Deepak Singh Kalhan, PhD candidate
David R. Cheriton School of Computer Science
Supervisors: Professors Stephen Watt, Robert Corless
Differentiation matrices based on orthogonal polynomial bases grow rapidly with degree, leading to numerical instability in high-order approximations. We study the effect of Sobolev regularization, which introduces derivative information into the inner product, on the behavior of these operators.
We show that Sobolev regularization reduces the growth rate of differentiation operators. In particular, for both Legendre–Sobolev and Chebyshev–Sobolev bases, the operator and Frobenius norms decrease by one order in the polynomial degree compared to the classical case. We further characterize a transition between unstable and stabilized regimes by the parameter $\mu$, with scaling controlled by $\mu N^3$. Although both polynomial families exhibit the same asymptotic behavior, their $\mu$-dependent mode mixing differs, reflecting structural differences in the underlying bases.
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