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| Natura: | Preprint |
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2025
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| Accesso online: | https://arxiv.org/abs/2509.14896 |
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| _version_ | 1866914045681991680 |
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| author | Croci, Matteo Haji-Ali, Abdul-Lateef Powell, Ian C. J. |
| author_facet | Croci, Matteo Haji-Ali, Abdul-Lateef Powell, Ian C. J. |
| contents | We propose a new numerical scheme for approximating level-sets of Lipschitz multivariate functions which is robust to stochastic noise. The algorithm's main feature is an adaptive grid-based stochastic approximation strategy which automatically refines the approximation over regions close to the level set. This strategy combines a local function approximation method with a noise reduction scheme and produces $\varepsilon$-accurate approximations with an expected cost complexity reduction of $\varepsilon^{-\left(\frac{p+1}{αp}\right)}$ compared to a non-adaptive scheme, where $α$ is the convergence rate of the function approximation method and we assume that the noise can be controlled in $L^p$. We provide numerical experiments in support of our theoretical findings. These include 2- and 3-dimensional functions with a complex level set structure, as well as a failure region estimation problem described by a hyperelasticity partial differential equation with random field coefficients. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_14896 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | An Adaptive Sampling Algorithm for Level-set Approximation Croci, Matteo Haji-Ali, Abdul-Lateef Powell, Ian C. J. Numerical Analysis 65Y20, 65D15, 65C05 We propose a new numerical scheme for approximating level-sets of Lipschitz multivariate functions which is robust to stochastic noise. The algorithm's main feature is an adaptive grid-based stochastic approximation strategy which automatically refines the approximation over regions close to the level set. This strategy combines a local function approximation method with a noise reduction scheme and produces $\varepsilon$-accurate approximations with an expected cost complexity reduction of $\varepsilon^{-\left(\frac{p+1}{αp}\right)}$ compared to a non-adaptive scheme, where $α$ is the convergence rate of the function approximation method and we assume that the noise can be controlled in $L^p$. We provide numerical experiments in support of our theoretical findings. These include 2- and 3-dimensional functions with a complex level set structure, as well as a failure region estimation problem described by a hyperelasticity partial differential equation with random field coefficients. |
| title | An Adaptive Sampling Algorithm for Level-set Approximation |
| topic | Numerical Analysis 65Y20, 65D15, 65C05 |
| url | https://arxiv.org/abs/2509.14896 |