72 research outputs found
The Jormungand Climate Model
The geological and paleomagnetic record indicate that around 750 million and 580 millions years ago glaciers grew near the equator, though as of yet we do not fully understand the nature of these glaciations. The well-known Snowball Earth Hypothesis states that the Earth was covered entirely by glaciers. However, it is hard for this hypothesis to account for certain aspects of the biological evidence such as the survival of photosynthetic eukaryotes. Thus the Jormungand Hypothesis was developed as an alternative to the Snowball Earth Hypothesis. In this paper we investigate previous models of the Jormungand state and look at the dynamics of the Hadley cells to develop a new model to represent the Jormungand Hypothesis. We end by solving for an analytical approximation to the model using a finite Legendre expansion and geometric singular perturbation theory. The resultant model gives a stable equilibrium point near the equator with strong hysteresis that satisfies the Jormungand Hypothesis
The Jormungand Climate Model
The geological and paleomagnetic record indicate that around 750 million and 580 millions years ago glaciers grew near the equator, though as of yet we do not fully understand the nature of these glaciations. The well-known Snowball Earth Hypothesis states that the Earth was covered entirely by glaciers. However, it is hard for this hypothesis to account for certain aspects of the biological evidence such as the survival of photosynthetic eukaryotes. Thus the Jormungand Hypothesis was developed as an alternative to the Snowball Earth Hypothesis. In this paper we investigate previous models of the Jormungand state and look at the dynamics of the Hadley cells to develop a new model to represent the Jormungand Hypothesis. We end by solving for an analytical approximation to the model using a finite Legendre expansion and geometric singular perturbation theory. The resultant model gives a stable equilibrium point near the equator with strong hysteresis that satisfies the Jormungand Hypothesis
The Jormungand Climate Model
The geological and paleomagnetic record indicate that around 750 million and 580 millions years ago glaciers grew near the equator, though as of yet we do not fully understand the nature of these glaciations. The well-known Snowball Earth Hypothesis states that the Earth was covered entirely by glaciers. However, it is hard for this hypothesis to account for certain aspects of the biological evidence such as the survival of photosynthetic eukaryotes. Thus the Jormungand Hypothesis was developed as an alternative to the Snowball Earth Hypothesis. In this paper we investigate previous models of the Jormungand state and look at the dynamics of the Hadley cells to develop a new model to represent the Jormungand Hypothesis. We end by solving for an analytical approximation to the model using a finite Legendre expansion and geometric singular perturbation theory. The resultant model gives a stable equilibrium point near the equator with strong hysteresis that satisfies the Jormungand Hypothesis
The Jormungand Climate Model
The geological and paleomagnetic record indicate that around 750 million and 580 millions years ago glaciers grew near the equator, though as of yet we do not fully understand the nature of these glaciations. The well-known Snowball Earth Hypothesis states that the Earth was covered entirely by glaciers. However, it is hard for this hypothesis to account for certain aspects of the biological evidence such as the survival of photosynthetic eukaryotes. Thus the Jormungand Hypothesis was developed as an alternative to the Snowball Earth Hypothesis. In this paper we investigate previous models of the Jormungand state and look at the dynamics of the Hadley cells to develop a new model to represent the Jormungand Hypothesis. We end by solving for an analytical approximation to the model using a finite Legendre expansion and geometric singular perturbation theory. The resultant model gives a stable equilibrium point near the equator with strong hysteresis that satisfies the Jormungand Hypothesis
Confederated Modular Differential Equation APIs for Accelerated Algorithm Development and Benchmarking
Performant numerical solving of differential equations is required for
large-scale scientific modeling. In this manuscript we focus on two questions:
(1) how can researchers empirically verify theoretical advances and
consistently compare methods in production software settings and (2) how can
users (scientific domain experts) keep up with the state-of-the-art methods to
select those which are most appropriate? Here we describe how the confederated
modular API of DifferentialEquations.jl addresses these concerns. We detail the
package-free API which allows numerical methods researchers to readily utilize
and benchmark any compatible method directly in full-scale scientific
applications. In addition, we describe how the complexity of the method choices
is abstracted via a polyalgorithm. We show how scientific tooling built on top
of DifferentialEquations.jl, such as packages for dynamical systems
quantification and quantum optics simulation, both benefit from this structure
and provide themselves as convenient benchmarking tools.Comment: 4 figures, 3 algorithm
Stochastic Optimal Control via Local Occupation Measures
Viewing stochastic processes through the lens of occupation measures has
proved to be a powerful angle of attack for the theoretical and computational
analysis of a wide range of stochastic optimal control problems. We present a
simple modification of the traditional occupation measure framework derived
from resolving the occupation measures locally on a partition of the control
problem's space-time domain. This notion of local occupation measures provides
fine-grained control over the construction of structured semidefinite
programming relaxations for a rich class of stochastic optimal control problems
with embedded diffusion and jump processes via the moment-sum-of-squares
hierarchy. As such, it bridges the gap between discretization-based
approximations to the solution of the Hamilton-Jacobi-Bellmann equations and
approaches based on convex optimization and the moment-sum-of-squares
hierarchy. We demonstrate with examples that this approach enables the
computation of high quality bounds on the optimal value for a large class of
stochastic optimal control problems with notable performance gains relative to
the traditional occupation measure framework.Comment: 20 pages, 4 figures, associated implementation:
https://github.com/FHoltorf/MarkovBounds.j
Extending JumpProcess.jl for fast point process simulation with time-varying intensities
Point processes model the occurrence of a countable number of random points
over some support. They can model diverse phenomena, such as chemical
reactions, stock market transactions and social interactions. We show that
JumpProcesses.jl is a fast, general-purpose library for simulating point
processes. JumpProcesses.jl was first developed for simulating jump processes
via stochastic simulation algorithms (SSAs) (including Doob's method,
Gillespie's methods, and Kinetic Monte Carlo methods). Historically, jump
processes have been developed in the context of dynamical systems to describe
dynamics with discrete jumps. In contrast, the development of point processes
has been more focused on describing the occurrence of random events. In this
paper, we bridge the gap between the treatment of point and jump process
simulation. The algorithms previously included in JumpProcesses.jl can be
mapped to three general methods developed in statistics for simulating
evolutionary point processes. Our comparative exercise revealed that the
library initially lacked an efficient algorithm for simulating processes with
variable intensity rates. We, therefore, extended JumpProcesses.jl with a new
simulation algorithm, Coevolve, that enables the rapid simulation of processes
with locally-bounded variable intensity rates. It is now possible to
efficiently simulate any point process on the real line with a non-negative,
left-continuous, history-adapted and locally bounded intensity rate coupled or
not with differential equations. This extension significantly improves the
computational performance of JumpProcesses.jl when simulating such processes,
enabling it to become one of the few readily available, fast, general-purpose
libraries for simulating evolutionary point processes
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