带记忆的演化方程中的自适应、快速且无记忆卷积
Adaptive, Fast and Oblivious Convolution in Evolution Equations with Memory
AI总结：
提出一种变步长算法，通过轮廓积分表示将卷积转化为微分方程并自适应求解，实现O(N log N)运算和O(log N)内存，替代直接实现的O(N^2)运算和O(N)内存。
摘要：
为了逼近带记忆项的演化方程中出现的卷积，提出了一种变步长算法，该算法推进N步仅需O(N log(N))次运算和O(log(N))活动内存，而直接实现需要O(N^2)次运算和O(N)内存。该快速算法的一个基本特征是通过轮廓积分表示将卷积简化为微分方程，并用自适应步长进行数值求解。算法中使用的是拉普拉斯变换而非核本身。通过三个例子说明该算法：源自具有集中非线性的薛定谔方程的爆破例子、具有抑制扩散的化学反应以及具有分数阶本构律的粘弹性。
英文摘要：
To approximate convolutions which occur in evolution equations with memory terms, a variable-stepsize algorithm is presented for which advancing N steps requires only O(N log(N)) operations and O(log(N)) active memory, in place of O(N^2) operations and O(N) memory for a direct implementation. A basic feature of the fast algorithm is the reduction, via contour integral representations, to differential equations which are solved numerically with adaptive step sizes. Rather than the kernel itself, its Laplace transform is used in the algorithm. The algorithm is illustrated on three examples: a blow-up example originating from a Schrödinger equation with concentrated nonlinearity, chemical reactions with inhibited diffusion, and viscoelasticity with a fractional order constitutive law.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
