Grid-Free Approach to Partial Differential Equations on Volumetric Domains [pdf]
http://rohansawhney.io/RohanSawhneyPhDThesis.pdf
#HackerNews #GridFree #PDEs #VolumetricDomains #MathematicalModeling #ComputationalPhysics
#PDEs
@bradheintz I feel like I had a similar attitude towards #probability in my undergrad. It wasn’t until later on in my PhD and even only as a researcher research that I came to appreciate the beauty. Especially through the connection to measure theory and through that #analysis. It is a very interesting field full of deep results and applications crossing to several fields like random #geometry and stochastic #PDEs. Many results in #numbertheory have even been proven using probabilistic methods, I.e. showing that properties hold for a set of numbers that has a positive measure in some sense.
A Distributed-memory Tridiagonal Solver Based on a Specialised Data Structure Optimised for CPU and GPU Architectures
'Boundary constrained Gaussian processes for robust physics-informed machine learning of linear partial differential equations', by David Dalton, Alan Lazarus, Hao Gao, Dirk Husmeier.
http://jmlr.org/papers/v25/23-1508.html
#boundary #pdes #gaussian
LINEAR TRANSPORT EQUATION
The linear transport equation (LTE) models the variation of the concentration of a substance flowing at constant speed and direction. It's one of the simplest partial differential equations (PDEs) and one of the few that admits an analytic solution.
Given \(\mathbf{c}\in\mathbb{R}^n\) and \(g:\mathbb{R}^n\to\mathbb{R}\), the following Cauchy problem models a substance flowing at constant speed in the direction \(\mathbf{c}\).
\[\begin{cases}
u_t+\mathbf{c}\cdot\nabla u=0,\ \mathbf{x}\in\mathbb{R}^n,\ t\in\mathbb{R}\\
u(\mathbf{x},0)=g(\mathbf{x}),\ \mathbf{x}\in\mathbb{R}^n
\end{cases}\]
If \(g\) is continuously differentiable, then \(\exists u:\mathbb{R}^n\times\mathbb{R}\to\mathbb{R}\) solution of the Cauchy problem, and it is given by
\[u(\mathbf{x},t)=g(\mathbf{x}-\mathbf{c}t)\]
#LinearTransportEquation #LinearTransport #Cauchy #CauchyProblem #PDE #PDEs #CauchyModel #Math #Maths #Mathematics #Linear #LinearPDE #TransportEquation #DifferentialEquations
'Multilevel CNNs for Parametric PDEs', by Cosmas Heiß, Ingo Gühring, Martin Eigel.
http://jmlr.org/papers/v24/23-0421.html
#pdes #solvers #deep
'Neural Q-learning for solving PDEs', by Samuel N. Cohen, Deqing Jiang, Justin Sirignano.
http://jmlr.org/papers/v24/22-1075.html
#pdes #pde #nonlinear
`The fast multipole method (FMM), introduced by Rokhlin Jr. and Greengard has been said to be one of the top ten #algorithms of the 20th century. The FMM algorithm reduces the complexity of matrix-vector multiplication involving a certain type of dense #matrix which can arise out of many #physical #systems.`
https://en.wikipedia.org/wiki/Fast_multipole_method
#physics #simulation #molecularDynamics #MD #dynamics #differentialEquations #ODEs #PDEs #ODE #PDE #Coulomb #Coulombic #CoulombForce #CoulombInteraction
in many ways, #mathematical #physics is based on the #smoothing properties of the #integral
#integrableSystems #mathematicalPhysics #complexAnalysis #spectralTheory #harmonicAnalysis #functionalAnalysis #mathematicalAnalysis #differentialEquations #ODEs #PDEs #SDEs #DEs #equations #BrownianMotion #LangevinDynamics #dynamics #Langevin #StochasticDifferentialEquations #StochasticProcesses #WienerProcess #OrnsteinUhlenbeck #HarmonicOscillator #WaveEquation #Newton #Newtonian #Maxwell #Einstein
Deep Operator Learning Lessens the Curse of Dimensionality for PDEs
Beautiful PDE visualization tool!
Here's a reaction-diffusion system with a pic of Turing himself as the initial condition.
Learning to correct spectral methods for simulating turbulent flows
Gideon Dresdner, Dmitrii Kochkov, Peter Christian Norgaard et al.
Action editor: Ivan Oseledets.
'Neural Operator: Learning Maps Between Function Spaces With Applications to PDEs', by Nikola Kovachki et al.
http://jmlr.org/papers/v24/21-1524.html
#discretization #operators #pdes
'Learning Partial Differential Equations in Reproducing Kernel Hilbert Spaces', by George Stepaniants.
http://jmlr.org/papers/v24/21-1363.html
#regression #pdes #kernel
Are you keen to read about results in #MathematicalPhysics and Analysis of #PDEs? Take a look at the paper of our previous workshop participant! 🧐
#FunctionalAnalysis #SpectralTheory
@univienna
https://arxiv.org/pdf/2303.04527.pdf
#SoftMatter have just published the results of a project that Renato Assante, Davide Marenduzzo, Alexander Morozov, and I recently worked on together! What did we do and what’s new? Briefly…
#Microswimmer suspensions behave in a similar way to fluids containing kinesin and microtubules. Both systems can be described by the same system of three coupled nonlinear #PDEs.
A #LinearStabilityAnalysis of these equations suggests that variations in concentration across the system don’t significantly affect emergent #phaseBehaviour. How then can we explain #experiments that show visible inhomogeneities in #microtubule–#kinesin mixtures, for instance?
With increasing activity, we move away from the quiescent regime, past the onset of #SpontaneousFlow, and deeper into the active phase, where #nonlinearities become more important. What role do concentration inhomogeneities play here?
We investigated these questions, taking advantage of the #openSource #Dedalus #spectral framework to simulate the full nonlinear time evolution. This led us to predict a #novel regime of #spontaneous #microphaseSeparation into active (nematically ordered) and passive domains.
Active flow arrests macrophase separation in this regime, counteracting domain coarsening due to thermodynamic coupling between active matter concentration and #nematic order. As a result, domains reach a characteristic size that decreases with increasing activity.
This regime is one part of the #PhaseDiagram we mapped out. Along with our other findings, you can read all about it here!
low #ReynoldsNumber #turbulence #ActiveTurbulence #CahnHilliard #ActiveMatter #NavierStokes #BerisEdwards #CondensedMatter #PhaseTransitions #TheoreticalPhysics #BioPhysics #StatisticalPhysics #FluidDynamics #ComputationalPhysics #Simulation #FieldTheory #paperthread #NewPaper #science #research #ActiveGel #activeNematic #analytic #cytoskeleton #hydrodynamics #MPI #theory
How do Maxwell's equations predict that the speed of light is constant?🤔 (photo credit: Fermat's Library)
Maxwell's equations, a set of coupled partial differential equations (PDEs), describe the behaviour of electric and magnetic fields and predict that the speed of light is constant in all reference frames, which is a fundamental principle of the theory of relativity. #Electromagnetism #Relativity #SpeedOfLight #LightSpeed #Light #ElectricField #MagneticField #PDEs #Maxwell #CoupledPDEs
#SoftMatter have just published the results of a project that Renato Assante, Davide Marenduzzo, Alexander Morozov, and I recently worked on together! What did we do and what’s new? Briefly…
The #hydrodynamic behaviour of inhomogeneous #activeNematic gels (such as extensile bundles of #cytoskeletal filaments or suspensions of low #ReynoldsNumber swimmers) can be described by the time evolution of three coupled #PDEs.
Standard #ActiveGel #theory concludes, from a #LinearStabilityAnalysis of these equations, that fluctuations in concentration don’t significantly affect emergent #phaseBehaviour. However, this leaves #experimental #observations of visible inhomogeneities in #microtubule–#kinesin mixtures unexplained. As we move away from the passive (quiescent) regime, past the onset of #SpontaneousFlow, and deeper into the active phase, #nonlinearities become more important. What role do concentration inhomogeneities play here?
Alongside #analytic techniques, we used an in-house #MPI-parallel code developed within the #Dedalus #spectral framework to investigate. We predict a #novel regime of #spontaneous #microphaseSeparation into active (nematically ordered) and passive domains. In this regime, active flow arrests macrophase separation, which is itself driven by the thermodynamic coupling between active matter concentration and #nematic order. As a result, domains do not #coarsen past a typical size, which decreases with increasing activity. This regime is one part of the #PhaseDiagram we mapped out.
Along with our other findings, you can read all about it here!
#CahnHilliard #ActiveMatter #NavierStokes #BerisEdwards #CondensedMatter #PhaseTransitions #TheoreticalPhysics #BioPhysics #StatisticalPhysics #FluidDynamics #ComputationalPhysics #Simulation #FieldTheory #paperthread #NewPaper #science #research
On my way to Perth today for the Trilateral meeting on #nonlinear #PDEs …
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