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Overcoming the curse of dimensionality in the numerical approximation of semilinear parabolic partial differential equations. (arXiv:1807.01212v1 [math.PR])
来源于:arXiv
For a long time it is well-known that high-dimensional linear parabolic
partial differential equations (PDEs) can be approximated by Monte Carlo
methods with a computational effort which grows polynomially both in the
dimension and in the reciprocal of the prescribed accuracy. In other words,
linear PDEs do not suffer from the curse of dimensionality. For general
semilinear PDEs with Lipschitz coefficients, however, it remained an open
question whether these suffer from the curse of dimensionality. In this paper
we partially solve this open problem. More precisely, we prove in the case of
semilinear heat equations with gradient-independent and globally Lipschitz
continuous nonlinearities that the computational effort of a variant of the
recently introduced multilevel Picard approximations grows polynomially both in
the dimension and in the reciprocal of the required accuracy. 查看全文>>