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The success of matrix factorizations such as the singular value decomposition (SVD)has motivated the search for even more factorizations. We catalog 53 matrix factorizations, most ofwhich we believe to be new. Our systematic approach, inspired by the generalized Cartan decom-position of the Lie theory, also encompasses known factorizations such as the SVD, the symmetriceigendecomposition, the CS decomposition, the hyperbolic SVD, structured SVDs, the Takagi factor-ization, and others thereby covering familiar matrix factorizations, as well as ones that were waitingto be discovered. We suggest that the Lie theory has one way or another been lurking hidden in thefoundations of the very successful field of matrix computations with applications routinely used in somany areas of computation. In this paper, we investigate consequences of the Cartan decompositionand the little known generalized Cartan decomposition for matrix factorizations. We believe thatthese factorizations once properly identified can lead to further work on algorithmic computationsand applications. 

We complete Dyson’s dream by cementing the links between symmetric spaces and classical random matrix ensembles. Previous work has focused on a one-to-one correspondence between symmetric spaces and many but not all of the classical random matrix ensembles. This work shows that we can completely capture all of the classical random matrix ensembles from Cartan’s symmetric spaces through the use of alternative coordinate systems. In the end, we have to let go of the notion of a one-to-one correspondence. We emphasize that the KAK decomposition traditionally favored by mathematicians is merely one coordinate system on the symmetric space, albeit a beautiful one. However, other matrix factorizations, especially the generalized singular value decomposition from numerical linear algebra, reveal themselves to be perfectly valid coordinate systems that one symmetric space can lead to many classical random matrix theories. We establish the connection between this numerical linear algebra viewpoint and the theory of generalized Cartan decompositions. This, in turn, allows us to produce yet more random matrix theories from a single symmetric space. Yet, again, these random matrix theories arise from matrix factorizations, though ones that we are not aware have appeared in the literature. 

As mathematical computing becomes more democratized in high-level languages, high-performance symbolic-numeric systems are necessary for domain scientists and engineers to get the best performance out of their machine without deep knowledge of code optimization. Naturally, users need different term types either to have different algebraic properties for them, or to use efficient data structures. To this end, we developed Symbolics.jl, an extendable symbolic system which uses dynamic multiple dispatch to change behavior depending on the domain needs. In this work we detail an underlying abstract term interface which allows for speed without sacrificing generality. We show that by formalizing a generic API on actions independent of implementation, we can retroactively add optimized data structures to our system without changing the pre-existing term rewriters. We showcase how this can be used to optimize term construction and give a 113x acceleration on general symbolic transformations. Further, we show that such a generic API allows for complementary term-rewriting implementations. Exploiting this feature, we demonstrate the ability to swap between classical term-rewriting simplifiers and e-graph-based term-rewriting simplifiers. We illustrate how this symbolic system improves numerical computing tasks by showcasing an e-graph ruleset which minimizes the number of CPU cycles during expression evaluation, and demonstrate how it simplifies a real-world reaction-network simulation to halve the runtime. Additionally, we show a reaction-diffusion partial differential equation solver which is able to be automatically converted into symbolic expressions via multiple dispatch tracing, which is subsequently accelerated and parallelized to give a 157x simulation speedup. Together, this presents Symbolics.jl as a next-generation symbolic-numeric computing environment geared towards modeling and simulation. 

Modern design, control, and optimization often require multiple expensive simulations of highly nonlinear stiff models. These costs can be amortized by training a cheap surrogate of the full model, which can then be used repeatedly. Here we present a general data-driven method, the continuous time echo state network (CTESN), for generating surrogates of nonlinear ordinary differential equations with dynamics at widely separated timescales. We empirically demonstrate the ability to accelerate a physically motivated scalable model of a heating system by 98x while maintaining relative error of within 0.2 %. We showcase the ability for this surrogate to accurately handle highly stiff systems which have been shown to cause training failures with common surrogate methods such as Physics-Informed Neural Networks (PINNs), Long Short Term Memory (LSTM) networks, and discrete echo state networks (ESN). We show that our model captures fast transients as well as slow dynamics, while demonstrating that fixed time step machine learning techniques are unable to adequately capture the multi-rate behavior. Together this provides compelling evidence for the ability of CTESN surrogates to predict and accelerate highly stiff dynamical systems which are unable to be directly handled by previous scientific machine learning techniques. 

The conservation field is experiencing a rapid increase in the amount, variety, and quality of spatial data that can help us understand species movement and landscape connectivity patterns. As interest grows in more dynamic representations of movement potential, modelers are often limited by the capacity of their analytic tools to handle these datasets. Technology developments in software and high-performance computing are rapidly emerging in many fields, but uptake within conservation may lag, as our tools or our choice of computing language can constrain our ability to keep pace. We recently updated Circuitscape, a widely used connectivity analysis tool developed by Brad McRae and Viral Shah, by implementing it in Julia, a high-performance computing language. In this initial re-code (Circuitscape 5.0) and later updates, we improved computational efficiency and parallelism, achieving major speed improvements, and enabling assessments across larger extents or with higher resolution data. Here, we reflect on the benefits to conservation of strengthening collaborations with computer scientists, and extract examples from a collection of 572 Circuitscape applications to illustrate how through a decade of repeated investment in the software, applications have been many, varied, and increasingly dynamic. Beyond empowering continued innovations in dynamic connectivity, we expect that faster run times will play an important role in facilitating co-production of connectivity assessments with stakeholders, increasing the likelihood that connectivity science will be incorporated in land use decisions.