Approximation Bounds for Model Reduction on Polynomially Mapped Manifolds

For projection-based linear-subspace model order reduction (MOR), it is well known that the Kolmogorov <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>n</mml:mi> </mml:math> -width describes the best-possible error for a reduced order model (ROM) of size <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>n</mml:mi> </mml:math> . In this paper, we provide approximation bounds for ROMs on polynomially mapped manifolds. In particular, we show that the approximation bounds depend on the polynomial degree <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>p</mml:mi> </mml:math> of the mapping function as well as on the linear Kolmogorov <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>n</mml:mi> </mml:math> -width for the underlying problem. This results in a Kolmogorov <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>n</mml:mi> <mml:mo>,</mml:mo> <mml:mi>p</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> -width, which describes a lower bound for the best-possible error for a ROM on polynomially mapped manifolds of polynomial degree <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>p</mml:mi> </mml:math> and reduced size <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>n</mml:mi> </mml:math> .