Abstract
A common interpretation is presented for four powerful modal decomposition techniques: “proper orthogonal decomposition,” “smooth orthogonal decomposition,” “state-variable decomposition,” and “dynamic mode decomposition.” It is shown that, in certain cases, each technique can be interpreted as an optimization problem and similarities between methods are highlighted. By interpreting each technique as an optimization problem, significant insight is gained toward the physical properties of the identified modes. This insight is strengthened by being consistent with cross-multiple decomposition techniques. To illustrate this, an inter-method comparison of synthetic hypersonic boundary layer stability data is presented.
Highlights
Modal decomposition techniques are powerful tools for the investigation of the dynamics underlying various systems
The main contribution of the present work is a unified interpretation of POD, SOD, S-VD, and DMD, in which each method can be interpreted as a constrained maximization problem
Let us begin with the POD technique, which is known as “Empirical Orthogonal Functions” (EOFs), “Principle Component Analysis” (PCA), and “Karhunen–Loève Decomposition” depending on the field of study
Summary
Modal decomposition techniques are powerful tools for the investigation of the dynamics underlying various systems. As most modal decomposition techniques are statistical in nature, caution must be practiced when attempting the latter In such cases, additional constraints are applied to the statistical methods in an effort to isolate the physically relevant aspects of the data. When performing a modal analysis, it is of great importance to have a fundamental understanding of the chosen method and ideally, this understanding should be consistent across a number of available methods to build intuition This manuscript provides a common interpretation of four popular (and powerful) modal decomposition techniques; the so-called “Proper Orthogonal Decomposition” (POD1,2), “Smooth Orthogonal Decomposition” (SOD3), “State-Variable Decomposition” (S-VD4,5), and “Dynamic Mode Decomposition” (DMD6), thereby allowing for improved mode interpretation and cross-method comparison. The focus will be on several works of Schmid and co-workers and Kutz. The main contribution of the present work is a unified (though only for restricted cases) interpretation of POD, SOD, S-VD, and DMD, in which each method can be interpreted as a constrained maximization problem
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