Title: Measure transport via polynomial density surrogates Show Chinese title

Title: 通过多项式密度代理进行测量传输

Abstract: We discuss an algorithm to compute transport maps that couple the uniform measure on $[0,1]^d$ with a specified target distribution $\pi$ on $[0,1]^d$. The primary objectives are either to sample from or to compute expectations w.r.t. $\pi$. The method is based on leveraging a polynomial surrogate of the target density, which is obtained by a least-squares or interpolation approximation. We discuss the design and construction of suitable sparse approximation spaces, and provide a complete error and cost analysis for target densities belonging to certain smoothness classes. Further, we explore the relation between our proposed algorithm and related approaches that aim to find suitable transports via optimization over a class of parametrized transports. Finally, we discuss the efficient implementation of our algorithm and report on numerical experiments which confirm our theory. Show Chinese abstract

Abstract: 我们讨论一种算法，用于计算将均匀测度在$[0,1]^d$上与指定的目标分布$\pi$在$[0,1]^d$上耦合的运输映射。主要目标是采样或计算相对于$\pi$的期望值。该方法基于利用目标密度的多项式近似，该近似通过最小二乘或插值逼近获得。我们讨论了适合的稀疏逼近空间的设计和构建，并提供了针对属于某些光滑性类别的目标密度的完整误差和成本分析。此外，我们探讨了我们提出的算法与旨在通过在参数化运输类中优化来寻找合适运输的相关方法之间的关系。最后，我们讨论了我们算法的有效实现，并报告了确认我们理论的数值实验。