This paper introduces a multifidelity formulation that reduces the computational cost of the proper orthogonal decomposition (POD) of a high-fidelity model by leveraging data from cheaper, lower-fidelity models. POD is a prevalent technique for extracting a low-dimensional basis from training data to achieve subsequent dimension reduction or reduced-order modeling. In scientific and engineering applications, the training data are typically numerical snapshot solutions of a high-fidelity model, and computation of a sufficiently rich snapshot set can be prohibitively expensive, especially when sampling over a high-dimensional parameter space. Insufficient snapshot training data risks overfitting and poor generalizability of the POD basis to outside the training regime. Our multifidelity POD (MFPOD) formulation reallocates computational budget to cheaper, low-fidelity models that can be sampled more extensively. MFPOD then weights high- and low-fidelity snapshot data via a control-variate formulation to guarantee an unbiased estimate of the expected high-fidelity least-squares projection error. The MFPOD subspace is chosen to minimize the estimate of this projection error, and converges in probability to the same subspace as single-fidelity POD in the limit of an arbitrarily large budget. For restrictive computational budgets, the MFPOD cost function has (under some assumptions) lower variance than the POD cost function, which makes the MFPOD subspace more robust against variations in the training data and thus less prone to overfitting. For a numerical example modeling the velocity of the Pine Island glacier, MFPOD achieves the same accuracy as single-fidelity POD with an order of magnitude reduction in the offline computational cost of snapshot generation.

翻译：本文提出了一种多保真度公式，通过利用更便宜的、低保真度模型的数据，降低了高保真度模型本征正交分解（POD）的计算成本。POD是一种从训练数据中提取低维基以用于后续降维或降阶建模的常用技术。在科学和工程应用中，训练数据通常是高保真度模型的数值快照解，计算足够丰富的快照集可能成本过高，尤其是在高维参数空间上进行采样时。训练快照数据不足可能导致POD基过拟合，并且对训练域外的泛化能力较差。我们的多保真度POD（MFPOD）公式将计算预算重新分配到更便宜、能够更广泛采样的低保真度模型上。然后，MFPOD通过控制变量公式对高保真度和低保真度快照数据进行加权，以确保对期望的高保真度最小二乘投影误差的无偏估计。MFPOD子空间被选为最小化该投影误差的估计值，并且在任意大预算的极限下，概率收敛到与单保真度POD相同的子空间。对于受限制的计算预算，MFPOD代价函数的方差（在某些假设下）低于POD代价函数的方差，这使得MFPOD子空间对训练数据的变化更加鲁棒，因此更不易过拟合。对于一个模拟松岛冰川流速的数值示例，MFPOD在快照生成的离线计算成本降低一个数量级的情况下达到了与单保真度POD相同的精度。