论文标题
半连接的部分局部整数差异方程及其复杂性分析的多级PICARD近似算法
Multilevel Picard approximation algorithm for semilinear partial integro-differential equations and its complexity analysis
论文作者
论文摘要
在本文中，我们介绍了半连接抛物线抛物线部分差异方程式（pides）的多级PICARD近似算法。我们证明，数值近似方案会收敛到所考虑的pide的独特粘度解。为此，我们为半线性脚步的独特粘度解提供了Feynman-kac表示，从而扩展了线性针的经典Feynman-kac表示。此外，我们表明该算法不会受到维数的诅咒，即算法的计算复杂性在多个尺寸$ d $和处方准确性$ \ varepsilon $的倒数中限制。我们还提供了多达10 000个维度的数值示例，以证明其适用性。
In this paper we introduce a multilevel Picard approximation algorithm for semilinear parabolic partial integro-differential equations (PIDEs). We prove that the numerical approximation scheme converges to the unique viscosity solution of the PIDE under consideration. To that end, we derive a Feynman-Kac representation for the unique viscosity solution of the semilinear PIDE, extending the classical Feynman-Kac representation for linear PIDEs. Furthermore, we show that the algorithm does not suffer from the curse of dimensionality, i.e. the computational complexity of the algorithm is bounded polynomially in the dimension $d$ and the reciprocal of the prescribed accuracy $\varepsilon$. We also provide a numerical example in up to 10'000 dimensions to demonstrate its applicability.
