SIAM Journal on Scientific Computing, Volume 42, Issue 2, Page A1269-A1288, January 2020.
In many inverse problems, a nonnegativity constraint is natural. Moreover, in some cases, we expect the vector of unknown parameters to have zero components. When a Bayesian approach is taken, this motivates a desire for prior probability density (and hence posterior probability density) functions that have positive mass at the boundary of the set $x$ in\mathbbR^N,|,$x$\geq$0$. Unfortunately, it is difficult to define a prior with this property that yields computationally tractable inference for large-scale inverse problems. In this paper, we use nonnegativity constrained optimization to define such prior and posterior density functions when the measurement error is either Gaussian or Poisson distributed. The numerical optimization methods we use are highly efficient, and hence our approach is computationally tractable even in large-scale cases. We embed our nonnegativity constrained optimization approach within a hierarchical framework, obtaining Gibbs samplers for both Gaussian and Poisson distributed measurement cases. Finally, we test the resulting Markov chain Monte Carlo methods on examples from both image deblurring and positron emission tomography.

中文翻译：

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非负约束线性反问题的计算UQ的MCMC算法

SIAM科学计算杂志，第42卷，第2期，第A1269-A1288页，2020年1月。
在许多反问题中，非负约束是很自然的。此外，在某些情况下，我们希望未知参数的向量具有零分量。当采用贝叶斯方法时，这激发了对在集合$ x $ in \ mathbbR ^ N，|，$ x $ \ geq $ 0的边界处具有正质量的先验概率密度（以及后验概率密度）函数的需求。 $。不幸的是，很难定义具有此性质的先验，该先验产生针对大规模反问题的计算上容易处理的推断。在本文中，当测量误差为高斯分布或泊松分布时，我们使用非负约束优化来定义此类先验和后验密度函数。我们使用的数值优化方法非常高效，因此即使在大规模情况下，我们的方法在计算上也易于处理。我们将非负约束优化方法嵌入到分层框架中，从而获得了针对高斯和泊松分布测量案例的吉布斯采样器。最后，我们以图像去模糊和正电子发射断层扫描为例，测试了所得的马尔可夫链蒙特卡罗方法。