## On stable numerical differentiation (1968)

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Venue: | Mathem. of Computation |

Citations: | 42 - 23 self |

### BibTeX

@ARTICLE{Ramm68onstable,

author = {Alexander G. Ramm and Alexandra and B. Smirnova},

title = {On stable numerical differentiation},

journal = {Mathem. of Computation},

year = {1968},

volume = {70},

pages = {1131--1153}

}

### Years of Citing Articles

### OpenURL

### Abstract

Abstract. A new approach to the construction of finite-difference methods is presented. It is shown how the multi-point 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.

### Citations

927 |
Solutions of Ill-Posed Problems
- Tikhonov, Arsenin
- 1977
(Show Context)
Citation Context ...t the specific class to which the function to be differentiated belongs. Most of the regularization procedures [4]-[7], [27], [28] that belong to the third category make use of the variational ([14], =-=[26]-=-) approach for solving ill-posed problems. These methods typically involve writing the derivative as the solution to an integral Volterra equation and then reducing the integral equation to a family o... |

121 |
A technique for the numerical solution of certain integral equations of the first kind
- Phillips
- 1962
(Show Context)
Citation Context ...n about the specific class to which the function to be differentiated belongs. Most of the regularization procedures [4]-[7], [27], [28] that belong to the third category make use of the variational (=-=[14]-=-, [26]) approach for solving ill-posed problems. These methods typically involve writing the derivative as the solution to an integral Volterra equation and then reducing the integral equation to a fa... |

72 |
The Radon Transform and Local Tomography
- Ramm, Katsevich
- 1996
(Show Context)
Citation Context ...ch yield satisfactory results when the function to be differentiated is given very precisely ([9], [2], [4]). In Sections 1 and 2 of our paper the other view on these methods, based on the works [16]-=-=[23]-=- and [12], is presented (see also [10] where the results and ideas of [16] are used). Namely, it Received by the editor August 5, 1999. 2000 Mathematics Subject Classification. Primary 65D25; Secondar... |

17 | Numerical differentiation procedures for non-exact data - Anderssen, Bloomfield |

17 |
Ill-Posed Problems with a Priori Information
- Vasin, Ageev
- 1995
(Show Context)
Citation Context ...ans that Volterra integral equations allow one to generate stable numerical methods by application of quadrature formula directly to the initial equation and the following theorem holds. Theorem 5.1 (=-=[29]-=-). Let (5.2) �x a K(x, s)z(s) ds = fδ(x), ||fδ − f|| C[a,b] ≤ δ, a ≤ x ≤ b,s1152 A. G. RAMM AND A. B. SMIRNOVA and assume 1. mina≤x≤b |K(x, x)| > 0, K ′ x (x, s) ∈ C(∆), � K′′ xs (x, s) ∈ C(∆), where ... |

15 | A numerical method for solving nonlinear ill-posed problems, Numerical Funct
- Ramm, Smirnova
- 1999
(Show Context)
Citation Context ... the result in [16] generalized and applied in [17]-[23]. In Section 4 we suggest an iteratively regularized scheme for solving a Volterra equation based on the idea of continuous regularization (see =-=[1]-=-), which is an alternative to the variational one. This procedure avoids some of the limitations in a choice of the regularization parameter mentioned above. In Section 5 the results of the numerical ... |

13 |
Numerical differentiation and regularization
- Cullum
- 1971
(Show Context)
Citation Context ...own. They have the advantage of simplicity and are considered by many authors to be the ones which yield satisfactory results when the function to be differentiated is given very precisely ([9], [2], =-=[4]-=-). In Sections 1 and 2 of our paper the other view on these methods, based on the works [16]-[23] and [12], is presented (see also [10] where the results and ideas of [16] are used). Namely, it Receiv... |

7 |
de Hoog, Finite difference methods for the numerical differentiation of nonexact data
- Anderssen, R
(Show Context)
Citation Context ... have been developed for numerical differentiation. They fall into three categories: difference methods, interpolation methods and regularization methods. The first two categories (see, for instance, =-=[3]-=-, [15], [24], [25] and others), especially the central difference formula that can be related to both of them, are well known. They have the advantage of simplicity and are considered by many authors ... |

5 |
On numerical differentiation
- Dolgopolova, Ivanov
- 1966
(Show Context)
Citation Context ...r is found, the corresponding well-posed problem is solved to obtain an estimate for the derivative. Unfortunately, the determination of the optimal parameter value is generally a nontrivial task. In =-=[6]-=- the authors propose the quasi-solution method (see [8]) for regularization, which can be described as follows: find the coefficients of the expansion of fδ in the Legendre polynomials Pk(x) (1.1) bk ... |

5 | A variational method for numerical differentiation
- Knowles, Wallace
- 1995
(Show Context)
Citation Context ...re well known. They have the advantage of simplicity and are considered by many authors to be the ones which yield satisfactory results when the function to be differentiated is given very precisely (=-=[9]-=-, [2], [4]). In Sections 1 and 2 of our paper the other view on these methods, based on the works [16]-[23] and [12], is presented (see also [10] where the results and ideas of [16] are used). Namely,... |

4 |
Estimates of the derivatives of random functions
- Miller
- 1985
(Show Context)
Citation Context ...satisfactory results when the function to be differentiated is given very precisely ([9], [2], [4]). In Sections 1 and 2 of our paper the other view on these methods, based on the works [16]-[23] and =-=[12]-=-, is presented (see also [10] where the results and ideas of [16] are used). Namely, it Received by the editor August 5, 1999. 2000 Mathematics Subject Classification. Primary 65D25; Secondary 65D05. ... |

4 | of the derivatives of random functions - Estimates - 1984 |

3 |
Numerical solution of the problem of reconstructing the derivative (in Russian
- Kolpakova
- 1976
(Show Context)
Citation Context ... function to be differentiated is given very precisely ([9], [2], [4]). In Sections 1 and 2 of our paper the other view on these methods, based on the works [16]-[23] and [12], is presented (see also =-=[10]-=- where the results and ideas of [16] are used). Namely, it Received by the editor August 5, 1999. 2000 Mathematics Subject Classification. Primary 65D25; Secondary 65D05. Key words and phrases. Numeri... |

3 |
Light-induced and H ∗ -ion fluxes and bioelectric phenomena in mesophyll cells of Atriplex Spongiosa, Zeit. fuer Pflanz. 62
- Pallaghy, Luttge
- 1970
(Show Context)
Citation Context ...ever, in many applications it is necessary to estimate the derivative of a function given the noisy values of this function. As an example we refer to the analysis of photoelectric response data (see =-=[13, 1970]-=-). The goal of this experiment is to determine the relationship between the intensity of light falling on certain plant cells and their rate of uptake of various substances in order to gain further in... |

3 |
solutions of some ill-posed problems
- Stable
- 1981
(Show Context)
Citation Context ...at f(x) has derivatives up to order m and ||f (m) || ≤ Mm, m>2. Let fδ ∈ L∞ (R) be given such that ||f − fδ|| ≤ δ, where the norm is calculated by (2.1). Suppose that m>2 is odd and define (see [17], =-=[19]-=-) (3.1) where the numbers A Q j (3.2) � � �T Q h fδ − f ′ �A j=−Q Q j T Q h fδ := h −1 Q� j=−Q A Q j fδ � x + jh � , Q (j = −Q, ..., Q) are to be determined. One has � � � � � ≤ �T Q h (fδ � � � � − f... |

2 | Finite dimensional regularization in the case of numerical differentiation of periodic functions - Dolgopolova - 1970 |

2 |
On a problem of numerical differentiation
- Egorov, Kondratiev
- 1989
(Show Context)
Citation Context ...se in the initial data, and b) one has to take into account a priori information about the specific class to which the function to be differentiated belongs. Most of the regularization procedures [4]-=-=[7]-=-, [27], [28] that belong to the third category make use of the variational ([14], [26]) approach for solving ill-posed problems. These methods typically involve writing the derivative as the solution ... |

2 |
Applied Analysis, Englewood Cliffs, N.J
- Lanczos
- 1956
(Show Context)
Citation Context ... +1) jh � (3.36) , Q or in the equivalent form (3.37) T Q ˜h fσ := � Q j=−Q j=−Q jfσ(x + j˜ h) 2˜ h �Q j=−Q j2 , h ˜ h := Q , which coincides with the least squares differentiator derived by Lanczos (=-=[11]-=-) and also investigated by Anderssen and de Hoog ([3]). However, neither in [11] nor in [3] the special choice of h by formula (3.32) was proposed to guarantee a better accuracy of the approximations.... |

2 |
Optimally stable lagrangian numerical differentiation
- Rivlin
- 1975
(Show Context)
Citation Context ...developed for numerical differentiation. They fall into three categories: difference methods, interpolation methods and regularization methods. The first two categories (see, for instance, [3], [15], =-=[24]-=-, [25] and others), especially the central difference formula that can be related to both of them, are well known. They have the advantage of simplicity and are considered by many authors to be the on... |

2 |
Regularization of numerical differentiation problem, Matem. zap. Ural’skii un-t. 7(2
- Vasin
- 1969
(Show Context)
Citation Context ...the initial data, and b) one has to take into account a priori information about the specific class, to which the function to be differentiated belongs. Most of the regularization procedures [4]-[7], =-=[27]-=-, [28] that belong to the third category make use of the variational ([14], [26]) approach for solving ill-posed problems. These methods typically involve writing the derivative as the solution to an ... |

1 |
On linear problems, which are not well-posed, Doklady Akad
- Ivanov
- 1962
(Show Context)
Citation Context ...ved to obtain an estimate for the derivative. Unfortunately, the determination of the optimal parameter value is generally a nontrivial task. In [6] the authors propose the quasi-solution method (see =-=[8]-=-) for regularization, which can be described as follows: find the coefficients of the expansion of fδ in the Legendre polynomials Pk(x) (1.1) bk = 2k +1 2 �1 −1 “choosing n so that ||qn+1 −fδ|| ≤ δ an... |

1 |
A new approach to numerical differentiation and regularization
- Qu
- 1996
(Show Context)
Citation Context ... been developed for numerical differentiation. They fall into three categories: difference methods, interpolation methods and regularization methods. The first two categories (see, for instance, [3], =-=[15]-=-, [24], [25] and others), especially the central difference formula that can be related to both of them, are well known. They have the advantage of simplicity and are considered by many authors to be ... |

1 | fields estimation theory, Longman Scientific and - Random - 1990 |

1 |
for the derivatives
- Inequalities
(Show Context)
Citation Context ...e numerical differentiation with various a priori information are given. 2. Inequalities for the derivatives 2.1. The main result. In this section we investigate and answer Questions 2.1 and 2.2 (see =-=[22]-=-). Question 2.1. Given fδ ∈ L ∞ (R) andf ∈ C 1 (R) such that inequalities ||fδ − f|| ≤ δ, ||f (m) || ≤ Mm < ∞, m =0, 1, hold with some known δ and unknown (or roughly estimated) Mm, can one compute f ... |

1 |
Some problems in optimally stable Lagrangian differentiation
- Salzer
- 1974
(Show Context)
Citation Context ...ped for numerical differentiation. They fall into three categories: difference methods, interpolation methods and regularization methods. The first two categories (see, for instance, [3], [15], [24], =-=[25]-=- and others), especially the central difference formula that can be related to both of them, are well known. They have the advantage of simplicity and are considered by many authors to be the ones whi... |

1 |
Regularization of a numerical differentiation problem
- Vasin
- 1969
(Show Context)
Citation Context ... the initial data, and b) one has to take into account a priori information about the specific class to which the function to be differentiated belongs. Most of the regularization procedures [4]-[7], =-=[27]-=-, [28] that belong to the third category make use of the variational ([14], [26]) approach for solving ill-posed problems. These methods typically involve writing the derivative as the solution to an ... |

1 |
The stable evaluation of a derivative in
- Vasin
- 1973
(Show Context)
Citation Context ...nitial data, and b) one has to take into account a priori information about the specific class to which the function to be differentiated belongs. Most of the regularization procedures [4]-[7], [27], =-=[28]-=- that belong to the third category make use of the variational ([14], [26]) approach for solving ill-posed problems. These methods typically involve writing the derivative as the solution to an integr... |

1 |
On a problem of numerical differentiation, Vestnik Moskov
- Egorov
- 1989
(Show Context)
Citation Context ...e in the initial data, and b) one has to take into account a priori information about the specific class, to which the function to be differentiated belongs. Most of the regularization procedures [4]-=-=[7]-=-, [27], [28] that belong to the third category make use of the variational ([14], [26]) approach for solving ill-posed problems. These methods typically involve writing the derivative as the solution ... |

1 |
On linear ill-posed problems, Doklady Akad
- Ivanov
- 1962
(Show Context)
Citation Context ...olved to obtain an estimate for the derivative. Unfortunately the determination of the optimal parameter value is generally a nontrivial task. In [6] the authors propose the quasisolution method (see =-=[8]-=-) for regularization, which can be described as follows: find the coefficients of the expansion of fδ in the Legendre polynomials Pk(x): (1.1) bk = 2k + 1 2 �1 −1 ”choosing n so that ||qn+1 −fδ|| ≤ δ ... |

1 |
Applied Analysis
- Lanczoc
- 1956
(Show Context)
Citation Context ... 1) jh � (3.36) , Q or in the equivalent form: (3.37) T Q ˜h fσ := � Q j=−Q j=−Q jfσ(x + j˜ h) 2˜ h �Q j=−Q j2 , h ˜ h := Q , which coincides with the least squares differentiator derived by Lanczos (=-=[11]-=-) and also investigated by Anderssen and de Hoog ([3]). However, neither in [11] nor in [3] the special choice of h by formula (3.32) was proposed to guarantee a better accuracy of the approximations.... |