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13
On stable numerical differentiation
 Mathem. of Computation
, 1968
"... Abstract. A new approach to the construction of finitedifference methods is presented. It is shown how the multipoint differentiators can generate regularizing algorithms with a stepsize h being a regularization parameter. The explicitly computable estimation constants are given. Also an iterative ..."
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Cited by 39 (22 self)
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Abstract. A new approach to the construction of finitedifference methods is presented. It is shown how the multipoint differentiators can generate regularizing algorithms with a stepsize h being a regularization parameter. The explicitly computable estimation constants are given. Also an iteratively regularized scheme for solving the numerical differentiation problem in the form of Volterra integral equation is developed. 1.
Stable numerical differentiation: when is it possible
 Jour. Korean SIAM
"... Abstract. Two principally different statements of the problem of stable numerical differentiation are considered. It is analyzed when it is possible in principle to get a stable approximation to the derivative f ′ given noisy data fδ. Computational aspects of the problem are discussed and illustrate ..."
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Cited by 11 (8 self)
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Abstract. Two principally different statements of the problem of stable numerical differentiation are considered. It is analyzed when it is possible in principle to get a stable approximation to the derivative f ′ given noisy data fδ. Computational aspects of the problem are discussed and illustrated by examples. These examples show the practical value of the new understanding of the problem of stable differentiation. 1.
Error Analysis for a Class of Numerical Differentiator: Application to State Observation
 in "48th IEEE Conference on Decision and Control
, 2009
"... Résumé Ce rapport est consacré aux estimations des dérivées. Contrairement à la procédure de régularisation de Tikhonov, nous utilisons un cadre algébrique récent qui conduit enfin à une projection dans la base de polynômes de Jacobi, afin d’estimer des dérivées des signaux bruités. Aucune informati ..."
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Cited by 5 (2 self)
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Résumé Ce rapport est consacré aux estimations des dérivées. Contrairement à la procédure de régularisation de Tikhonov, nous utilisons un cadre algébrique récent qui conduit enfin à une projection dans la base de polynômes de Jacobi, afin d’estimer des dérivées des signaux bruités. Aucune information sur les propriétés statistiques du bruit est requise. Nous donnons quelques résultats concernant le choix des paramètres dans cette méthode de manière à minimiser l’erreur due au bruit et les erreurs d’approximation. De plus, deux nouveaux estimateurs centraux fondés sur telles techniques algébriques de la différenciation sont introduits. Une comparaison est faite entre ces estimations et quelques méthodes classiques de la différentiation numérique. inria00439386, version 1 7 Dec 2009 1
A Variational Method for Numerical Differentiation
, 1995
"... this paper, we suggest a method for numerical differentiation that we believe avoids some of the limitations mentioned above. Given a real valued, smooth function u defined on the closed interval [a; b], we construct an associated functional, ..."
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Cited by 4 (0 self)
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this paper, we suggest a method for numerical differentiation that we believe avoids some of the limitations mentioned above. Given a real valued, smooth function u defined on the closed interval [a; b], we construct an associated functional,
A scheme for stable numerical differentiation
 J. COMP. APPL. MATH.
, 2006
"... A method for stable numerical differentiation of noisy data is proposed. The method requires solving a Volterra integral equation of the second kind. This equation is solved analytically. In the examples considered its solution is computed analytically. Some numerical results of its application are ..."
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Cited by 2 (1 self)
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A method for stable numerical differentiation of noisy data is proposed. The method requires solving a Volterra integral equation of the second kind. This equation is solved analytically. In the examples considered its solution is computed analytically. Some numerical results of its application are presented. These examples show that the proposed method for stable numerical differentiation is numerically more efficient than some other methods, in particular, than variational regularization.
ON COMPLEXVALUED 2D EIKONALS. PART FOUR: CONTINUATION PAST A CAUSTIC
, 905
"... Abstract. Theories of monochromatic highfrequency electromagnetic fields have been designed by Felsen, Kravtsov, Ludwig and others with a view to portraying features that are ignored by geometrical optics. These theories have recourse to eikonals that encode information on both phase and amplitude ..."
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Abstract. Theories of monochromatic highfrequency electromagnetic fields have been designed by Felsen, Kravtsov, Ludwig and others with a view to portraying features that are ignored by geometrical optics. These theories have recourse to eikonals that encode information on both phase and amplitude — in other words, are complexvalued. The following mathematical principle is ultimately behind the scenes: any geometric optical eikonal, which conventional rays engender in some light region, can be consistently continued in the shadow region beyond the relevant caustic, provided an alternative eikonal, endowed with a nonzero imaginary part, comes on stage. In the present paper we explore such a principle in dimension 2. We investigate a partial differential system that governs the real and the imaginary parts of complexvalued twodimensional eikonals, and an initial value problem germane to it. In physical terms, the problem in hand amounts to detecting waves that rise beside, but on the dark side of, a given caustic. In mathematical terms, such a problem shows two main peculiarities: on the one hand, degeneracy near the initial curve; on the other hand, illposedness in the sense of Hadamard. We benefit from using a number of technical devices: hodograph transforms, artificial viscosity, and a suitable discretization. Approximate differentiation and a parody of the quasireversibility method are also involved. We offer an algorithm that restrains instability and produces effective approximate solutions. 1.
Acknowledgments
, 2008
"... careful guidance and unwavering support, and for his continual encouragement. He introduced me to many advanced spectral methods and suggested the topic of this thesis. I enjoyed the many discussions we had, both mathematical and nonmathematical, which I will always keep in good memory. I wish to ex ..."
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careful guidance and unwavering support, and for his continual encouragement. He introduced me to many advanced spectral methods and suggested the topic of this thesis. I enjoyed the many discussions we had, both mathematical and nonmathematical, which I will always keep in good memory. I wish to express my gratitude to Professor Tom Y. Hou for encouraging me to join the Applied and Computational Mathematics Program at Caltech and for his kind support especially during the first graduate year. Special thanks are due to Professor Christophe Geuzaine, for providing me with Gmsh, a threedimensional finite element mesh generator he and his coworkers developed, which can be downloaded from
Approximate Solution of LargeScale Linear Inverse Problems with Monte Carlo Simulation ∗
, 2009
"... We consider the approximate solution of linear illposed inverse problems of high dimension with a simulationbased algorithm that approximates the solution within a lowdimensional subspace. The algorithm uses Tikhonov regularization, regression, and lowdimensional linear algebra calculations and ..."
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We consider the approximate solution of linear illposed inverse problems of high dimension with a simulationbased algorithm that approximates the solution within a lowdimensional subspace. The algorithm uses Tikhonov regularization, regression, and lowdimensional linear algebra calculations and storage. For sampling efficiency, we use variance reduction/importance sampling schemes, specially tailored to the structure of inverse problems. We demonstrate the implementation of our algorithm in a series of practical largescale examples arising from Fredholm integral equations of the first kind. 1