Uploaded August 2021 | Updated September 2026, 1 week ago
Hello everyone!
I'm gearing up for teaching in person (in Florida) next week, after nearly 2 years of teaching online. This is amid the worst numbers we have ever seen for COVID here, and Florida is one of the three hot spots of the world right now. It's a bit daunting, but I'm doing my best to leverage online platforms to run a flipped classroom and record everything to help students who are concerned and want to distance themselves.
But I'm not here to talk about any of that.
If you're only vaguely aware of Hilbert space theory and Operators, then the video linked below is a start on discussing the Spectral Theorem for Operators, where we go over the theory for Self Adjoint Compact Operators (following Lang's Real and Functional Analysis). I am motivating it with Dynamic Mode Decompositions, where we absolutely need this theory to establish convergence of models obtained from DMD algorithms to those of the true dynamics.
This is continuing my course on Data Driven Methods in Dynamical Systems that I started in spring, and this series expands on our discussion of Dynamic Mode Decompositions. The video linked below is setting us up to talk about how to get convergent routines, and it comes down to some 101 theorems from Functional Analysis.
Right now, the literature has settled on purely heuristic motivations for DMD, where convergence theories haven't been strong enough to get the convergence of the spectra in DMD. Some recent work of me and my colleagues have begun to illuminate how you can achieve actual convergence, and this video series is intended to build up to that new theory.
0:00 Start
1:06 Back in the saddle again
1:26 Matrices and Operator Representation
2:35 Limitations of DMD Convergence Theory
3:23 Defining Bounded Operators
4:59 Self Adjoint Compact Operators Basic Properties
9:39 Compact Operators and And Equivalent Definition
10:30 Existence of Eigenfunctions
14:25 The Spectral Theorem for Self Adjoint Compact Operators
17:45 The Matrix Representation of Self Adjoint Compact Operators
19:47 Closing Remarks and Scaled Liouville Operators
Hello everyone!
I'm gearing up for teaching in person (in Florida) next week, after nearly 2 years of teaching online. This is amid the worst numbers we have ever seen for COVID here, and Florida is one of the three hot spots of the world right now. It's a bit daunting, but I'm doing my best to leverage online platforms to run a flipped classroom and record everything to help students who are concerned and want to distance themselves.
But I'm not here to talk about any of that.
If you're only vaguely aware of Hilbert space theory and Operators, then the video linked below is a start on discussing the Spectral Theorem for Operators, where we go over the theory for Self Adjoint Compact Operators (following Lang's Real and Functional Analysis). I am motivating it with Dynamic Mode Decompositions, where we absolutely need this theory to establish convergence of models obtained from DMD algorithms to those of the true dynamics.
This is continuing my course on Data Driven Methods in Dynamical Systems that I started in spring, and this series expands on our discussion of Dynamic Mode Decompositions. The video linked below is setting us up to talk about how to get convergent routines, and it comes down to some 101 theorems from Functional Analysis.
Right now, the literature has settled on purely heuristic motivations for DMD, where convergence theories haven't been strong enough to get the convergence of the spectra in DMD. Some recent work of me and my colleagues have begun to illuminate how you can achieve actual convergence, and this video series is intended to build up to that new theory.
0:00 Start
1:06 Back in the saddle again
1:26 Matrices and Operator Representation
2:35 Limitations of DMD Convergence Theory
3:23 Defining Bounded Operators
4:59 Self Adjoint Compact Operators Basic Properties
9:39 Compact Operators and And Equivalent Definition
10:30 Existence of Eigenfunctions
14:25 The Spectral Theorem for Self Adjoint Compact Operators
17:45 The Matrix Representation of Self Adjoint Compact Operators
19:47 Closing Remarks and Scaled Liouville Operators
