Exact-Residual Certified Reduced Basis Methods Based on Least-Squares Variational Principles

Many chemical, physical, and engineering problems can be modeled by PDEs. It is only possible to solve these PDEs numerically most of time. Thus, it is a central research challenge to design efficient and accurate numerical methods. Solving PDEs numerically often leads to very high-dimensional problems requiring a high computational power. These high demanding tasks pose a serious problem for multi-query or real-time scenarios. Such scenarios appear in inverse problems, optimization, design, parameter studies, and statistical analysis. The reduced basis method (RBM) is a modern model reduction method for parametric PDEs, which is very efficient for these scenarios.For many challenging problems, due to the lack of the variational structure and a rigorous error estimator, a more expensive POD approach other than the greedy algorithm has to be used. For those problems which do have a variational structure, often special cares need to paid to ensure the stability of the RB formulation. Besides these, a classic RBM uses the residual error estimator. The error is estimated in some parameter independent genetic norm and an estimation of the stability constant is also needed. This type of error estimator measures only the difference between the RB solution and an unrealistic "truth" discrete solution. The choice of the RB error tolerance is often heuristic and will cause under or over-computing.To overcome these shortcomings, we will develop an exact-residual certified reduced basis method for a wide range of problems. We use the least-squares (LS) variational principle as the brute-force general energy-type minimization principle to define the variational problem and the Galerkin projection. The natural exact residual in the least-squares formulation is used as the true-error a posteriori estimator of the RBM and the related underlying finite element method. The LS formulation is automatically stable, and can separate the continuity requirements and impose boundary conditions easily.We propose two approaches in the project: one is that the LS principle is used for both the FEM and RBM, and the other is a non-intrusive approach with the LS method only used for the RBM while the discrete truth solver being user-defined. Furthermore, for the LS-RBM for nonlinear equations, we propose a new artificial neuron network method to generate matrices and vectors appeared in the nonlinear RB iterations. We will test the methods developed in the project on the second order elliptic equations, Stokes equation, transport problems, time-dependent problems, and nonlinear problems including Navier-Stokes problems.