Reduced basis methods for time-dependent problems

Numerical simulation of parametrized differential equations is crucial importance in the study real-world phenomena applied science and engineering. Computational methods for real-time many-query such problems often require prohibitively high computational costs to achieve sufficiently accurate numerical solutions. During last few decades, model order reduction has proved successful providing low-complexity high-fidelity surrogate models that allow rapid simulations under parameter variation, thus enabling increasingly complex problems. However, many challenges remain secure robustness efficiency needed nonlinear time-dependent The purpose this article survey state art reduced basis draw together recent advances three main directions. First, we discuss structure-preserving designed retain key physical properties continuous problem. Second, localized adaptive based on approximations solution space. Finally, consider data-driven techniques non-intrusive which an approximation map between space coefficients learned. Within each class methods, describe different approaches provide a comparative discussion lends insights advantages, disadvantages potential open questions.