Projection-based Reduced Order Models minimize the discrete residual of a "full order model" (FOM) while constraining the unknowns to a reduced dimension space. For problems with symmetric positive definite (SPD) Jacobians, this is optimally achieved by projecting the full order residual onto the approximation basis (Galerkin Projection). This is sub-optimal for non-SPD Jacobians as it only minimizes the projection of the residual, not the residual itself. An alternative is to directly minimize the 2-norm of the residual, achievable using QR factorization or the method of the normal equations (LSPG). The first approach involves constructing and factorizing a large matrix, while LSPG avoids this but requires constructing a product element by element, necessitating a complementary mesh and adding complexity to the hyper-reduction process. This work proposes an alternative based on Petrov-Galerkin minimization. We choose a left basis for a least-squares minimization on a reduced problem, ensuring the discrete full order residual is minimized. This is applicable to both SPD and non-SPD Jacobians, allowing element-by-element assembly, avoiding the use of a complementary mesh, and simplifying finite element implementation. The technique is suitable for hyper-reduction using the Empirical Cubature Method and is applicable in nonlinear reduction procedures.

翻译：基于投影的降阶模型在将未知量约束至低维空间的同时，最小化“全阶模型”（FOM）的离散残差。对于雅可比矩阵对称正定（SPD）的问题，通过将全阶残差投影到近似基上（伽辽金投影）可最优实现该目标。但对于非对称正定雅可比矩阵，该方法仅能最小化残差的投影而非残差本身，因此存在次优性。另一替代方案是直接最小化残差的2-范数，可通过QR分解或正规方程法（LSPG）实现。前者需构造并分解大型矩阵，而LSPG虽避免了该过程，但需逐元素构造乘积项，要求使用互补网格并增加超缩减过程的复杂度。本文提出基于Petrov-Galerkin最小化的替代方案。我们为降阶问题的最小二乘最小化选择左基，确保离散全阶残差被最小化。该方法适用于对称正定与非对称正定雅可比矩阵，支持逐元素组装，无需使用互补网格，且简化了有限元实现。该技术适用于基于经验立方体求积法的超缩减，并可用于非线性降阶过程。