Abstract
The article presents a stable algorithm of solving an integral equation of the first kind taking into account the a priori information on the sought solution.
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Abbreviations
- ϕ (x):
-
sought solutlon of the integral equation
- K(y, x):
-
kernel of the integral equation
- f(y):
-
exact right-hand side of the integral equation
- \(\varphi ^{(l_i )} (x_i^* )(i = 1,...,N_r )\) :
-
derivative of li-th order of the function ϕ(x) at the point x *1
- d1 (i=1, ..., Nr):
-
constraints imposed on the values\(\varphi ^{(l_i )} (x)\)
- \(\tilde f_i (i = 1,...,n)\) :
-
measured values of the righthand side of the integral equation
- ξi :
-
measurement noise
- Pm,z,v :
-
domain of the descriptive solutions
- \(\tilde x_i (i = 1,...,N + m + 1)\) :
-
nodes of the B-spline Bj,m of the m-th degree
- N:
-
dimensionality of the domain of the solutions
- αj :
-
coefficients of the solution in the base of the B-splines
- φ(α):
-
quadratic functional
- pi :
-
weighting factors contained in φ(α)
- gi(α) (i=1, ..., Nr):
-
linear form determining the permissible domain Ω dN for the coefficientsα j
- Nopt :
-
optimum dimensionality of the domain of the solutions
- ei(N) (i=1, ..., n):
-
discrepancy of the i-th measurement
- ϕN(x),α * :
-
rms solution
- ϕN(x, α**):
-
descriptive solution of the integral equation
Literature cited
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V. A. Morozov, N. L. Gol'dman, and M. K. Samarin, “Methods of descriptive regularization and the quality of approximate solutions,” Inzh.-Fiz. Zh.,33, No. 6, 1117–1123 (1977).
C. Boor, A Practical Guide to Spline Function, Springer-Verlag, New York (1978).
Yu. S. Zav'yalov, B. I. Kvasov, and V. L. Miroshnichenko, Methods of Spline Functions [in Russian], Nauka, Moscow (1980).
F. P. Vasil'ev, Numerical Methods of Solving Extremum Problems [in Russian], Nauka, Moscow (1980).
Yu. E. Voskoboinikov, “Construction of descriptive approximations of experimental data by B-splines,” in: Algorithmic and instrumental means of information processing, Institut Teplofizika Sib. Otd. Akad. Nauk SSSR, Novosibirsk (1981), pp. 36–46.
Yu. E. Voskoboinikov, “Criteria and algorithms for selecting parameters in smoothing by spline functions,” in: Algorithms for Processing and Means of Automating Thermophysical Experiments, Institut Teplofiziki Sib. Otd. Akad. Nauk SSSR, Novosibirsk (1978), pp. 30–45.
G. H. Goluband M. Heath, “Generalized cross-validation as a method for choosing a good ridge parameter,” Technometrics,21, No. 2, 215–223 (1979).
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Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 45, No. 5, pp. 760–765, November, 1983.
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Voskoboinikov, Y.E., Preobrazhenskii, N.G. Descriptive solution of the inverse heat-conduction problem in the B-spline basis. Journal of Engineering Physics 45, 1254–1258 (1983). https://doi.org/10.1007/BF01254728
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DOI: https://doi.org/10.1007/BF01254728