反问题的随机正则化方法

Computational lager scale inverse problem has been attracted a lot of attentions in signal processing, computed tomography, machine learning, parameter identification in recent years. The main difficult comes from the ill-posedness of the inverse problem, and the computational cost for the large scale data. We will study the randomized regularization method for this kind of problems, which combines the variational regularization (or iterative regularization) technique for the inverse problem and the randomized strategy to reduce the computational cost. First, constructing an accurate but cheaper surrogate to the linear operator by randomized projection or randomized SVD, we consider the variational regularization to the surrogate problem. Its convergence and optimal convergence rate will be proved under a proper regularization parameter selection rule. Second, we can show that the stochastic gradient descent algorithm and randomized Gauss-Newton algorithm are iterative regularization method in the expectation sense, when they are equipped with a variance reduction strategy and a properly chosen stopping condition. Last, we will give the pre-asymptotic error analysis for stochastic gradient descent and randomized Gauss-Newton method. Some new randomized iterative algorithm for the inverse problem will be proposed to ensure the fast convergence before the stopping condition is fulfilled. These algorithms will be applied to computed tomography and other problems.

反问题的大规模计算在信号处理、CT成像、机器学习、参数识别等领域具有广泛的应用，其具有反问题的不适定性和计算规模大的困难。本项目主要研究该类反问题的随机正则化方法，将反问题的正则化方法与随机性算法进行结合处理反问题的大规模计算问题。为了处理反问题的不适定性，我们采用变分正则化或者迭代正则化；为了减少计算代价，我们采用随机替代函数方法或者随机梯度下降等随机优化算法。首先我们研究线性反问题采用随机替代函数后的变分正则化理论，包括正则化参数的选择，正则化解的收敛性和最佳收敛阶。其次对于反问题的随机梯度下降和随机Gauss-Newton方法我们给出合适的方差控制技术以及恰当的停机准则，使该随机迭代方法具有期望意义下的正则化效应。最后我们通过研究误差的初始传播情况，结合迭代正则化的早停机策略，构造适用于反问题的随机迭代方法，并将其应用于CT成像等具体问题中。

不适定反问题在信号处理，机器学习，图像恢复，高维统计数据分析，微分方程参数识别等领域有着广阔的应用。本项目主要研究正则化方法的理论和应用，在反问题的正则化理论、带稀疏约束线性回归问题的快速计算方法和应用、机器学习中的模型和算法中取得了一系列有意义的成果。给出了随机梯度下降方法的正则化效应，在变分正则化中建立启发式参数选择方法，给出了稀疏约束问题的牛顿型算法，并分析了深度学习求解偏微分方程的误差阶，在项目的支持下，在相关领域的主流期刊上发表了22篇论文。