On unifying randomized methods for inverse problems
On unifying randomized methods for inverse problems
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DOI：
10.1088/1361-6420/acd36e
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发表时间：
2023-01
期刊：
影响因子：
2.1
通讯作者：
J. Wittmer;K. Giri;Hai Viet Nguyen;T. Bui-Thanh
中科院分区：
文献类型：
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作者：
J. Wittmer;K. Giri;Hai Viet Nguyen;T. Bui-Thanh
这项工作统一了各种随机方法的分析，解决线性和非线性反问题与高斯先验的随机优化设置的问题。通过这样做，我们表明，许多随机化方法的样本平均近似（SAA）的变体。更重要的是，我们能够证明一个单一的理论结果，保证了各种随机化方法的渐近收敛。此外，将随机化方法视为SAA使我们能够首次证明考虑中的随机化方法的单个非渐近误差结果。我们的统一框架的另一个重要结果是，它允许我们发现新的随机化方法。我们提出了各种数值结果的线性，非线性，代数和偏微分方程约束的反问题，验证了理论的收敛结果，并提供了一个明显不同的收敛速度和各种随机化方法的行为的讨论。
This work unifies the analysis of various randomized methods for solving linear and nonlinear inverse problems with Gaussian priors by framing the problem in a stochastic optimization setting. By doing so, we show that many randomized methods are variants of a sample average approximation (SAA). More importantly, we are able to prove a single theoretical result that guarantees the asymptotic convergence for a variety of randomized methods. Additionally, viewing randomized methods as an SAA enables us to prove, for the first time, a single non-asymptotic error result that holds for randomized methods under consideration. Another important consequence of our unified framework is that it allows us to discover new randomization methods. We present various numerical results for linear, nonlinear, algebraic, and PDE-constrained inverse problems that verify the theoretical convergence results and provide a discussion on the apparently different convergence rates and the behavior for various randomized methods.