Regularization of ill-posed problems in Banach spaces: convergence rates

Resmerita, Elena

doi:10.1088/0266-5611/21/4/007

Cited by 140 publications

(146 citation statements)

References 21 publications

(45 reference statements)

Supporting

Mentioning

143

Contrasting

Unclassified

Order By: Relevance

“…Solving the problem is computationally feasible. We refer to Eggermont (1993), Burger and Osher (2004), and Resmerita (2005) for regularization with general convex regularization functional. Furthermore, we refer to Newey and Powell (2003) for the related nonparametric IV problem.…”

Section: Nonparametric Estimatormentioning

confidence: 99%

Nonparametric identification of endogenous and heterogeneous aggregate demand models: complements, bundles and the market level

Dunker¹,

Hoderlein²,

Kaido³

2015

View full text Add to dashboard Cite

This paper studies nonparametric identification in market level demand models for differentiated products. We generalize common models by allowing for the distribution of heterogeneity parameters (random coefficients) to have a nonparametric distribution across the population and give conditions under which the density of the random coefficients is identified. We show that key identifying restrictions are provided by (i) a set of moment conditions generated by instrumental variables together with an inversion of aggregate demand in unobserved product characteristics; and (ii) an integral transform (Radon transform) that maps the random coefficient density to the aggregate demand.This feature is shown to be common across a wide class of models, and we illustrate this by studying leading demand models. Our examples include demand models based on the multinomial choice (Berry, Levinsohn, Pakes, 1995), the choice of bundles of goods that can be substitutes or complements, and the choice of goods consumed in multiple units.

show abstract

Section: Nonparametric Estimatormentioning

confidence: 99%

Nonparametric identification of endogenous and heterogeneous aggregate demand models: complements, bundles and the market level

Dunker¹,

Hoderlein²,

Kaido³

2015

View full text Add to dashboard Cite

show abstract

“…To account of the sensitivity to noise a common practice is to invoke a regularization method to tackle this sort of ill-posed problems (Kunisch & Zou, 1998;Resmerita, 2005;Wang & Xiao, 2001;Xie & Zou, 2002), where a suitable regularization parameter is utilized to depress the bias in the computed solution via a better balance of the approximation error and the propagated data error. Developed were several techniques after the pioneering work of Tikhonov & Arsenin (1977).…”

Section: Ill-posed Problems and Remedymentioning

confidence: 99%

Accelerated Bidirectional Modifications of the Steepest Descent Method for Ill-posed Linear Algebraic Systems with Dynamics-theoretical and Optimization Interpretation

Liu¹,

Hong²,

Liu³

2015

JMR

View full text Add to dashboard Cite

It is well known that the classical numerical algorithm of the steepest descent method (SDM) is effective for well-posed linear systems, but performs poorly for ill-posed ones. In this paper we propose accelerated and/or bidirectional modifications of SDM, namely the accelerated steepest descent method (ASDM), the bidirectional method (2DM), and the accelerated bidirectional method (A2DM). The starting point is a manifold defined in terms of a minimum functional and a fictitious time variable; nevertheless, in the end the fictitious time variable disappears so that we arrive at purely iterative algorithms. The proposed algorithms are justified by dynamics-theoretical and optimization interpretation. The accelerator plays a prominent role of switching from the situation of slow convergence to a new situation that the functional tends to decrease stepwise in an intermittent and ceaseless manner. Three examples of solving ill-posed systems are examined and comparisons are made with exact solutions and with the existing algorithms of the SDM, the Barzilai-Borwein method, and the random SDM, revealing that the new algorithms of ASDM and A2DM have better computational efficiency and accuracy even for the highly ill-posed systems.

show abstract

“…Many practical problems can be converted into solving this problem, such as ill-posed problems, inverse problems, some constrained optimization problems, and model parameter estimation [27,31,28,29,14,15].…”

Section: Numerical Experimentsmentioning

confidence: 99%

Global Convergence of a New Hybrid Gauss–Newton Structured BFGS Method for Nonlinear Least Squares Problems

Zhou¹,

Chen²

2010

SIAM J. Optim.

View full text Add to dashboard Cite

Abstract. In this paper, we propose a hybrid Gauss-Newton structured BFGS method with a new update formula and a new switch criterion for the iterative matrix to solve nonlinear least squares problems. We approximate the second term in the Hessian by a positive definite BFGS matrix. Under suitable conditions, global convergence of the proposed method with a backtracking line search is established. Moreover, the proposed method automatically reduces to the GaussNewton method for zero residual problems and the structured BFGS method for nonzero residual problems in a neighborhood of an accumulation point. A locally quadratic convergence rate for zero residual problems and a locally superlinear convergence rate for nonzero residual problems are obtained for the proposed method. Some numerical results are given to compare the proposed method with some existing methods.

show abstract

Regularization of ill-posed problems in Banach spaces: convergence rates

Cited by 140 publications

References 21 publications

Nonparametric identification of endogenous and heterogeneous aggregate demand models: complements, bundles and the market level

Nonparametric identification of endogenous and heterogeneous aggregate demand models: complements, bundles and the market level

Accelerated Bidirectional Modifications of the Steepest Descent Method for Ill-posed Linear Algebraic Systems with Dynamics-theoretical and Optimization Interpretation

Global Convergence of a New Hybrid Gauss–Newton Structured BFGS Method for Nonlinear Least Squares Problems

Contact Info

Product

Resources

About