Non-dissipative and structure-preserving emulators via spherical optimization
Non-dissipative and structure-preserving emulators via spherical optimization
复制标题
通过球形优化实现非耗散且结构保持的模拟器
DOI：
10.1093/imaiai/iaac021
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发表时间：
2023
期刊：
影响因子：
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通讯作者：
Narayan, Akil
中科院分区：
文献类型：
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作者：
Dai, Dihan;Epshteyn, Yekaterina;Narayan, Akil
用有限级数逼近函数，例如，涉及多项式或三角函数，是计算和数据分析中的关键工具。通过现在的标准方法（如最小二乘法或压缩采样）构建这种近似并不能确保近似符合某些凸线性结构约束，如正性或单调性。确保这种结构的现有方法是规范耗散的，并且这在应用这些方法时可能具有有害影响，例如，当数值求解偏微分方程时。我们提出了一个新的框架，通过优化这种结构的近似值，同时保持范数。这导致在球面上的概念上简单的凸优化问题，但这类问题的可行集可能非常复杂。我们通过球面凸性的结果建立了优化问题的适定性，并设计了几个基于球面投影的算法来数值计算解决方案。最后，我们通过几个数值例子证明了这种方法的有效性。
Approximating a function with a finite series, e.g., involving polynomials or trigonometric functions, is a critical tool in computing and data analysis. The construction of such approximations via now-standard approaches like least squares or compressive sampling does not ensure that the approximation adheres to certain convex linear structural constraints, such as positivity or monotonicity. Existing approaches that ensure such structure are norm-dissipative and this can have a deleterious impact when applying these approaches, e.g., when numerical solving partial differential equations. We present a new framework that enforces via optimization such structure on approximations and is simultaneously norm-preserving. This results in a conceptually simple convex optimization problem on the sphere, but the feasible set for such problems can be very complex. We establish well-posedness of the optimization problem through results on spherical convexity and design several spherical-projection-based algorithms to numerically compute the solution. Finally, we demonstrate the effectiveness of this approach through several numerical examples.