Background

I had some confusion on SVD yesterday about when and where we introduce approximation on the DMD algorithm. My thought was that, intuitively, we cannot find an exact linear map to transform one set of points to another random set of points. Because of the randomness, the exact map most likely is nonlinear and nonlocal. But, in the 2nd step of DMD where we learn the linear dynamics from data, when and where do we make an approximation? This exercise was created to clear that confusion.

Problem Description

In this exercise, I would like apply the Dynamic Mode Decomposition (DMD) algorithm on a simple problem to familiarize myself with DMD. I will first generate the evolution history of a 2D oscillating system. I will then embed the 2D dynamics into a 3D space (put a planar curve into a 3D space) and corrupt the 3D signal with noise. The noisy 3D data will be fed into the DMD algorithm. I want to see if I can recover the true 2D dynamics from the corrupted 3D signal. The code below is written in MATLAB. I created the toy problem myself and modified the DMD codes in ref 1 to make it work for the problem.

Background

I bought a book on data-driven dynamical systems (ref. 1) earlier this year right before the pandemic began to spread across the US. The book was writen by two professors from University of Washington and was published quite recently in 2019. The book talks about dynamical systems, but instead of focusing on the classical theories that we learnt in school, it focuses more on how data can help us identify and reduce the order of the systems. Session 7.2 of the book introduces a method named “Dynamic Mode Decomposition” (DMD. I find it interesting because, essentially, the method enables us to use data to discover, characterize and predict how a dynamical system evolves. More importantly the characterization could be done in low order (i.e., with few degrees of freedom), which means if the data is from a high fidelity model, the method gives us a way to produce a reduced order model. This post summarizes my current high-level understanding of DMD after reading Session 7.2 of the book. Here I will not cite the papers already cited in the book, if interested, please look up the original research papers from the book.

Background

I’ve been working with Professor Y. Marzouk and his postdoc P. Rubio on a Schlumberger project for the past year. One interesting technique I learnt from them is the so-called transport map. It is not about how to fit all MBTA routes onto an A4 paper, although everytime I saw the MBTA map on a bus I wonder if there is any math behind the design of that map. The transport map I learnt during the past year is about how to represent a complex probability distribution. Without sharing any Schlumberger related things, this post tries to document my understanding of the basics of transport maps (all have been published by Marzouk earlier, see ref.1 for example).

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