A methodology for using random sketching in the context of model order reduction for high-dimensional parameter-dependent systems of equations was introduced in Balabanov and Nouy (Part I; Advances in Computational Mathematics 45:2969–3019, 2019). Following this framework, we here construct a reduced model from a small, efficiently computable random object called a sketch of a reduced model, using minimal residual methods. We introduce a sketched version of the minimal residual based projection as well as a novel nonlinear approximation method, where for each parameter value, the solution is approximated by minimal residual projection onto a subspace spanned by several vectors picked (online) from a dictionary of candidate basis vectors. It is shown that random sketching technique can improve not only efficiency but also numerical stability. A rigorous analysis of the conditions on the random sketch required to obtain a given accuracy is presented. These conditions may be ensured a priori with high probability by considering for the sketching matrix an oblivious embedding of sufficiently large size. Furthermore, a simple and reliable procedure for a posteriori verification of the quality of the sketch is provided. This approach can be used for certification of the approximation as well as for adaptive selection of the size of the random sketching matrix.

在Balabanov和Nouy中介绍了在模型维数减少的情况下使用随机素描进行建模的方法，该方程系统由Balabanov和Nouy撰写（第一部分；计算数学的进展）45：2969–3019，2019）。按照这个框架，我们在这里使用最小的残差方法，从一个小的，可有效计算的随机对象构造一个简化模型，该模型称为简化模型的草图。我们介绍了基于最小残差的投影的草绘版本以及一种新颖的非线性逼近方法，其中对于每个参数值，通过最小残差投影到子空间上的近似解，该子空间由从候选字典中选取（在线）的多个向量构成基本向量。结果表明，随机素描技术不仅可以提高效率，而且可以提高数值稳定性。对获得给定精度所需的随机草图上的条件进行了严格的分析。通过为草图矩阵考虑足够大的尺寸的不明显的嵌入，可以以较高的概率先验地确保这些条件。此外，提供了用于草图质量的后验验证的简单而可靠的过程。该方法可用于近似的证明以及随机草图矩阵大小的自适应选择。