Convex optimization-based structure-preserving filtering for polynomial-based numerical methods

Vidhi Zala (University of Utah, USA)

In the simulation sciences, capturing real-world problem features as accurately as possible is desirable. Methods popular for scientific simulations such as the finite element method (FEM) and the finite volume method (FVM) use piecewise polynomials to approximate various characteristics of the problem, such as the concentration profile of chemicals and the temperature distribution across the domain. Polynomial-based approximations of functions with finite data often do not respect certain structural properties of the functions. “Structure” in our context refers to fairly general types of linear inequality constraints, such as positivity, monotonicity, maximum principle, flux, and integral conservation, etc. In addition, polynomials are prone to creating artifacts such as Gibbs oscillations while capturing a complex profile. An efficient and accurate approach must be applied to deal with such inconsistencies in order to obtain accurate simulations that match the real-world expectations of the data. This often entails dealing with the occurrence of negative values while simulating the concentration of chemicals, a percentage value over 100, and other such issues. We consider these inconsistencies in the context of partial differential equations (PDEs). The solution we propose is the application of an innovative filter based on a convex optimization approach to deal with the structure-preservation problems observed in polynomial-based methods. We provide a variety of tests on various multivariate function approximations and time-dependent PDEs that demonstrate the efficacy of the method. We empirically show that rates of convergence and solution accuracy are unaffected by the inclusion of the structural constraints.