Geodesic
An efficient direct method for the numerical solution to the Cauchy problem for the Laplace equation
Sibirskij žurnal vyčislitelʹnoj matematiki, Tome 22 (2019) no. 1, pp. 99-117

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One of widespread approaches to solving the Cauchy problem for the Laplace equation is to reduce it to the inverse problem. As a rule, an iterative procedure to solve the latter is used. In this study, an efficient direct method for the numerical solution of the inverse problem in the rectangular form is described. The main idea is based on the expansion of the desired solution with respect to a basis consisting of eigenfunctions of a difference analogue of the Laplace operator. 
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     author = {S. B. Sorokin},
     title = {An efficient direct method for the numerical solution to the {Cauchy} problem for the {Laplace} equation},
     journal = {Sibirskij \v{z}urnal vy\v{c}islitelʹnoj matematiki},
     pages = {99--117},
     publisher = {mathdoc},
     volume = {22},
     number = {1},
     year = {2019},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/SJVM_2019_22_1_a6/}
}
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S. B. Sorokin. An efficient direct method for the numerical solution to the Cauchy problem for the Laplace equation. Sibirskij žurnal vyčislitelʹnoj matematiki, Tome 22 (2019) no. 1, pp. 99-117. http://geodesic.mathdoc.fr/item/SJVM_2019_22_1_a6/