63 research outputs found
Solving periodic semilinear stiff PDEs in 1D, 2D and 3D with exponential integrators
Dozens of exponential integration formulas have been proposed for the
high-accuracy solution of stiff PDEs such as the Allen-Cahn, Korteweg-de Vries
and Ginzburg-Landau equations. We report the results of extensive comparisons
in MATLAB and Chebfun of such formulas in 1D, 2D and 3D, focusing on fourth and
higher order methods, and periodic semilinear stiff PDEs with constant
coefficients. Our conclusion is that it is hard to do much better than one of
the simplest of these formulas, the ETDRK4 scheme of Cox and Matthews
Computability of differential equations
In this chapter, we provide a survey of results concerning the computability and computational complexity of differential equations. In particular, we study the conditions which ensure computability of the solution to an initial value problem for an ordinary differential equation (ODE) and analyze the computational complexity of a computable solution. We also present computability results concerning the asymptotic behaviors of ODEs as well as several classically important partial differential equations.info:eu-repo/semantics/acceptedVersio
VisualPDE: rapid interactive simulations of partial differential equations
Computing has revolutionised the study of complex nonlinear systems, both by
allowing us to solve previously intractable models and through the ability to
visualise solutions in different ways. Using ubiquitous computing
infrastructure, we provide a means to go one step further in using computers to
understand complex models through instantaneous and interactive exploration.
This ubiquitous infrastructure has enormous potential in education, outreach
and research. Here, we present VisualPDE, an online, interactive solver for a
broad class of 1D and 2D partial differential equation (PDE) systems. Abstract
dynamical systems concepts such as symmetry-breaking instabilities, subcritical
bifurcations and the role of initial data in multistable nonlinear models
become much more intuitive when you can play with these models yourself, and
immediately answer questions about how the system responds to changes in
parameters, initial conditions, boundary conditions or even spatiotemporal
forcing. Importantly, VisualPDE is freely available, open source and highly
customisable. We give several examples in teaching, research and knowledge
exchange, providing high-level discussions of how it may be employed in
different settings. This includes designing web-based course materials
structured around interactive simulations, or easily crafting specific
simulations that can be shared with students or collaborators via a simple URL.
We envisage VisualPDE becoming an invaluable resource for teaching and research
in mathematical biology and beyond. We also hope that it inspires other efforts
to make mathematics more interactive and accessible.Comment: 19 pages, 7 figures. This is a companion paper to the website
https://visualpde.com
Unsupervised physics-informed neural network in reaction-diffusion biology models (Ulcerative colitis and Crohn's disease cases) A preliminary study
We propose to explore the potential of physics-informed neural networks
(PINNs) in solving a class of partial differential equations (PDEs) used to
model the propagation of chronic inflammatory bowel diseases, such as Crohn's
disease and ulcerative colitis. An unsupervised approach was privileged during
the deep neural network training. Given the complexity of the underlying
biological system, characterized by intricate feedback loops and limited
availability of high-quality data, the aim of this study is to explore the
potential of PINNs in solving PDEs. In addition to providing this exploratory
assessment, we also aim to emphasize the principles of reproducibility and
transparency in our approach, with a specific focus on ensuring the robustness
and generalizability through the use of artificial intelligence. We will
quantify the relevance of the PINN method with several linear and non-linear
PDEs in relation to biology. However, it is important to note that the final
solution is dependent on the initial conditions, chosen boundary conditions,
and neural network architectures
Computing Solution Operators of Boundary-value Problems for Some Linear Hyperbolic Systems of PDEs
We discuss possibilities of application of Numerical Analysis methods to
proving computability, in the sense of the TTE approach, of solution operators
of boundary-value problems for systems of PDEs. We prove computability of the
solution operator for a symmetric hyperbolic system with computable real
coefficients and dissipative boundary conditions, and of the Cauchy problem for
the same system (we also prove computable dependence on the coefficients) in a
cube . Such systems describe a wide variety of physical
processes (e.g. elasticity, acoustics, Maxwell equations). Moreover, many
boundary-value problems for the wave equation also can be reduced to this case,
thus we partially answer a question raised in Weihrauch and Zhong (2002).
Compared with most of other existing methods of proving computability for PDEs,
this method does not require existence of explicit solution formulas and is
thus applicable to a broader class of (systems of) equations.Comment: 31 page
Computability and analysis: the legacy of Alan Turing
We discuss the legacy of Alan Turing and his impact on computability and
analysis.Comment: 49 page
State-dependent Importance Sampling for a Slow-down Tandem Queue
In this paper we investigate an advanced variant of the classical (Jackson) tandem queue, viz. a two-node system with server slow-down. The slow-down mechanism has the primary objective to protect the downstream queue from frequent overflows, and it does so by reducing the service speed of the upstream queue as soon as the number of jobs in the downstream queue reaches some pre-specified threshold. To assess the efficacy of such a policy, techniques are needed for evaluating overflow metrics of the second queue. We focus on the estimation of the probability of the following rare event: overflow in the downstream queue before exhausting the system, starting from any given state in the state space.\ud
Due to the rarity of the event under consideration, naive, direct Monte Carlo simulation is often infeasible. We therefore rely on the application of importance sampling to obtain variance reduction. The principal contribution of this paper is that we construct an importance sampling scheme that is asymptotically efficient. In more detail, the paper addresses the following issues. (i) We rely on powerful heuristics to identify the exponential decay rate of the probability under consideration, and verify this result by applying sample-path large deviations techniques. (2) Immediately from these heuristics, we develop a proposal for a change of measure to be used in importance sampling. (3) We prove that the resulting algorithm is asymptotically efficient, which effectively means that the number of runs required to obtain an estimate with fixed precision grows subexponentially in the buffer size. We stress that our method to prove asymptotic efficiency is substantially shorter and more straightforward than those usually provided in the literature. Also our setting is more general than the situations analyzed so far, as we allow the process to start off at any state of the state space, and in addition we do not impose any conditions on the values of the arrival rate and service rates, as long as the underlying queueing system is stable
Hyperbolic Techniques in Modelling, Analysis and Numerics
Several research areas are flourishing on the roots of the breakthroughs in conservation laws that took place in the last two decades. The meeting played a key role in providing contacts among the different branches that are currently developing. All the invitees shared the same common background that consists of the analytical and numerical techniques for nonlinear hyperbolic balance laws. However, their fields of applications and their levels of abstraction are very diverse. The workshop was the unique opportunity to share ideas about analytical issues like the fine-structure of singular solutions or the validity of entropy solution concepts. It turned out that generalized hyperbolic techniques are able to handle the challenges posed by new applications. The design of efficient structure preserving methods turned out to be the major line of development in numerical analysis
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