Title: A sparse approximation of the Lieb functional with moment constraints Show Chinese title

Title: 具有矩约束的Lieb泛函的稀疏逼近

Abstract: The aim of this paper is to present new sparsity results about the so-called Lieb functional, which is a key quantity in Density Functional Theory for electronic structure calculations of molecules. The Lieb functional was actually shown by Lieb to be a convexification of the so-called L\'evy-Lieb functional. Given an electronic density for a system of $N$ electrons, which may be seen as a probability density on $\mathbb{R}^3$, the value of the Lieb functional for this density is defined as the solution of a quantum multi-marginal optimal transport problem, which reads as a minimization problem defined on the set of trace-class operators acting on the space of electronic wave-functions that are anti-symmetric $L^2$ functions of $\mathbb{R}^{3N}$, with partial trace equal to the prescribed electronic density. We introduce a relaxation of this quantum optimal transport problem where the full partial trace constraint is replaced by a finite number of moment constraints on the partial trace of the set of operators. We show that, under mild assumptions on the electronic density, there exist sparse minimizers to the resulting moment constrained approximation of the Lieb (MCAL) functional that read as operators with rank at most equal to the number of moment constraints. We also prove under appropriate assumptions on the set of moment functions that the value of the MCAL functional converges to the value of the exact Lieb functional as the number of moments go to infinity. We also prove some rates of convergence on the associated approximation of the ground state energy. We finally study the mathematical properties of the associated dual problem and introduce a suitable numerical algorithm in order to solve some simple toy models. Show Chinese abstract

Abstract: 本文的目的是提出关于所谓的李偏函数的新稀疏性结果，该函数是分子电子结构计算中密度泛函理论中的一个关键量。李偏函数实际上被李证明是所谓的勒维-李偏函数的凸化。给定一个由$N$个电子组成的系统的电子密度，可以看作是$\mathbb{R}^3$上的概率密度，该密度的李偏函数值定义为一个量子多边缘最优传输问题的解，该问题可表述为在一个作用于电子波函数空间上的迹类算子集合上定义的最小化问题，这些波函数是$\mathbb{R}^{3N}$的反对称$L^2$函数，且部分迹等于指定的电子密度。 我们引入了这个量子最优传输问题的一个松弛版本，在这个版本中，完整的部分迹约束被有限数量的部分迹矩约束所取代。我们证明了，在电子密度满足温和假设的情况下，存在稀疏的最小化器来近似李偏（MCAL）泛函，这些最小化器表示为秩不超过矩约束数量的算子。我们也证明了，在适当的矩函数集假设下，当矩的数量趋于无穷时，MCAL 泛函的值收敛到精确李偏泛函的值。我们还证明了与基态能量相关近似的一些收敛速度。最后，我们研究了相关对偶问题的数学性质，并引入了一个合适的数值算法以求解一些简单的玩具模型。