Bayesian inference provides a systematic framework for integration of data with mathematical models to quantify the uncertainty in the solution of the inverse problem. However, the solution of Bayesian inverse problems governed by complex forward models described by partial differential equations (PDEs) remains prohibitive with black-box Markov chain Monte Carlo (MCMC) methods. We present hIPPYlib-MUQ, an extensible and scalable software framework that contains implementations of state-of-the art algorithms aimed to overcome the challenges of high-dimensional, PDE-constrained Bayesian inverse problems. These algorithms accelerate MCMC sampling by exploiting the geometry and intrinsic low-dimensionality of parameter space via derivative information and low rank approximation. The software integrates two complementary open-source software packages, hIPPYlib and MUQ. hIPPYlib solves PDE-constrained inverse problems using automatically-generated adjoint-based derivatives, but it lacks full Bayesian capabilities. MUQ provides a spectrum of powerful Bayesian inversion models and algorithms, but expects forward models to come equipped with gradients and Hessians to permit large-scale solution. By combining these two complementary libraries, we created a robust, scalable, and efficient software framework that realizes the benefits of each and allows us to tackle complex large-scale Bayesian inverse problems across a broad spectrum of scientific and engineering disciplines. To illustrate the capabilities of hIPPYlib-MUQ, we present a comparison of a number of MCMC methods available in the integrated software on several high-dimensional Bayesian inverse problems. These include problems characterized by both linear and nonlinear PDEs, various noise models, and different parameter dimensions. The results demonstrate that large (∼ 50×) speedups over conventional black box and gradient-based MCMC algorithms can be obtained by exploiting Hessian information (from the log-posterior), underscoring the power of the integrated hIPPYlib-MUQ framework.

贝叶斯推理提供了一个将数据与数学模型集成的系统框架，以量化反问题解决方案中的不确定性。然而，由偏微分方程 (PDE)描述的复杂正向模型控制的贝叶斯逆问题的解决方案仍然无法通过黑盒马尔可夫链蒙特卡罗 (MCMC)解决方法。我们推出了 hIPPYlib-MUQ，这是一个可扩展且可扩展的软件框架，其中包含最先进算法的实现，旨在克服高维、偏微分方程约束贝叶斯逆问题的挑战。这些算法通过导数信息和低秩近似来利用参数空间的几何结构和固有低维性，从而加速 MCMC 采样。该软件集成了两个互补的开源软件包：hIPPYlib 和 MUQ。hIPPYlib 使用自动生成的基于伴随的导数来解决偏微分方程约束的反问题，但它缺乏完整的贝叶斯功能。MUQ 提供了一系列强大的贝叶斯反演模型和算法，但期望正向模型配备梯度和 Hessian，以允许大规模解决方案。通过结合这两个互补的库，我们创建了一个强大、可扩展且高效的软件框架，该框架实现了每个库的优点，并使我们能够跨广泛的科学和工程学科解决复杂的大规模贝叶斯逆问题。为了说明 hIPPYlib-MUQ 的功能，我们对集成软件中可用的多种 MCMC 方法在几个高维贝叶斯逆问题上进行了比较。其中包括以线性和非线性偏微分方程、各种噪声模型和不同参数维度为特征的问题。结果表明，通过利用 Hessian 信息（来自对数后验），可以比传统黑盒和基于梯度的 MCMC 算法获得大幅（~ 50×）加速，