scholarly journals Some transpose-free CG-like solvers for nonsymmetric ill-posed problems

2020 ◽  
Vol 28 (1) ◽  
pp. 15-32
Author(s):  
Silvia Gazzola ◽  
Paolo Novati
Keyword(s):  
Krylov Subspace ◽  
Main Idea ◽  
System Matrix ◽  
Numerical Tests ◽  
New Methods ◽  
Arnoldi Algorithm ◽  
Ill Posed ◽  
Efficient Alternative ◽  
Original Class ◽  
Theoretical Insight

AbstractThis paper introduces and analyzes an original class of Krylov subspace methods that provide an efficient alternative to many well-known conjugate-gradient-like (CG-like) Krylov solvers for square nonsymmetric linear systems arising from discretizations of inverse ill-posed problems. The main idea underlying the new methods is to consider some rank-deficient approximations of the transpose of the system matrix, obtained by running the (transpose-free) Arnoldi algorithm, and then apply some Krylov solvers to a formally right-preconditioned system of equations. Theoretical insight is given, and many numerical tests show that the new solvers outperform classical Arnoldi-based or CG-like methods in a variety of situations.

A Refined Arnoldi Algorithm Based Krylov Subspace Technique for MEMS Model Order Reduction

2012 ◽  
Vol 503 ◽  
pp. 260-265
Author(s):  
Le Guan ◽  
Jia Li Gao ◽  
Zhi Wen Wang ◽  
Guo Qing Zhang ◽  
Jin Kui Chu
Keyword(s):  
Model Order Reduction ◽  
Krylov Subspace ◽  
Order Reduction ◽  
Original System ◽  
Reduced Order Model ◽  
System Matrix ◽  
Order Model ◽  
Model Order ◽  
Reduced Order ◽  
Arnoldi Algorithm

A refined approach producing MEMS numerical macromodels is proposed in this paper by generating the iterative Krylov subspace using a refined Arnoldi algorithm, which can reduce the degrees of freedom of the original system equations described by the state space method. Projection of the original system matrix onto the Krylov subspace which is spanned by a refined Arnoldi algorithm is still based on the transfer function moment matching principle. The idea of the iterative version is to expect that a new initial vector will contain more and more information on the required eigenvectors that is called refined vector. The refined approach improves approximation accuracy of the system matrix eigenvalues equivalent to a more accurate approximation to the poles of the system transfer function, obtaining a more accurate reduced-order model. The clamped beam model and the FOM model are reduced order by classical Arnoldi and refined Arnoldi algorithm in numerical experiments. From the computing result it is concluded that the refined Arnoldi algorithm based Krylov subspace technique for MEMS model order reduction has more accuracy and reaches lower order number of reduced order model than the classical Arnoldi process.

scholarly journals Regularization of backward time-fractional parabolic equations by Sobolev-type equations

2020 ◽  
Vol 28 (5) ◽  
pp. 659-676
Author(s):  
Dinh Nho Hào ◽  
Nguyen Van Duc ◽  
Nguyen Van Thang ◽  
Nguyen Trung Thành
Keyword(s):  
Error Estimates ◽  
Parabolic Equations ◽  
Regularization Parameter ◽  
A Priori ◽  
Sobolev Type ◽  
A Posteriori ◽  
Parameter Choice ◽  
Numerical Tests ◽  
Ill Posed ◽  
Parameter Choice Rules

AbstractThe problem of determining the initial condition from noisy final observations in time-fractional parabolic equations is considered. This problem is well known to be ill-posed, and it is regularized by backward Sobolev-type equations. Error estimates of Hölder type are obtained with a priori and a posteriori regularization parameter choice rules. The proposed regularization method results in a stable noniterative numerical scheme. The theoretical error estimates are confirmed by numerical tests for one- and two-dimensional equations.

scholarly journals Error Analysis for a Noisy Lacunary Cubic Spline Interpolation and a Simple Noisy Cubic Spline Quasi Interpolation

2014 ◽  
Vol 2014 ◽  
pp. 1-8
Author(s):  
Feng-Gong Lang ◽  
Xiao-Ping Xu
Keyword(s):  
Error Analysis ◽  
Error Bounds ◽  
Interpolation Method ◽  
Spline Interpolation ◽  
Cubic Spline ◽  
The Other ◽  
Cubic Spline Interpolation ◽  
Numerical Tests ◽  
New Methods ◽  
Noisy Function

We mainly present the error analysis for two new cubic spline based methods; one is a lacunary interpolation method and the other is a very simple quasi interpolation method. The new methods are able to reconstruct a function and its first two derivatives from noisy function data. The explicit error bounds for the methods are given and proved. Numerical tests and comparisons are performed. Numerical results verify the efficiency of our methods.

scholarly journals Robust simulation of mineral precipitation–dissolution problems with variable mineral surface area

2021 ◽  
Vol 129 (1) ◽  
Author(s):  
Serge Kräutle ◽  
Jan Hodai ◽  
Peter Knabner
Keyword(s):  
Surface Area ◽  
Ad Hoc ◽  
Chemical Species ◽  
Special Focus ◽  
Proof Of Concept ◽  
Mineral Surface ◽  
Mineral Precipitation ◽  
Numerical Tests ◽  
Ill Posed ◽  
Mineral Surface Area

AbstractWe consider a macroscale model of transport and reaction of chemical species in a porous medium with a special focus on mineral precipitation–dissolution processes. In the literature, it is frequently proposed that the reaction rate should depend on the reactive mineral surface area, and so on the amount of mineral. We point out that a frequently used model is ill posed in the sense that it admits non-unique solutions. We investigate what consequences this non-uniqueness has on the numerical solution of the model. The main novelty in this article is our proposal of a certain substitution which removes the ill-posedness from the system and which leads to better numerical results than some “ad hoc methods.” We think that the proposed substitution is a rather elegant way to get rid of the non-uniqueness and the numerical difficulties and is much less technical than other ideas. As a proof of concept, we present some numerical tests and simulations for the new model.

Refined Algorithm for Solving High-Order Harmonics of Multi Group Neutron Diffusion Equation

2017 ◽  
Author(s):  
Gaoxin Zhou ◽  
Zhi Gang
Keyword(s):  
Diffusion Equation ◽  
Iteration Method ◽  
Krylov Subspace ◽  
Krylov Subspace Method ◽  
Neutron Diffusion ◽  
Subspace Method ◽  
Arnoldi Method ◽  
Arnoldi Algorithm ◽  
Eigenvalue And Eigenvector ◽  
Neutron Diffusion Equation

In recent years, high order harmonic (or eigenvector) of neutron diffusion equation has been widely used in on-line monitoring system of reactor power. There are two kinds of calculation method to solve the equation: corrected power iteration method and Krylov subspace methods. Fu Li used the corrected power iteration method. When solving for the ith harmonic, it tries to eliminate the influence of the front harmonics using the orthogonality of the harmonic function. But its convergence speed depends on the occupation ratio. When the dominant ratios equal to 1 or close to 1, convergence speed of fixed source iteration method is slow or convergence can’t be achieved. Another method is the Krylov subspace method, the main idea of this method is to project the eigenvalue and eigenvector of large-scale matrix to a small one. Then we can solve the small matrix eigenvalue and eigenvector to get the large ones. In recent years, the restart Arnoldi method emerged as a development of Krylov subspace method. The method uses continuous reboot Arnoldi decomposition, limiting expanding subspace, and the orthogonality of the subspace is guaranteed using orthogonalization method. This paper studied the refined algorithms, a method based on the Krylov subspace method of solving eigenvalue problem for large sparse matrix of neutron diffusion equation. Two improvements have been made for a restarted Arnoldi method. One is that using an ingenious linear combination of the refined Ritz vector forms an initial vector and then generates a new Krylov subspace. Another is that retaining the refined Ritz vector in the new subspace, called, augmented Krylov subspace. This way retains useful information and makes the resulting algorithm converge faster. Several numerical examples are the new algorithm with the implicitly restart Arnoldi algorithm (IRA) and the implicitly restarted refined Arnoldi algorithm (IRRA). Numerical results confirm efficiency of the new algorithm.

scholarly journals A Wavelet Method for the Cauchy Problem for the Helmholtz Equation

2012 ◽  
Vol 2012 ◽  
pp. 1-18
Author(s):  
Fang-Fang Dou ◽  
Chu-Li Fu
Keyword(s):  
Cauchy Problem ◽  
Exact Solution ◽  
Helmholtz Equation ◽  
Error Estimates ◽  
Fixed Frequency ◽  
Wavelet Method ◽  
Numerical Tests ◽  
The Cauchy Problem ◽  
Ill Posed

We consider a Cauchy problem for the Helmholtz equation at a fixed frequency. The problem is severely ill posed in the sense that the solution (if it exists) does not depend continuously on the data. We present a wavelet method to stabilize the problem. Some error estimates between the exact solution and its approximation are given, and numerical tests verify the efficiency and accuracy of the proposed method.

EXPLICIT EIGHTH ORDER NUMEROV-TYPE METHODS WITH REDUCED NUMBER OF STAGES FOR OSCILLATORY IVPs

2008 ◽  
Vol 19 (06) ◽  
pp. 957-970 ◽  
Author(s):  
I. Th. FAMELIS
Keyword(s):  
Initial Value Problems ◽  
Computational Cost ◽  
Phase Lag ◽  
Initial Value ◽  
Eighth Order ◽  
Oscillatory Solutions ◽  
Numerical Tests ◽  
New Methods ◽  
Oscillating Solutions ◽  
Algebraic Order

Using a new methodology for deriving hybrid Numerov-type schemes, we present new explicit methods for the solution of second order initial value problems with oscillating solutions. The new methods attain algebraic order eight at a cost of eight function evaluations per step which is the most economical in computational cost that can be found in the literature. The methods have high amplification and phase-lag order characteristics in order to suit to the solution of problems with oscillatory solutions. The numerical tests in a variety of problems justify our effort.

scholarly journals Scattering-based focusing for imaging in highly complex media from band-limited, multi-component data

Geophysics ◽  
2021 ◽  
pp. 1-64
Author(s):  
David Vargas ◽  
Ivan Vasconcelos ◽  
Yanadet Sripanich ◽  
Matteo Ravasi
Keyword(s):  
A Priori ◽  
Alternative Form ◽  
Complex Media ◽  
Subsurface Structures ◽  
The Past ◽  
Numerical Tests ◽  
New Methods ◽  
Band Limited ◽  
Multicomponent Data ◽  
Realistic Data

Reconstructing the details of subsurface structures deep beneath complex overburden structures, such as sub-salt, remains a challenge for seismic imaging. Over the past years, the Marchenko redatuming approach has proven to reliably retrieve full-wavefield information in the presence of complex overburden effects. When used for redatuming, current practical Marchenko schemes cannot make use of a priori subsurface models with sharp contrasts because of their requirements regarding initial focusing functions, which for sufficiently complex media can result in redatumed fields with significant waveform inaccuracies. Using a scattering framework, we present an alternative form of the Marchenko representation that aims at retrieving only the unknown perturbations to both focusing functions and redatumed fields. From this framework, we propose a two-step practical focusing-based redatuming scheme that first solves an inverse problem for the background focusing functions, which are then used to estimate the perturbations to focusing functions and redatumed fields. In our scheme, initial focusing functions are significantly different from previous approaches since they contain complex waveforms encoding the full transmission response of the a priori model. Our goal is the handling of not only highly complex media but also realistic data - band-limited, unevenly sampled, free-surface-multiple contaminated data. To that end, we combine the versatility of Rayleigh-Marchenko redatuming with the proposed scattering-based scheme allowing an extended version of the method able to handle single-sided band-limited multicomponent data. This Scattering-Rayleigh-Marchenko strategy accurately retrieves wavefields while requiring minimum pre-processing of the data. In support of the new methods, we present a comprehensive set of numerical tests using a complex 2D subsalt model. Our numerical results show that the scattering approaches retrieve accurate redatumed fields that appropriately account for the complexity of the a priori model. We show that the improvements in wavefield retrieval translate into measurable improvements in our subsalt images.

scholarly journals A Krylov subspace type method for Electrical Impedance Tomography

2021 ◽  
Author(s):  
Mirjeta Pasha ◽  
Shyla Kupis ◽  
Sanwar Ahmad ◽  
Taufiquar Khan
Keyword(s):  
Electrical Impedance Tomography ◽  
Electrical Impedance ◽  
Krylov Subspace ◽  
Low Frequency ◽  
Small Dimension ◽  
Impedance Tomography ◽  
Type Method ◽  
Newton Algorithm ◽  
Ill Posed ◽  
Internal Conductivity

Electrical Impedance Tomography (EIT) is a well-known imaging technique for detecting the electrical properties of an object in order to detect anomalies, such as conductive or resistive targets. More specifically, EIT has many applications in medical imaging for the detection and location of bodily tumors since it is an affordable and non-invasive method, which aims to recover the internal conductivity of a body using voltage measurements resulting from applying low frequency current at electrodes placed at its surface. Mathematically, the reconstruction of the internal conductivity is a severely ill-posed inverse problem and yields a poor quality image reconstruction. To remedy this difficulty, at least in  part, we regularize and solve the nonlinear minimization problem by the aid of a Krylov subspace-type method for the linear sub problem during each iteration.  In EIT, a tumor or general anomaly can be modeled as a piecewise constant perturbation of a smooth background, hence, we solve the regularized problem on a subspace of relatively small dimension by the Flexible Golub-Kahan process that provides solutions that have sparse representation. For comparison, we use a well-known modified Gauss-Newton algorithm as a benchmark. Using simulations, we demonstrate the effectiveness of the proposed method. The obtained reconstructions indicate that the Krylov subspace method is better adapted to solve the ill-posed EIT problem and results in higher resolution images and faster convergence compared to reconstructions using the modified Gauss-Newton algorithm.

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

2021 ◽  
Vol 5 (3) ◽  
pp. 125
Author(s):  
Alicia Cordero ◽  
Cristina Jordán ◽  
Esther Sanabria-Codesal ◽  
Juan R. Torregrosa
Keyword(s):  
Fourth Order ◽  
Chaotic Behavior ◽  
Basins Of Attraction ◽  
Order Convergence ◽  
Iterative Schemes ◽  
Numerical Tests ◽  
New Methods ◽  
Stable Elements ◽  
Parameter Values ◽  
Theoretical Results

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.

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