A masked projection approximates a vector-valued function from only a few of its components: a selection operator P made of columns of the identity samples m entries, and the approximation f-tilde = U1 (P^T U1)^-1 P^T f is reconstructed inside the m-dimensional subspace spanned by an orthonormal basis U1. When the number of sampled components equals the subspace dimension the projection is uniquely determined. Writing the orthonormal matrix built from P and the basis in its cosine-sine decomposition makes the singular values sigma_i of P^T U1 the natural error parameters: the pointwise squared error is bounded by (1 + sum_i (1-sigma_i^2)/sigma_i^2) times the orthogonal-projection error, and the average error over N samples is bounded more sharply using the eigenvalues of the sample Gram matrix X X^T with X = U2^T [f1 ... fN]. Both bounds hold for any selection operator, independent of how it was generated, so they cover DEIM and its variants, and they are numerically one to two orders of magnitude tighter than the QDEIM bound.
In nonlinear model order reduction, a large parameterized dynamical system dx/dt = f(x(t;mu)) with state x in R^n is approximated in a p-dimensional subspace, but projecting the state alone does not …