TY - JOUR
T1 - Stability properties of a projector-splitting scheme for dynamical low rank approximation of random parabolic equations
AU - Kazashi, Yoshihito
AU - Nobile, Fabio
AU - Vidličková, Eva
N1 - Kazashi, Y., Nobile, F. & Vidličková, E. Stability properties of a projector-splitting scheme for dynamical low rank approximation of random parabolic equations. Numer. Math. 149, 973–1024 (2021). https://doi.org/10.1007/s00211-021-01241-4
© The Author(s) 2021
PY - 2021/11/17
Y1 - 2021/11/17
N2 - We consider the Dynamical Low Rank (DLR) approximation of random parabolic equations and propose a class of fully discrete numerical schemes. Similarly to the continuous DLR approximation, our schemes are shown to satisfy a discrete variational formulation. By exploiting this property, we establish stability of our schemes: we show that our explicit and semi-implicit versions are conditionally stable under a “parabolic” type CFL condition which does not depend on the smallest singular value of the DLR solution; whereas our implicit scheme is unconditionally stable. Moreover, we show that, in certain cases, the semi-implicit scheme can be unconditionally stable if the randomness in the system is sufficiently small. Furthermore, we show that these schemes can be interpreted as projector-splitting integrators and are strongly related to the scheme proposed in [29, 30], to which our stability analysis applies as well. The analysis is supported by numerical results showing the sharpness of the obtained stability conditions.
AB - We consider the Dynamical Low Rank (DLR) approximation of random parabolic equations and propose a class of fully discrete numerical schemes. Similarly to the continuous DLR approximation, our schemes are shown to satisfy a discrete variational formulation. By exploiting this property, we establish stability of our schemes: we show that our explicit and semi-implicit versions are conditionally stable under a “parabolic” type CFL condition which does not depend on the smallest singular value of the DLR solution; whereas our implicit scheme is unconditionally stable. Moreover, we show that, in certain cases, the semi-implicit scheme can be unconditionally stable if the randomness in the system is sufficiently small. Furthermore, we show that these schemes can be interpreted as projector-splitting integrators and are strongly related to the scheme proposed in [29, 30], to which our stability analysis applies as well. The analysis is supported by numerical results showing the sharpness of the obtained stability conditions.
KW - stability properties
KW - random parabolic equations
KW - dynamical low rank approximation
KW - projector-splitting scheme
KW - fully discrete numerical schemes
KW - implicit scheme
KW - semi-implicit scheme
KW - numerical results
U2 - 10.1007/s00211-021-01241-4
DO - 10.1007/s00211-021-01241-4
M3 - Article
SN - 0029-599X
VL - 149
SP - 973
EP - 1024
JO - Numerische Mathematik
JF - Numerische Mathematik
ER -