论文标题
通过Wasserstein梯度流求解第一类的Fredholm积分方程
Solving Fredholm Integral Equations of the First Kind via Wasserstein Gradient Flows
论文作者
论文摘要
在应用科学的许多领域，求解第一类的弗雷德方程至关重要。在这项工作中，我们通过考虑在概率措施中使用熵正则化的概率问题来采用概率和变化的观点。与离散解决方案域的经典方法相反，我们引入了一种算法，以从正则最小化问题的独特溶液中渐近地采样。结果，我们的估计器不依赖于任何潜在的网格，并且比大多数现有方法具有更好的可伸缩性属性。我们的算法基于McKean-vlasov随机微分方程的粒子近似，与我们的变异配方的Wasserstein梯度流有关。我们证明了对最小化器的融合，并为其数值实施提供了实用的准则。最后，将我们的方法与其他几个示例中的方法进行了比较，包括密度反卷积和流行病学。
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.
