Reduced Order Models Sample Clauses

Reduced Order Models. ‌ When using a mathematical model to describe a complex system, in order to have an accurate solution, the number of variables (and equations) must usually be fairly large and, in addition, may have to be solved several times, as often happens in optimization problems. This means that tackling the problem is expensive both from the storage and, most importantly, the computational point of view. To reduce the computational costs, there are two main strategies: - reduce the complexity of the model - reduce the size of the space where a solution is sought. In the first case, further assumptions are made, so that the set of equations to be solved simplifies to a smaller set of equations or to a different (and simpler) type of equation. This is precisely what we presented in Section 2.4, where a 2D model has been used for the vessel wall instead of a full 3D one. However, the model cannot be simplified beyond a certain level, otherwise the quality and reliability of the so- lution would deteriorate. In the example presented in (2.3), a simplification of the model corresponds to an increase of the model error term εn. If the model cannot be further simplified without losing reliability, then addi- tional reduction in the computational costs can be achieved by reducing the di- mension of the space of solutions. To explain the concept, consider the simple al- gebraic problem Apµqx bpµq, (4.1) where µ P R is a parameter, A P Rn n and x, b P Rn. In general, different values of µ would lead to different solutions. It is convenient at this point to think of x as a function of the parameter µ. Then, the map that at each parameter associates a so- lution defines a trajectory (or a hyperplane, if µ is a vector) in the space of solutions Rn. Clearly, each point along this trajectory has a representation with respect to the canonical basis, of the form n xpµq xipµqei. (4.2)