Regularization of some linear ill-posed problems with discretized random noisy data

Mathé, Peter; Pereverzev, Sergei V.

doi:10.1090/s0025-5718-06-01873-4

Cited by 91 publications

(86 citation statements)

References 15 publications

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“…Thus, one needs to find a balance between the approximation and the noise propagation errors. This is achieved by presenting the a posteriori parameter choice rule, which is based on the so-called balancing principle that has been extensively studied (see, for example, [11], [20] and references therein).…”

Section: Adaptive Parameter Choicementioning

confidence: 99%

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Filtered Legendre expansion method for numerical differentiation at the boundary point with application to blood glucose predictions

Mhaskar

Naumova

Pereverzyev

2013

Applied Mathematics and Computation

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Let f : [−1, 1] → R be continuously differentiable. We consider the question of approximating f ′ (1) from given data of the form (t j , f (t j )) M j=1 where the points t j are in the interval [−1, 1]. It is well known that the question is ill-posed, and there is very little literature on the subject known to us. We consider a summability operator using Legendre expansions, together with high order quadrature formulas based on the points t j 's to achieve the approximation. We also estimate the effect of noise on our approximation. The error estimates, both with or without noise, improve upon those in the existing literature, and appear to be unimprovable. The results are applied to the problem of short term prediction of blood glucose concentration, yielding better results than other comparable methods.

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Section: Adaptive Parameter Choicementioning

confidence: 99%

“…Definition 2 Following [20], we say that a function ϕ(n) = ϕ(n; f, δ) is admissible for given f and δ if the following holds…”

Section: Adaptive Parameter Choicementioning

confidence: 99%

Filtered Legendre expansion method for numerical differentiation at the boundary point with application to blood glucose predictions

Mhaskar

Naumova

Pereverzyev

2013

Applied Mathematics and Computation

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“…Since then it has been developed further for regularization of linear inverse problems [50,135,105,106,18] and nonlinear inverse problems [12,13] in deterministic and stochastic settings. The notation we will use is taken from [9].…”

Section: Balancing Principlementioning

confidence: 99%

Comparingparameter choice methods for regularization of ill-posed problems

Bauer

Lukas

2011

Mathematics and Computers in Simulation

252

111

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In the literature on regularization, many different parameter choice methods have been proposed in both deterministic and stochastic settings. However, based on the available information, it is not always easy to know how well a particular method will perform in a given situation and how it compares to other methods. This paper reviews most of the existing parameter choice methods, and evaluates and compares them in a large simulation study for spectral cut-off and Tikhonov regularization. The test cases cover a wide range of linear inverse problems with both white and colored stochastic noise. The results show some marked differences between the methods, in particular, in their stability with respect to the noise and its type. We conclude with a table of properties of the methods and a summary of the simulation results, from which we identify the best methods.

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“…The idea of our adaptive principle has its origin in the paper [11], devoted to statistical estimation of function y(t) with unknown Hölder smoothness from direct observation blurred by Gaussian white noise. In the context of general ill-posed problems this idea has been realized in [5], [25], [8], [15]- [17], [20]. We use this idea for the adaptive choice of the stepsize in finite-difference methods because the structure of the error estimate (2.5) is very similar to the loss function of statistical estimation, where some parameter always controls the tradeoff between the bias and the variance of the risk.…”

Section: How Do We Approximate a Derivative Y (T) Of A Smooth Functiomentioning

confidence: 99%

Numerical differentiation from a viewpoint of regularization theory

Lü

Pereverzev

2006

Math. Comp.

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Abstract. In this paper, we discuss the classical ill-posed problem of numerical differentiation, assuming that the smoothness of the function to be differentiated is unknown. Using recent results on adaptive regularization of general ill-posed problems, we propose new rules for the choice of the stepsize in the finite-difference methods, and for the regularization parameter choice in numerical differentiation regularized by the iterated Tikhonov method. These methods are shown to be effective for the differentiation of noisy functions, and the order-optimal convergence results for them are proved.

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Regularization of some linear ill-posed problems with discretized random noisy data

Cited by 91 publications

References 15 publications

Filtered Legendre expansion method for numerical differentiation at the boundary point with application to blood glucose predictions

Filtered Legendre expansion method for numerical differentiation at the boundary point with application to blood glucose predictions

Comparingparameter choice methods for regularization of ill-posed problems

Numerical differentiation from a viewpoint of regularization theory

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