Smoothing approximations for piecewise smooth functions: a probabilistic approach. (English) Zbl 1497.65032
Summary: In this article, we present a new approach to construct smoothing approximations for piecewise smooth functions. This approach proposes to formulate any piecewise smooth function as the expectation of a random variable. Based on this formulation, we show that smoothing all elements of a defined space of piecewise smooth functions is equivalent to smooth a single probability distribution. Furthermore, we propose to use the Boltzmann distribution as a smoothing approximation for this probability distribution. Moreover, we present the theoretical results, error estimates, and some numerical examples for this new smoothing method in both one-dimensional and multiple-dimensional cases.
MSC:
| 65D10 | Numerical smoothing, curve fitting |
| 26A27 | Nondifferentiability (nondifferentiable functions, points of nondifferentiability), discontinuous derivatives |
| 49J52 | Nonsmooth analysis |
| 65K05 | Numerical mathematical programming methods |
| 90C30 | Nonlinear programming |
Keywords:
piecewise smooth functions; smoothing approximations; smoothing techniques; non-smooth analysis; Boltzmann distributionReferences:
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