On enforcing non-negativity in polynomial approximations in high dimensions
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arXiv
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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2024
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| _version_ | 1866909381224824832 |
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| author | Chen, Yuan Xiu, Dongbin Zhang, Xiangxiong |
| author_facet | Chen, Yuan Xiu, Dongbin Zhang, Xiangxiong |
| contents | Polynomial approximations of functions are widely used in scientific computing. In certain applications, it is often desired to require the polynomial approximation to be non-negative (resp. non-positive), or bounded within a given range, due to constraints posed by the underlying physical problems. Efficient numerical methods are thus needed to enforce such conditions. In this paper, we discuss effective numerical algorithms for polynomial approximation under non-negativity constraints. We first formulate the constrained optimization problem, its primal and dual forms, and then discuss efficient first-order convex optimization methods, with a particular focus on high dimensional problems. Numerical examples are provided, for up to $200$ dimensions, to demonstrate the effectiveness and scalability of the methods. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2401_12415 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On enforcing non-negativity in polynomial approximations in high dimensions Chen, Yuan Xiu, Dongbin Zhang, Xiangxiong Numerical Analysis 42C05, 41A10, 65K10, 90C25 Polynomial approximations of functions are widely used in scientific computing. In certain applications, it is often desired to require the polynomial approximation to be non-negative (resp. non-positive), or bounded within a given range, due to constraints posed by the underlying physical problems. Efficient numerical methods are thus needed to enforce such conditions. In this paper, we discuss effective numerical algorithms for polynomial approximation under non-negativity constraints. We first formulate the constrained optimization problem, its primal and dual forms, and then discuss efficient first-order convex optimization methods, with a particular focus on high dimensional problems. Numerical examples are provided, for up to $200$ dimensions, to demonstrate the effectiveness and scalability of the methods. |
| title | On enforcing non-negativity in polynomial approximations in high dimensions |
| topic | Numerical Analysis 42C05, 41A10, 65K10, 90C25 |
| url | https://arxiv.org/abs/2401.12415 |