Numerical Integration in Multiple Dimensions with Designed Quadrature
Numerical Integration in Multiple Dimensions with Designed Quadrature
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DOI：
10.1137/17m1137875
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发表时间：
2018-04
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影响因子：
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通讯作者：
Vahid Keshavarzzadeh;R. Kirby;A. Narayan
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作者：
Vahid Keshavarzzadeh;R. Kirby;A. Narayan
我们提出了一个系统的计算框架，产生积极的求积规则在一般几何形状的多维。一个直接的时刻匹配制定，强制执行多项式子空间的精确积分产生非线性条件和几何约束的节点和权重。我们使用惩罚方法来解决几何约束，并随后通过高斯-牛顿方法解决二次最小化问题。我们的分析为给定多项式子空间的求积规则的必要大小提供了指导，并在多项式矩条件被少量违反的情况下，对求积规则中的误差给出了有用的用户端稳定性界限，例如，有限的精度限制或优化过程的停滞。我们提出了几个数值例子调查最佳的低程度的求积规则，勒贝格常数，和100维求积。我们的顶点的例子比较我们的正交方法流行的替代品，如稀疏网格和准蒙特卡罗方法，在线性弹性和拓扑优化的问题。
We present a systematic computational framework for generating positive quadrature rules in multiple dimensions on general geometries. A direct moment-matching formulation that enforces exact integration on polynomial subspaces yields nonlinear conditions and geometric constraints on nodes and weights. We use penalty methods to address the geometric constraints, and subsequently solve a quadratic minimization problem via the Gauss-Newton method. Our analysis provides guidance on requisite sizes of quadrature rules for a given polynomial subspace, and furnishes useful user-end stability bounds on error in the quadrature rule in the case when the polynomial moment conditions are violated by a small amount due to, e.g., finite precision limitations or stagnation of the optimization procedure. We present several numerical examples investigating optimal low-degree quadrature rules, Lebesgue constants, and 100-dimensional quadrature. Our capstone examples compare our quadrature approach to popular alternatives, such as sparse grids and quasi-Monte Carlo methods, for problems in linear elasticity and topology optimization.