  Low-rank approximation using time-dependent bases (TDBs) has proven effective
for reduced-order modeling of stochastic partial differential equations
(SPDEs). In these techniques, the random field is decomposed to a set of
deterministic TDBs and time-dependent stochastic coefficients. When applied to
SPDEs with non-homogeneous stochastic boundary conditions (BCs), appropriate BC
must be specified for each of the TDBs. However, determining BCs for TDB is not
trivial because: (i) the dimension of the random BCs is different than the rank
of the TDB subspace; (ii) TDB in most formulations must preserve orthonormality
or orthogonality constraints and specifying BCs for TDB should not violate
these constraints in the space-discretized form. In this work, we present a
methodology for determining the boundary conditions for TDBs at no additional
computational cost beyond that of solving the same SPDE with homogeneous BCs.
Our methodology is informed by the fact the TDB evolution equations are the
optimality conditions of a variational principle. We leverage the same
variational principle to derive an evolution equation for the value of TDB at
the boundaries. The presented methodology preserves the orthonormality or
orthogonality constraints of TDBs. We present the formulation for both the
dynamically bi-orthonormal (DBO) decomposition as well as the dynamically
orthogonal (DO) decomposition. We show that the presented methodology can be
applied to stochastic Dirichlet, Neumann, and Robin boundary conditions. We
assess the performance of the presented method for linear advection-diffusion
equation, Burgers' equation, and two-dimensional advection-diffusion equation
with constant and temperature-dependent conduction coefficient.
