In this paper we propose a new class of iterative regularization methods for solving ill-posed linear operator equations. The prototype of these iterative regularization methods is in the form of second order evolution equation with a linear vanishing damping term, which can be viewed not only as an extension of the asymptotical regularization, but also as a continuous analog of the Nesterov's acceleration scheme. New iterative regularization methods are derived from this continuous model in combination with damped symplectic numerical schemes. The regularization property as well as convergence rates and acceleration effects under the Holder-type source conditions of both continuous and discretized methods are proven. The second part of this paper is concerned with the application of the newly developed accelerated iterative regularization methods to the diffusion-based bioluminescence tomography, which is modeled as an inverse source problem in elliptic partial differential equations with both Dirichlet and Neumann boundary data. A relaxed mathematical formulation is proposed so that the discrepancy principle can be applied to the iterative scheme without the usage of Sobolev embedding constants. Several numerical examples, as well as a comparison with the state-of-the-art methods, are given to show the accuracy and the acceleration effect of the new methods.

中文翻译：

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一类新的加速正则化方法，适用于生物发光断层扫描

在本文中，我们提出了一类新的迭代正则化方法，用于求解不适定线性算子方程。这些迭代正则化方法的原型是具有线性消失阻尼项的二阶演化方程，它不仅可以看作是渐近正则化的扩展，而且可以看作是 Nesterov 加速方案的连续模拟。新的迭代正则化方法是从这个连续模型与阻尼辛数值方案相结合推导出来的。证明了连续和离散方法在Holder型源条件下的正则化特性以及收敛速度和加速效果。本文的第二部分涉及新开发的加速迭代正则化方法在基于扩散的生物发光断层扫描中的应用，该方法被建模为具有 Dirichlet 和 Neumann 边界数据的椭圆偏微分方程中的逆源问题。提出了一种宽松的数学公式，以便在不使用 Sobolev 嵌入常数的情况下将差异原理应用于迭代方案。给出了几个数值例子，以及与最先进方法的比较，以显示新方法的准确性和加速效果。提出了一种宽松的数学公式，以便在不使用 Sobolev 嵌入常数的情况下将差异原理应用于迭代方案。给出了几个数值例子，以及与最先进方法的比较，以显示新方法的准确性和加速效果。提出了一种宽松的数学公式，以便在不使用 Sobolev 嵌入常数的情况下将差异原理应用于迭代方案。给出了几个数值例子，以及与最先进方法的比较，以显示新方法的准确性和加速效果。