通过Wasserstein梯度流求解第一类Fredholm积分方程
Solving Fredholm Integral Equations of the First Kind via Wasserstein Gradient Flows
AI总结：
针对第一类Fredholm积分方程求解问题，提出基于Wasserstein梯度流对应McKean–Vlasov方程粒子近似的无网格采样算法，证明收敛性并在多类算例中验证了更优的可扩展性。
摘要：
求解第一类Fredholm方程在应用科学的诸多领域中至关重要。在本研究中，我们采用概率与变分视角，考虑带熵正则项的概率测度空间极小化问题。与对解的定义域进行离散化的经典方法不同，我们提出一种算法，可从正则化极小化问题的唯一解中进行渐近采样。因此，我们的估计量不依赖任何底层网格，且比大多数现有方法具备更优的可扩展性。我们的算法基于McKean–Vlasov随机微分方程解的粒子近似，该方程与我们变分公式的Wasserstein梯度流相关。我们证明了算法向极小值点收敛，并为其数值实现提供了实用指导准则。最后，我们在密度反卷积、流行病学等多个算例上将该方法与其他方法进行了对比。
英文摘要：
Solving Fredholm equations of the first kind is crucial in many areas of the applied sciences. In this work we adopt a probabilistic and variational point of view by considering a minimization problem in the space of probability measures with an entropic regularization. Contrary to classical approaches which discretize the domain of the solutions, we introduce an algorithm to asymptotically sample from the unique solution of the regularized minimization problem. As a result our estimators do not depend on any underlying grid and have better scalability properties than most existing methods. Our algorithm is based on a particle approximation of the solution of a McKean--Vlasov stochastic differential equation associated with the Wasserstein gradient flow of our variational formulation. We prove the convergence towards a minimizer and provide practical guidelines for its numerical implementation. Finally, our method is compared with other approaches on several examples including density deconvolution and epidemiology.
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