The chapter introduces the discrete variant of the proper orthogonal decomposition (POD) method. The key points are:
Given a finite set of snapshot vectors {yk
j} in a separable complex Hilbert space X, the goal is to find an orthonormal set {ψi}ℓ
i=1 that best approximates the snapshots in a least-squares sense, where the dimension ℓ is as small as possible.
It is shown that the optimal POD basis {ψi}ℓ
i=1 is given by the eigenvectors of a specific linear operator associated with the snapshot data. The corresponding eigenvalues provide an "energy" measure that can be used to determine the appropriate dimension ℓ.
The chapter considers various cases, including when the Hilbert space X has finite or infinite dimension, as well as situations involving unitary, Euclidean, and weighted inner products. The POD method is also discussed for finite-dimensional dynamical systems.
The continuous variant of the POD method is introduced, where the snapshot data depends on a continuous parameter. The resulting POD basis is shown to be related to the discrete case through a specific linear operator.
A perturbation analysis is provided, showing the stability of the discrete POD basis with respect to approximations of the continuous snapshot data.
The case where the Hilbert space X is part of a Gelfand triple is discussed, allowing for the derivation of convergence rates for the POD approximation.
The chapter lays the theoretical foundation for the application of POD in model reduction and optimal control problems, which are covered in the subsequent chapters.
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