Dynamical Low Rank approximation of PDEs with random parameters

In this work, we focus on the Dynamical Low Rank (DLR) approximation of PDEs equations with random parameters. This can be interpreted as a reduced basis method, where the approximate solution is expanded in separable form over a set of few deterministic basis functions at each time, with the peculiarity that both the deterministic modes and the stochastic coefficients are computed on the fly and are free to adapt in time so as best describe the structure of the random solution. Our first goal is to generalize and reformulate in a variational setting the Dynamically Orthogonal (DO) method, proposed by Sapsis and Lermusiaux (2009) for the approximation of fluid dynamic problems with random initial conditions. The DO method is reinterpreted as a Galerkin projection of the governing equations onto the tangent space along the approximate trajectory to the manifold M_S , given by the collection of all functions which can be expressed as a sum of S linearly independent deterministic modes combined with S linearly independent stochastic modes. Depending on the parametrization of the tangent space, one obtains a set of nonlinear differential equations, suitable for numerical integration, for both the coefficients and the basis functions of the approximate solution. By formalizing the DLR variational principle for parabolic PDEs with random parameters we establish a precise link with similar techniques developed in different contexts such as the Multi-Configuration Time-Dependent Hartree method in quantum dynamics and the Dynamical Low-Rank approximation in the finite dimensional setting. By the use of curvature estimates for the approximation manifold M_S , we derive a theoretical bound for the approximation error of the S-terms DO solution by the corresponding S-terms best approximation at each time instant. The bound is applicable for full rank DLR approximate solutions on the largest time interval in which the best S-terms approximation is continuously differentiable in time. Secondly, we focus on parabolic equations, especially incompressible Navier-Stokes equations, with random Dirichlet boundary conditions and we propose a DLR technique which allows for the strong imposition of such boundary conditions. We show that the DLR variational principle can be set in the constrained manifold of all S rank random fields with a prescribed value on the boundary, expressed in low-rank format, with rank M smaller than S. We characterize the tangent space to the constrained manifold by means of the Dual Dynamically Orthogonal formulation, in which the stochastic modes are kept orthonormal and the deterministic modes satisfy suitable boundary conditions, consistent with the original problem. The same formulation is also used to conveniently include the incompressibility constraint when dealing with incompressible Navier-Stokes equations with random parameters. Finally, we extend the DLR approach for the approximation of wave equations with random parameters. We propose the Symplectic DO method, according to which the governing equation is rewritten in Hamiltonian form and the approximate solution is sought in the low dimensional manifold of all complex-valued random fields with fixed rank. Recast in the real setting, the approximate solution is expanded over a set of a few dynamical symplectic deterministic modes and satisfies the symplectic projection of the Hamiltonian system into the tangent space of the approximation manifold along the trajectory.