Finding the most probable non-negative solution of systems of linear algebraic equations by the likelihood method
Sibirskij žurnal vyčislitelʹnoj matematiki, Tome 16 (2013) no. 3, pp. 217-228.

Voir la notice de l'article provenant de la source Math-Net.Ru

A probabilistic method for regularization is proposed. This method enables one to obtain a non-negative solution to systems of linear algebraic equations. A theorem of existence of the best possible solution is proved. A numerical example of the method application is given.
@article{SJVM_2013_16_3_a2,
     author = {V. S. Antyufeev},
     title = {Finding the most probable non-negative solution of systems of linear algebraic equations by the likelihood method},
     journal = {Sibirskij \v{z}urnal vy\v{c}islitelʹnoj matematiki},
     pages = {217--228},
     publisher = {mathdoc},
     volume = {16},
     number = {3},
     year = {2013},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/SJVM_2013_16_3_a2/}
}
TY  - JOUR
AU  - V. S. Antyufeev
TI  - Finding the most probable non-negative solution of systems of linear algebraic equations by the likelihood method
JO  - Sibirskij žurnal vyčislitelʹnoj matematiki
PY  - 2013
SP  - 217
EP  - 228
VL  - 16
IS  - 3
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/SJVM_2013_16_3_a2/
LA  - ru
ID  - SJVM_2013_16_3_a2
ER  - 
%0 Journal Article
%A V. S. Antyufeev
%T Finding the most probable non-negative solution of systems of linear algebraic equations by the likelihood method
%J Sibirskij žurnal vyčislitelʹnoj matematiki
%D 2013
%P 217-228
%V 16
%N 3
%I mathdoc
%U http://geodesic.mathdoc.fr/item/SJVM_2013_16_3_a2/
%G ru
%F SJVM_2013_16_3_a2
V. S. Antyufeev. Finding the most probable non-negative solution of systems of linear algebraic equations by the likelihood method. Sibirskij žurnal vyčislitelʹnoj matematiki, Tome 16 (2013) no. 3, pp. 217-228. http://geodesic.mathdoc.fr/item/SJVM_2013_16_3_a2/

[1] Tikhonov A. N., “O nekorrektnykh zadachakh lineinoi algebry i ustoichivom metode ikh resheniya”, Dokl. AN SSSR, 163:6 (1965), 591–595

[2] Tikhonov A. N., “O regulyarizatsii nekorrektno postavlennykh zadach”, Dokl. AN SSSR, 153:1 (1963), 49–52 | MR | Zbl

[3] Tikhonov A. N., Arsenin V. Ya., Metody resheniya nekorrektnykh zadach, Nauka, M., 1979 | MR

[4] Turchin V. F., “Reshenie uravneniya Fredgolma 1-go roda v statisticheskom ansamble gladkikh funktsii”, ZhVMiMF, 7:6 (1967), 1270–1284 | MR | Zbl

[5] Kozlov V. L., Malkevich S. M., Turchin V. F., “Ispolzovanie metodov matematicheskoi statistiki dlya resheniya nekorrektnykh zadach”, UFN, 3:2 (1970), 345–386

[6] Turchin V. F., Nozik V. Z., “Statisticheskaya regulyarizatsiya resheniya nekorrektnykh zadach”, Izv. AN SSSR. Ser. FAO, 5:1 (1969), 255–267

[7] Butler J. P., Reeds J. A., Dawson S. V., “Estimating solutions of first kind integral equations with nonnegative constraints and optimal smoothing”, SIAM J. on Numerical Analysis, 18:3 (1981), 410–421 | DOI | MR

[8] Mosegaard K., Tarantola A., “Monte Carlo sampling of solutions to inverse problems”, J. Geophys. Res., 100:7 (1995), 12431–12447 | DOI

[9] Kramer G., Matematicheskie metody statistiki, Mir, M., 1975 | MR

[10] Anderson T., Vvedenie v mnogomernyi statisticheskii analiz, Fizmatgiz, M., 1963

[11] Ivanov V. K., Vasin V. V., Tanana V. P., Teoriya lineinykh nekorrektnykh zadach i ee prilozheniya, Nauka, M., 1978 | MR

[12] Moiseev N. N., Ivanilov Yu. P., Stolyarova E. M., Metody optimizatsii, Nauka, M., 1978

[13] Vasilev V. P., Chislennye metody resheniya ekstremalnykh zadach, Nauka, M., 1980 | MR

[14] Ermakov S. M., Mikhailov G. A., Kurs statisticheskogo modelirovaniya, Nauka, M., 1976 | MR | Zbl