetization ◽  
Large Scale ◽  
Magnetic Data ◽  
Inversion Problem ◽  
Aeromagnetic Data ◽  
Inversion Algorithm ◽  
Rapid Convergence ◽  
Magnetization Direction ◽  
Amplitude Inversion ◽  
Nonlinear Conjugate Gradient

We have developed a general 3D amplitude inversion algorithm for magnetic data in the presence of self-demagnetization and remanent magnetization. The algorithm uses a nonlinear conjugate gradient (NLCG) scheme to invert the amplitude of the magnetic anomaly vector within a partial differential equation framework. Three quantities— the amplitude of the anomalous magnetic field, the analytic signal, and the normalized source strength, defined as the amplitudes of magnetic data that are weakly dependent on the magnetization direction — are inverted to recover the 3D distribution of the subsurface magnetic susceptibility. Numerical experiments indicate that our NLCG amplitude inversion algorithm has a rapid convergence rate that provides a reasonable inversion solution in the absence of knowing the total magnetization direction. High-resolution aeromagnetic data collected from the Pea Ridge iron oxide-apatite-rare earth element deposit, southeast Missouri, USA, are used to illustrate the efficacy of our amplitude inversion algorithm. This algorithm is generally applicable for tackling the large-scale inversion problem in the presence of self-demagnetization and remanent magnetization.

scholarly journals Recovering multiple fractional orders in time-fractional diffusion in an unknown medium

2021 ◽  
Vol 477 (2253) ◽  
pp. 20210468
Author(s):  
Bangti Jin ◽  
Yavar Kian
Keyword(s):  
Single Point ◽  
Numerical Procedure ◽  
Nonlinear Least Squares ◽  
Fractional Diffusion ◽  
Small Time ◽  
Initial Condition ◽  
Model Function ◽  
Multiple Orders ◽  
Full Knowledge ◽  
Fractional Orders

In this work, we investigate an inverse problem of recovering multiple orders in a time-fractional diffusion model from the data observed at one single point on the boundary. We prove the unique recovery of the orders together with their weights, which does not require a full knowledge of the domain or medium properties, e.g. diffusion and potential coefficients, initial condition and source in the model. The proof is based on Laplace transform and asymptotic expansion. Furthermore, inspired by the analysis, we propose a numerical procedure for recovering these parameters based on a nonlinear least-squares fitting with either fractional polynomials or rational approximations as the model function, and provide numerical experiments to illustrate the approach for small time t .

scholarly journals Seismic reflectivity inversion using an L1-norm basis-pursuit method and GPU parallelisation

2020 ◽  
Author(s):  
Ruo Wang ◽  
Yanghua Wang ◽  
Ying Rao
Keyword(s):  
Large Scale ◽  
Seismic Inversion ◽  
Computational Time ◽  
Basis Pursuit ◽  
Inversion Problem ◽  
Inversion Algorithm ◽  
L1 Norm ◽  
The Matrix ◽  
Seismic Reflectivity ◽  
Reflectivity Inversion

Abstract Seismic reflectivity inversion problem can be formulated using a basis-pursuit method, aiming to generate a sparse reflectivity series of the subsurface media. In the basis-pursuit method, the reflectivity series is composed by large amounts of even and odd dipoles, thus the size of the seismic response matrix is huge and the matrix operations involved in seismic inversion are very time-consuming. In order to accelerate the matrix computation, a basis-pursuit method-based seismic inversion algorithm is implemented on Graphics Processing Unit (GPU). In the basis-persuit inversion algorithm, the problem is imposed with a L1-norm model constraint for sparsity, and this L1-norm basis-pursuit inversion problem is reformulated using a linear programming method. The core problems in the inversion are large-scale linear systems, which are resolved by a parallelised conjugate gradient method. The performance of this fully parallelised implementation is evaluated and compared to the conventional serial coding. Specifically, the investigation using several field seismic data sets with different sizes indicates that GPU-based parallelisation can significantly reduce the computational time with an overall factor up to 145. This efficiency improvement demonstrates a great potential of the basis-pursuit inversion method in practical application to large-scale seismic reflectivity inversion problems.

scholarly journals Inversion of hyperoutput time-delay systems

2019 ◽  
Vol 484 (5) ◽  
pp. 538-541
Author(s):  
A. V. Il’in ◽  
E. I. Atamas ◽  
V. V. Fomichev
Keyword(s):  
Time Delay ◽  
Canonical Form ◽  
Delay System ◽  
Delay Systems ◽  
Zero Dynamics ◽  
Inversion Problem ◽  
Time Delay Systems ◽  
Inversion Algorithm ◽  
Time Delay System

An inversion problem for LTI hyperoutput time-delay system is considered. For such systems canonical form with isolated zero dynamics is obtained, system invariant zeros and their relation to spectral observability of zero dynamics subsystem are investigated. Using this results, inversion algorithm is suggested for time-delay systems.

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