A multilevel augmentation method for solving ill-posed operator equations

Chen, Zhongying; Xu, Yuesheng; Yang, Hongqi

doi:10.1088/0266-5611/22/1/009

Cited by 61 publications

(63 citation statements)

References 31 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…We first prove inequality (12). Noting that Q n,i+1 := P n,i+1 − P n,i for i ∈ N 0 , by the triangle inequality, we get that…”

Section: Lemma 4 If Is Given and λ > 0 Then There Exists A Positive mentioning

confidence: 97%

“…Substituting (14) into (13) and using the summability of the geometric series, we obtain (12). From the definition of −1 n , we have that…”

Section: Lemma 4 If Is Given and λ > 0 Then There Exists A Positive mentioning

confidence: 99%

“…Since matrix U L is full and matrix V L is spare, we use Gaussian elimination algorithm to solve Eq. 20 and use the multilevel augmentation method in [12,15] In this experiment, we use multiparameter regularization with on parameter for each order term. For each of Eqs.…”

Section: Approximation By a Compressed Matrixmentioning

confidence: 99%

“…Increase in order of kernels results in huge increase in the computational cost. To overcome this difficulty, we propose to employ the fast multiscale collocation method developed in [11,13,14,46] to compress the matrix obtained from identifying the Volterra kernels and use the augmentation method developed in [12] to solve the resulting linear system.…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Multiparameter regularization for Volterra kernel identification via multiscale collocation methods

Brenner

Jiang

2008

Adv Comput Math

Self Cite

View full text Add to dashboard Cite

Identification of the Volterra system is an ill-posed problem. We propose a regularization method for solving this ill-posed problem via a multiscale collocation method with multiple regularization parameters corresponding to the multiple scales. Many highly nonlinear problems such as flight data analysis demand identifying the system of a high order. This task requires huge computational costs due to processing a dense matrix of a large order. To overcome this difficulty a compression strategy is introduced to approximate the full matrix resulted in collocation of the Volterra kernel by an appropriate sparse matrix. A numerical quadrature strategy is designed to efficiently compute the entries of the compressed matrix. Finally, numerical results of three simulation experiments are presented to demonstrate the accuracy and efficiency of the proposed method. Communicated by Juan Manuel Peña. 422 M. Brenner et al.

show abstract

“…We first prove inequality (12). Noting that Q n,i+1 := P n,i+1 − P n,i for i ∈ N 0 , by the triangle inequality, we get that…”

Section: Lemma 4 If Is Given and λ > 0 Then There Exists A Positive mentioning

confidence: 97%

“…Substituting (14) into (13) and using the summability of the geometric series, we obtain (12). From the definition of −1 n , we have that…”

Section: Lemma 4 If Is Given and λ > 0 Then There Exists A Positive mentioning

confidence: 99%

Section: Approximation By a Compressed Matrixmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Multiparameter regularization for Volterra kernel identification via multiscale collocation methods

Brenner

Jiang

2008

Adv Comput Math

Self Cite

View full text Add to dashboard Cite

show abstract

“…Moreover, solving this linear system requires O(n log 3 2 n) number of multiplications by the multilevel augmentation method proposed in [7]. Hence, it is a fast fully discrete algorithm.…”

Section: The Numerical Integration Scheme For Vector Fmentioning

confidence: 99%

A fast numerical solution for the first kind boundary integral equation for the Helmholtz equation

Cai

2012

Bit Numer Math

View full text Add to dashboard Cite

The main purpose of this paper is to develop a fast numerical method for solving the first kind boundary integral equation, arising from the two-dimensional interior Dirichlet boundary value problem for the Helmholtz equation with a smooth boundary. This method leads to a fully discrete linear system with a sparse coefficient matrix. We observe that it requires a nearly linear computational cost to produce and then solve this system, and the corresponding approximate solution obtained by this proposed method preserves the optimal convergence order. One numerical example demonstrates the efficiency and accuracy of the proposed method.

show abstract

Multilevel augmentation methods for solving the Burgers' equation

Chen

Cheng

et al. 2015

Numer. Methods Partial Differential Eq.

Self Cite

View full text Add to dashboard Cite

In this article, multilevel augmentation method (MAM) for solving the Burgers' equation is developed. The Crank-Nicolson-Galerkin scheme of the Burgers' equation results in nonlinear algebraic systems at each time step, the computational cost for solving these nonlinear systems is huge. The MAM allows us to solve the nonlinear system at a fixed initial lower level and then compensate the error by solving a linear system at the higher level. We prove that the method has the same optimal convergence order as the projection method, while reducing the computational complexity greatly. Finally, numerical experiments are presented to confirm the theoretical analysis and illustrate the efficiency of the proposed method.

show abstract

A multilevel augmentation method for solving ill-posed operator equations

Cited by 61 publications

References 31 publications

Multiparameter regularization for Volterra kernel identification via multiscale collocation methods

Multiparameter regularization for Volterra kernel identification via multiscale collocation methods

A fast numerical solution for the first kind boundary integral equation for the Helmholtz equation

Multilevel augmentation methods for solving the Burgers' equation

Contact Info

Product

Resources

About