Some topics I plan to write about (or never).
Interesting Topics
Development of modern university.
The precursor appears to be cathedral school with faculty of arts. Maybe start with examining the early universities and colleges in the US, such as UPenn. Another start could be the Humboldt University of Berlin, which appears to have the first modern university system. A third start is the Bologna University, which is among the earliest universities.
Parkinson’s law and Peter’s principle
Kakeya set
A Kakeya set, or Besicovitch set, is a set of points in Euclidean space which contains a unit line segment in every direction. Besicovitch found a surprising result that the minimum area, or measure, of the set is zero.
A Kakeya needle set is a set in the plane with a stronger property, that a unit line segment can be rotated continuously. A yet surprising result is that the measure of such set can be arbitrarily small.
Miura-ori and Theorema Egregium
How to fold an A4 paper to reduce its area by an order of magnitude, while still easy to unfold? Miura folding is one answer. An introductory article is found here. More interesting applications are found here, where a scholar developed some computational origami software.
Concerning computational origami, there is another guy that must be mentioned: Robert Lang. He was a NASA physicist, but later became an origami artist. Now he is recognized as one of the leading theorists of the mathematics of origami and has developed ways to algorithmetize the design process for origami.
Another seemingly vague connection to origami is the shell structure. This starts with Theorema Egregium, or Remarkable Theorem. It is stated by Carl Friedrich Gauss that If a curved surface is developed upon any other surface whatever, the measure of curvature in each point remains unchanged. Essentially, the so-called Gaussian curvature is an intrinsic invariant of a surface. The curvature does not change if the surface deforms by bending only and not stretching. The implication of this result in engineering is the application of shell structure - thin and light, but strong. A flat paper can be folded into complex curved surfaces using generalized Miura-ori. Such folded structure is orders of magnitude stronger than the flat paper.
SPRITE
SPRITE for Sample Parameter Reconstruction via Iterative TEchniques. It is an awesome tool (pre-print) that enables the examination of statistical results in research papers that do not include the raw data - are they made up or fudged? One case found by the authors of SPRITE is related to the carrots.
Game of life on manifold
Conway’s Game of Life is initially defined on a 2D flat discrete grid. However, nothing prevents it from generalization to grids of higher dimensions, or continous Euclidean spaces, or any proper metric spaces. A good example is found here and here.
NSF Survey of Science & Engineering Doctorates
It turns out NSF has been surveying the doctorates earned each year in the US. It would be fun to dig into the data and explore.
Optimal control for frying steaks
A tasty steak is made by precisely controlling the temperature distribution of the steak on the frying pan. The evolution of the temperature in the steak is governed by the heat conduction equation. The control variables are the duration and magnitude of the heat flux from the frying pan. The question is, is it possible to achieve the given temperature response in the steak using an optimal control approach?
More Serious Topics
DMD and Higher-Order DMD
Dynamic Mode Decomposition is a useful tool for post-processing spatiotemporal data (resulting from generally nonlinear dynamics) as an expansion of spatial modes times exponentials in the time variable, which exhibit generally nonzero growth rates. It has extentions like Higher Order Dynamic Mode Decomposition.
Polynomial rational interpolation and iSINDy
Polynomial curve fitting is one of the simplest tools for data modeling. However, it suffers from various issues, such as discussed here. A better model would be polynomial rationals. But its mathematical properties are not as clear as polynomials - some materials are found here. Also, practical algorithms are developed only recently, i.e. the Barycentric interpolation.
The iSINDy method is the implicit version of the SINDy (Sparse Identification of Nonlinear Dynamics) method. As their names imply, the methods are used to infer the analytical form of a nonlinear dynamical system from the samples of its response. The sparsity ensures that the analytical form is as simple as possible. In SINDy, the system equation is represented by a linear combination of nonlinear features, e.g. polynomials, that are constructed from the state variables. In iSINDy, the system equation is represented by a rational fraction, similar to Barycentric interpolation. However, the difference is that in iSINDy the number of terms in the rational fraction is minimized, while in the latter all the possible terms might be involved.
Multi-fidelity surrogate
Some surrogate models take in data from both high- and low-fidelity models. Usually such a “multi-fidelity” model is hierarchical, e.g. a fusion of multiple separate surrogates. This paper proposed a unified formulation for multi-fidelity surrogate, which seems to have some values in applications like surrogate-based optimization.
Gaussian process regression with discrete variables
There is an approach based on latent variables that makes the discrete variables smooth. An application of this approach found here shows that it is better than some conventional approaches.
Non-Intrusive Polynomial Chaos Expansion
Polynomial chaos expansion (PCE) is one approach for uncertainty quantification. However, PCE is intrusive as it requires the modification of the deterministic version of the analysis code. One way to mitigate this problem is the non-intrusive version of PCE, as discussed in this paper. One example of application is found here.
Methodology for uncertainty quantification
Nowadays people expect not only high-fidelity results from CFD-based simulation, but also its uncertainty, which may be quantified through rigorous verification, validation and uncertainty quantification (VVUQ) procedures. One such procedural framework is proposed by Roy and Oberkampf.
In general, a solution for UQ is provided by the DAKOTA software from the Sandia National Lab. One application of Dakota is found here.
Sobol sampling v.s. Latin hypercube sampling
Two sampling methods for Gaussian process regression. Also Halton sampling.
Non-Intrusive Least Squares Shadowing
Optimization of a chaotic system can be challenging, because the gradients w.r.t. to the design variables computed naively could be “noisy”. This can be mitigated by the shadowing approach, which generates smooth gradients suitable for optimization. A non-intrusive version of the shadowing approach has been developed in this paper.
OpenMDAO
The OpenMDAO package has evolved in the recent years into a quite comprehensive framework for large-scale multi-disciplinary optimization problems. Besides conventional aerostructural optimization, it has encompassed some interesting capabilities, e.g. optimal control and topology optimization. It seems Bayesian optimization is supported too, at least in some of the versions and some branches.