Lihong Feng

Harshit Kapadia, Lihong Feng, Peter Benner

When repeated evaluations for varying parameter configurations of a high-fidelity physical model are required, surrogate modeling techniques based on model order reduction are desired. In absence of the governing equations describing the dynamics, we need to construct the parametric reduced-order surrogate model in a non-intrusive fashion. In this setting, the usual residual-based error estimate for optimal parameter sampling associated with the reduced basis method is not directly available. Our work provides a non-intrusive optimality criterion to efficiently populate the parameter snapshots, thereby, enabling us to effectively construct a parametric surrogate model. We consider separate parameter-specific proper orthogonal decomposition (POD) subspaces and propose an active-learning-driven surrogate model using kernel-based shallow neural networks, abbreviated as ActLearn-POD-KSNN surrogate model. To demonstrate the validity of our proposed ideas, we present numerical experiments using two physical models, namely Burgers' equation and shallow water equations. Both the models have mixed -- convective and diffusive -- effects within their respective parameter domains, with each of them dominating in certain regions. The proposed ActLearn-POD-KSNN surrogate model efficiently predicts the solution at new parameter locations, even for a setting with multiple interacting shock profiles.

Cleophas Kweyu, Lihong Feng, Matthias Stein et al.

The Poisson-Boltzmann equation (PBE) is a nonlinear elliptic PDE that arises in biomolecular modeling and is a fundamental tool for structural biology. It is used to calculate electrostatic potentials around an ensemble of fixed charges immersed in an ionic solution. Efficient numerical computation of the PBE yields a high number of degrees of freedom in the resultant algebraic system of equations. Coupled with the fact that in most cases the PBE requires to be solved multiple times for a large number of system configurations, this poses great computational challenges to conventional numerical techniques. To accelerate such computations, we here present the reduced basis method (RBM) which greatly reduces this computational complexity by constructing a reduced order model of typically low dimension. In this study, we employ a simple version of the PBE for proof of concept and discretize the linearized PBE (LPBE) with a centered finite difference scheme. The resultant linear system is solved by the aggregation-based algebraic multigrid method at different samples of ionic strength on a three-dimensional Cartesian grid. The discretized LPBE, which we call the high-fidelity full order model (FOM), yields solution as accurate as other LPBE solvers. We then apply the RBM to FOM. The discrete empirical interpolation method (DEIM) is applied to the Dirichlet boundary conditions which are nonaffine with the parameter, to reduce the complexity of the reduced order model (ROM). From the numerical results, we notice that the RBM reduces the model order from $\mathcal{N} = 2\times 10^{6}$ to $N = 6$ at an accuracy of $10^{-9}$ and reduces computational time by a factor of approximately $7,600$. DEIM, on the other hand, is also used in the offline-online phase of solving the ROM for different values of parameters which provides a speed-up of $20$ for a single iteration of the greedy algorithm.