论文标题
用牛顿的方法求解激光脉冲检索的非线性积分方程
Solving non-linear integral equations for laser pulse retrieval with Newton's method
论文作者
论文摘要
我们提出了一种基于数值技术的算法，该算法已成为求解非线性积分方程的标准：牛顿的方法，同型连续性，多级方法和随机投影，以求解在检索超级强度图中的超音脉冲的超级强度图（频率均匀）的（均与元素）中（均为元素）的频率（frestical gitional-gestions gitions-nougy-distical-cordical-nitugient）（frestical-cordical cordional gitional）脉冲（有效）（frestical n em-distical gational gitions gitions gations gations gations gitions gations）时出现的反转问题（实验。在这里，我们将求解器应用于青蛙并为类似积分指定必要的修改。与其他方法不同，我们会在时间域中改变积分和工作，在时间域中可以将积分作为过度确定的多项式系统离散，并通过列表自动相关进行评估。解决方案曲线是部分继续并部分随机的，由小的链接路径段组成，并可以计算噪声礼物中最佳溶液。有趣的是，这是一种发现多项式系统的真实解决方案的新方法，众所周知，这些解决方案很难找到。我们展示了如何实现自适应tikhonov型正则化以在处理嘈杂数据时平滑解决方案，我们将噪声测试数据的结果与最小二乘求解器进行比较，并提出L-Curve方法来微调正则化参数。
We present an algorithm based on numerical techniques that have become standard for solving nonlinear integral equations: Newton's method, homotopy continuation, the multilevel method and random projection to solve the inversion problem that appears when retrieving the electric field of an ultrashort laser pulse from a 2-dimensional intensity map measured with Frequency-resolved optical gating (FROG), dispersion-scan or amplitude-swing experiments. Here we apply the solver to FROG and specify the necessary modifications for similar integrals. Unlike other approaches we transform the integral and work in time-domain where the integral can be discretised as an over-determined polynomial system and evaluated through list auto-correlations. The solution curve is partially continues and partially stochastic, consisting of small linked path segments and enables the computation of optimal solutions in the presents of noise. Interestingly, this is a novel method to find real solutions of polynomial systems which are notoriously difficult to find. We show how to implement adaptive Tikhonov-type regularization to smooth the solution when dealing with noisy data, we compare the results for noisy test data with a least-squares solver and propose the L-curve method to fine-tune the regularization parameter.
