Proper Orthogonal Decomposition Sample Clauses

Proper Orthogonal Decomposition. ‌ The Proper Orthogonal Decomposition (POD) approaches the problem of the con- struction of a RB from a different that can be regarded as the dual of that of the GRB method. The GRB method starts from an empty basis and progressively adds the vectors that help to improve the approximation error estimates. On the other hand, the POD starts from the set of all the solutions corresponding to the training set parameters, and discards redundant information. POD is a technique that have been used for decades in different fields, and has been named in different ways, depending on the field: in statistics it is often called Principal Component Analysis (PCA) or Hotelling transform [47], in the theory of stochastic processes it is known as Karhunen-Loéve transform [57], in whether fore- cast and meteorology is known as Empirical Orthogonal Functions analysis [58], in matrix analysis it is known as Xxxxxx-Xxxxx theorem [43] (although the result was first shown by Xxxxxxx [86]). To see how the POD method works, assume that we have a given set of values for the parameter S tµ1, . . . , µmu, which we will call training set, and that we have solved the problem (4.1) for all the parameters in S. We can gather the solutions in the so called snapshots matrix X rx1, . . . , xms, (4.20)