  We present a new methodology for the real-time reduced-order modeling of
stochastic partial differential equations called the dynamically/bi-orthonormal
(DBO) decomposition. In this method, the stochastic fields are approximated by
a low-rank decomposition to spatial and stochastic subspaces. Each of these
subspaces is represented by a set of orthonormal time-dependent modes. We
derive exact evolution equations of these time-dependent modes and the
evolution of the factorization of the reduced covariance matrix. We show that
DBO is equivalent to the dynamically orthogonal (DO) and bi-orthogonal (BO)
decompositions via linear and invertible transformation matrices that connect
DBO to DO and BO. However, DBO shows several improvements compared to DO and
BO: (i) DBO performs better than DO and BO for cases with ill-conditioned
covariance matrix; (ii) In contrast to BO, the issue of eigenvalue crossing is
not present in the DBO formulation; (iii) In contrast to DO, the stochastic
modes are orthonormal, which leads to more accurate representation of the
stochastic subspace. We study the convergence properties of the method and
compare it to the DO and BO methods. For demonstration, we consider three
cases: (i) stochastic linear advection equation, (ii) stochastic Burgers'
equation, and (iii) stochastic incompressible flow over a bump in a channel.
Overall we observe improvements in the numerical accuracy of DBO compared
against DO and BO.
