  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.
