Tootfinder

This https://arxiv.org/abs/2402.07665 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_mat…

Uniqueness of weak solutions for Hamilton-Jacobi equations
It is well known that when the nonlinearity is convex, the Hamilton-Jacobi PDE admits a unique semi-convex weak solution, which is the viscosity solution. In this paper, motivated by problems arising from spin glasses, we show that if the Hamilton-Jacobi PDE with strictly convex nonlinearity and regular enough initial condition admits a semi-concave weak solution, then this solution is the viscosity solution.

New instanton on a warped Kerr spacetime
Reinoud Jan Slagter
https://arxiv.org/abs/2404.08346 https://arxiv.org/pdf/2404.08346…

New instanton on a warped Kerr spacetime
We find an exact time-dependent instanton solution on a vacuum Kerr-like warped spacetime in conformal dilaton gravity. Remarkably, the metric solution results from a first-order PDE, allowing the connection with self-duality. The antipodal boundary condition on the hypersurface of a Klein bottle $\sim \mathbb{C}^1\times\mathbb{C}^1$ is applied to describe the Hawking particles. We used the Hopf fibration to get $S^2$ as the black hole horizon, where the centrix is not in a torus but in the Kle…

Stabilization and Optimal Control of Interconnected SDE - Scalar PDE System
Gabriel Velho, Jean Auriol, Riccardo Bonalli, Islam Boussaada
https://arxiv.org/abs/2405.08600

This https://arxiv.org/abs/2305.12017 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_mat…

Decay of correlations in stochastic quantization: the exponential Euclidean field in two dimensions
We present two approaches to establish the exponential decay of correlation functions of Euclidean quantum field theories (EQFTs) via stochastic quantization (SQ). In particular we consider the elliptic stochastic quantization of the Høegh--Krohn (or $\exp (αϕ)_2$) EQFT in two dimensions. The first method is based on a path-wise coupling argument and PDE apriori estimates while the second on estimates of the Malliavin derivative of the solution to the SQ equation.

This https://arxiv.org/abs/2206.07303 has been replaced.
link: https://scholar.google.com/scholar?q=a

Energetic Variational Neural Network Discretizations of Gradient Flows
We present a structure-preserving Eulerian algorithm for solving $L^2$-gradient flows and a structure-preserving Lagrangian algorithm for solving generalized diffusions. Both algorithms employ neural networks as tools for spatial discretization. Unlike most existing methods that construct numerical discretizations based on the strong or weak form of the underlying PDE, the proposed schemes are constructed based on the energy-dissipation law directly. This guarantees the monotonic decay of the s…

This https://arxiv.org/abs/2402.07665 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_mat…

Uniqueness of weak solutions for Hamilton-Jacobi equations
It is well known that when the nonlinearity is convex, the Hamilton-Jacobi PDE admits a unique semi-convex weak solution, which is the viscosity solution. In this paper, motivated by problems arising from spin glasses, we show that if the Hamilton-Jacobi PDE with strictly convex nonlinearity and regular enough initial condition admits a semi-concave weak solution, then this solution is the viscosity solution.

This https://arxiv.org/abs/2404.08346 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_hept…

New instanton on a warped Kerr spacetime
We find an exact time-dependent instanton solution on a vacuum Kerr-like warped spacetime in conformal dilaton gravity. Remarkably, the metric solution results from a first-order PDE, allowing the connection with self-duality. The antipodal boundary condition on the hypersurface of a Klein bottle $\sim \mathbb{C}^1\times\mathbb{C}^1$ is applied to describe the Hawking particles. We used the Hopf fibration to get $S^2$ as the black hole horizon, where the centrix is not in a torus but in the Kle…

This https://arxiv.org/abs/2404.11766 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_csLG_…

End-to-End Mesh Optimization of a Hybrid Deep Learning Black-Box PDE Solver
Deep learning has been widely applied to solve partial differential equations (PDEs) in computational fluid dynamics. Recent research proposed a PDE correction framework that leverages deep learning to correct the solution obtained by a PDE solver on a coarse mesh. However, end-to-end training of such a PDE correction model over both solver-dependent parameters such as mesh parameters and neural network parameters requires the PDE solver to support automatic differentiation through the iterativ…

This https://arxiv.org/abs/2401.15728 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_qfi…

Analytic Pricing of SOFR Futures Contracts with Smile and Skew
We introduce a perturbative formalism to solve the backward-looking futures pricing problem. The formalism is based on a time-ordered exponential series which allows to derive the functional form of the integral kernel associated to the backward-Kolmogorov diffusion PDE. We present an analytic pricing formula for SOFR futures contracts under an extension of the Hull-White model which incorporates not only the intrinsic convexity adjustments captured by Mercurio [2018], but also the skew and smi…

This https://arxiv.org/abs/2202.09894 has been replaced.
link: https://scholar.google.com/scholar?q=a

Third-order affine-invariant (systems of) PDEs in two independent variables as vanishing of the Fubini-Pick invariant
If one imposes that a $3^{\mathrm{rd}}$ order PDE $\mathcal{E}:=\{F(x,y,u,u_x,\dots,u_{xyy},u_{yyy})=0\}$ in two independent variables $x,y$ and one unknown function $u$ be invariant with respect to the group of affine transformation $\mathrm{Aff}(3)$ of $\mathbb{R}^3=\{(x,y,u)\}$, then it turns out that there is (up to a nowhere zero factor) only one choice of $F$. In this paper we study more in depth the geometry behind the occurrence of such a PDE: after proving the relationship of $F$ with …

This https://arxiv.org/abs/2207.07698 has been replaced.
link: https://scholar.google.com/scholar?q=a

Quasi-Monte Carlo and discontinuous Galerkin
In this study, we consider the development of tailored quasi-Monte Carlo (QMC) cubatures for non-conforming discontinuous Galerkin (DG) approximations of elliptic partial differential equations (PDEs) with random coefficients. We consider both the affine and uniform and the lognormal models for the input random field, and investigate the use of QMC cubatures to approximate the expected value of the PDE response subject to input uncertainty. In particular, we prove that the resulting QMC converg…

This https://arxiv.org/abs/2208.11551 has been replaced.
link: https://scholar.google.com/scholar?q=a

PDE characterisation of geometric distribution functions and quantiles
We show that in any Euclidean space, an arbitrary probability measure can be reconstructed explicitly by its geometric (or spatial) distribution function. The reconstruction takes the form of a (potentially fractional) linear PDE, where the differential operator is given in closed form. This result implies that, contrary to a common belief in the statistical depth community, geometric cdf's in principle provide exact control over the probability content of all depth regions. We present a compre…

Approximation of non-linear SPDEs with additive noise via weighted interacting particles systems: the stochastic McKean-Vlasov equation
Letizia Angeli, Dan Crisan, Martin Kolodziejczyk, Michela Ottobre
https://arxiv.org/abs/2404.07488

Approximation of non-linear SPDEs with additive noise via weighted interacting particles systems: the stochastic McKean-Vlasov equation
This paper is devoted to the problem of approximating non-linear Stochastic Partial Differential Equations (SPDEs) via interacting particle systems. In particular, we consider the Stochastic McKean-Vlasov equation, which is the McKean-Vlasov (MKV) PDE, perturbed by additive trace class noise. As is well-known, the MKV PDE can be obtained as mean field limit of the empirical measure of a stochastic system of interacting particles, where particles are subject to independent sources of noise. Ther…

This https://arxiv.org/abs/2305.04024 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_mat…

Differential-Equation Constrained Optimization With Stochasticity
Most inverse problems from physical sciences are formulated as PDE-constrained optimization problems. This involves identifying unknown parameters in equations by optimizing the model to generate PDE solutions that closely match measured data. The formulation is powerful and widely used in many sciences and engineering fields. However, one crucial assumption is that the unknown parameter must be deterministic. In reality, however, many problems are stochastic in nature, and the unknown paramete…

A fully differentiable GNN-based PDE Solver: With Applications to Poisson and Navier-Stokes Equations
Tianyu Li, Yiye Zou, Shufan Zou, Xinghua Chang, Laiping Zhang, Xiaogang Deng
https://arxiv.org/abs/2405.04466

A fully differentiable GNN-based PDE Solver: With Applications to Poisson and Navier-Stokes Equations
In this study, we present a novel computational framework that integrates the finite volume method with graph neural networks to address the challenges in Physics-Informed Neural Networks(PINNs). Our approach leverages the flexibility of graph neural networks to adapt to various types of two-dimensional unstructured grids, enhancing the model's applicability across different physical equations and boundary conditions. The core innovation lies in the development of an unsupervised training algor…

The Generalized Riemann Zeta heat flow
V\'ictor Castillo, Claudio Mu\~noz, Felipe Poblete, Vicente Salinas
https://arxiv.org/abs/2402.10154 https://

The Generalized Riemann Zeta heat flow
We consider the PDE flow associated to Riemann zeta and general Dirichlet $L$-functions. These are models characterized by nonlinearities appearing in classical number theory problems, and generalizing the classical holomorphic Riemann flow studied by Broughan and Barnett. Each zero of a Dirichlet $L$-function is an exact solution of the model. In this paper, we first show local existence of bounded continuous solutions in the Duhamel sense to any Dirichlet $L$-function flow with initial condit…

Statistical algorithms for low-frequency diffusion data: A PDE approach
Matteo Giordano, Sven Wang
https://arxiv.org/abs/2405.01372 https://

Statistical algorithms for low-frequency diffusion data: A PDE approach
We consider the problem of making nonparametric inference in multi-dimensional diffusion models from low-frequency data. Statistical analysis in this setting is notoriously challenging due to the intractability of the likelihood and its gradient, and computational methods have thus far largely resorted to expensive simulation-based techniques. In this article, we propose a new computational approach which is motivated by PDE theory and is built around the characterisation of the transition dens…

Rectified deep neural networks overcome the curse of dimensionality in the numerical approximation of gradient-dependent semilinear heat equations
Ariel Neufeld, Tuan Anh Nguyen
https://arxiv.org/abs/2403.09200

Rectified deep neural networks overcome the curse of dimensionality in the numerical approximation of gradient-dependent semilinear heat equations
Numerical experiments indicate that deep learning algorithms overcome the curse of dimensionality when approximating solutions of semilinear PDEs. For certain linear PDEs and semilinear PDEs with gradient-independent nonlinearities this has also been proved mathematically, i.e., it has been shown that the number of parameters of the approximating DNN increases at most polynomially in both the PDE dimension $d\in \mathbb{N}$ and the reciprocal of the prescribed accuracy $ε\in (0,1)$. The main c…

Gain-Only Neural Operator Approximators of PDE Backstepping Controllers
Rafael Vazquez, Miroslav Krstic
https://arxiv.org/abs/2403.19344 https://

Gain-Only Neural Operator Approximators of PDE Backstepping Controllers
For the recently introduced deep learning-powered approach to PDE backstepping control, we present an advancement applicable across all the results developed thus far: approximating the control gain function only (a function of one variable), rather than the entire kernel function of the backstepping transformation (a function of two variables). We introduce this idea on a couple benchmark (unstable) PDEs, hyperbolic and parabolic. We alter the approach of quantifying the effect of the approxim…

This https://arxiv.org/abs/2311.05306 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_mat…

The Exponential Stabilization of a Heat and Piezoelectric Beam Interaction with Static or Hybrid Feedback Controllers
This study investigates a strongly-coupled system of partial differential equations (PDE) governing heat transfer in a copper rod, longitudinal vibrations, and total charge accumulation at electrodes within a magnetizable piezoelectric beam. Conducted within the transmission line framework, the analysis reveals profound interactions between traveling electromagnetic and mechanical waves in magnetizable piezoelectric beams, despite disparities in their velocities. Findings suggest that in the op…

The Generalized Riemann Zeta heat flow
V\'ictor Castillo, Claudio Mu\~noz, Felipe Poblete, Vicente Salinas
https://arxiv.org/abs/2402.10154 https://

The Generalized Riemann Zeta heat flow
We consider the PDE flow associated to Riemann zeta and general Dirichlet $L$-functions. These are models characterized by nonlinearities appearing in classical number theory problems, and generalizing the classical holomorphic Riemann flow studied by Broughan and Barnett. Each zero of a Dirichlet $L$-function is an exact solution of the model. In this paper, we first show local existence of bounded continuous solutions in the Duhamel sense to any Dirichlet $L$-function flow with initial condit…

This https://arxiv.org/abs/2309.13986 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_mat…

Boundary Output Feedback Stabilization for a Novel Magnetizable Piezoelectric Beam Model
A magnetizable piezoelectric beam model, free at both ends, is considered. Piezoelectric materials have a strong interaction of electromagnetic and acoustic waves, whose wave propagation speeds differ substantially. The corresponding strongly-coupled PDE model describes the longitudinal vibrations and the total charge accumulation at the electrodes of the beam. It is known that the PDE model with appropriately chosen collocated state feedback controllers is known to have exponentially stable so…

This https://arxiv.org/abs/2308.05882 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_csCE_…

GPLaSDI: Gaussian Process-based Interpretable Latent Space Dynamics Identification through Deep Autoencoder
Numerically solving partial differential equations (PDEs) can be challenging and computationally expensive. This has led to the development of reduced-order models (ROMs) that are accurate but faster than full order models (FOMs). Recently, machine learning advances have enabled the creation of non-linear projection methods, such as Latent Space Dynamics Identification (LaSDI). LaSDI maps full-order PDE solutions to a latent space using autoencoders and learns the system of ODEs governing the l…

This https://arxiv.org/abs/2403.01667 has been replaced.
link: https://scholar.google.com/scholar?q=a

Reaction-diffusion models of biological invasion: Open source computational tools, key concepts and analysis
This review provides open-access computational tools that support a range of mathematical approaches to analyse three related scalar reaction-diffusion models used to study biological invasion. Starting with the classic Fisher-Kolmogorov (Fisher-KPP) model, we illustrate how computational methods can be used to explore time-dependent partial differential equation (PDE) solutions in parallel with phase plane and regular perturbation techniques to explore invading travelling wave solutions moving…

This https://arxiv.org/abs/2403.02224 has been replaced.
link: https://scholar.google.com/scholar?q=a

Understanding Latent Timescales in Neural Ordinary Differential Equation Models for Advection-Dominated Dynamical Systems
The neural ordinary differential equation (ODE) framework has shown promise in developing accelerated surrogate models for complex systems described by partial differential equations (PDEs). In PDE-based systems, neural ODE strategies use a two-step approach for acceleration: a nonlinear dimensionality reduction via an autoencoder, and a time integration step through a neural-network based model (neural ODE). This study explores the effectiveness of autoencoder-based neural ODE strategies for a…

This https://arxiv.org/abs/2208.08626 has been replaced.
link: https://scholar.google.com/scholar?q=a

CP-PINNs: Data-Driven Changepoints Detection in PDEs Using Online Optimized Physics-Informed Neural Networks
We investigate the inverse problem for Partial Differential Equations (PDEs) in scenarios where the parameters of the given PDE dynamics may exhibit changepoints at random time. We employ Physics-Informed Neural Networks (PINNs) - universal approximators capable of estimating the solution of any physical law described by a system of PDEs, which serves as a regularization during neural network training, restricting the space of admissible solutions and enhancing function approximation accuracy. …

Analysis of the SQP Method for Hyperbolic PDE-Constrained Optimization in Acoustic Full Waveform Inversion
Luis Ammann, Irwin Yousept
https://arxiv.org/abs/2405.05158

Analysis of the SQP Method for Hyperbolic PDE-Constrained Optimization in Acoustic Full Waveform Inversion
In this paper, the SQP method applied to a hyperbolic PDE-constrained optimization problem is considered. The model arises from the acoustic full waveform inversion in the time domain. The analysis is mainly challenging due to the involved hyperbolicity and second-order bilinear structure. This notorious character leads to an undesired effect of loss of regularity in the SQP method, calling for a substantial extension of developed parabolic techniques. We propose and analyze a novel strategy fo…

An asynchronous discontinuous Galerkin method for massively parallel PDE solvers
Shubham Kumar Goswami, Konduri Aditya
https://arxiv.org/abs/2404.03399 htt…

An asynchronous discontinuous Galerkin method for massively parallel PDE solvers
The discontinuous Galerkin (DG) method is widely being used to solve hyperbolic partial differential equations (PDEs) due to its ability to provide high-order accurate solutions in complex geometries, capture discontinuities, and exhibit high arithmetic intensity. However, the scalability of DG-based solvers is impeded by communication bottlenecks arising from the data movement and synchronization requirements at extreme scales. To address these challenges, recent studies have focused on the de…

This https://arxiv.org/abs/2305.09792 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_mat…

Score Operator Newton transport
We propose a new approach for sampling and Bayesian computation that uses the score of the target distribution to construct a transport from a given reference distribution to the target. Our approach is an infinite-dimensional Newton method, involving a linear PDE, for finding a zero of a ``score-residual'' operator. We prove sufficient conditions for convergence to a valid transport map. Our Newton iterates can be computed by exploiting fast solvers for elliptic PDEs, resulting in new algorith…

Optimal Control of a Diffusive Epidemiological Model Involving the Caputo-Fabrizio Fractional Time-Derivative
Achraf Zinihi, Moulay Rchid Sidi Ammi, Matthias Ehrhardt
https://arxiv.org/abs/2403.00364

Optimal Control of a Diffusive Epidemiological Model Involving the Caputo-Fabrizio Fractional Time-Derivative
In this work we study a fractional SEIR biological model of a reaction-diffusion, using the non-singular kernel Caputo-Fabrizio fractional derivative in the Caputo sense and employing the Laplacian operator. In our PDE model, the government seeks immunity through the vaccination program, which is considered a control variable. Our study aims to identify the ideal control pair that reduces the number of infected/infectious people and the associated vaccine and treatment costs over a limited time…

A gluing construction of singular solutions for a fully non-linear equation in conformal geometry
Mar\'ia Fernanda Espinal, Mar\'ia del Mar Gonz\'alez
https://arxiv.org/abs/2404.05965

A gluing construction of singular solutions for a fully non-linear equation in conformal geometry
In this paper we produce families of complete, non-compact Riemannian metrics with positive constant $σ_2$--curvature on the sphere $\mathbb S^n$, $n>4$, with a prescribed singular set $Λ$ given by a disjoint union of closed submanifolds whose dimension is positive and strictly less than $(n-\sqrt{n}-2)/2$. The $σ_2$--curvature in conformal geometry is defined as the second elementary symmetric polynomial of the eigenvalues of the Schouten tensor, which yields a fully non-linear PDE for the …

This https://arxiv.org/abs/2308.15755 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_ees…

Density Stabilization Strategies for Nonholonomic Agents on Compact Manifolds
In this article, we consider the problem of stabilizing stochastic processes, which are constrained to a bounded Euclidean domain or a compact smooth manifold, to a given target probability density. Most existing works on modeling and control of robotic swarms that use PDE models assume that the robots' dynamics are holonomic, and hence, the associated stochastic processes have generators that are elliptic. We relax this assumption on the ellipticity of the generator of the stochastic processes…

This https://arxiv.org/abs/2211.00079 has been replaced.
link: https://scholar.google.com/scholar?q=a

A dual variational principle for nonlinear dislocation dynamics
A dual variational principle is defined for the nonlinear system of PDE describing the dynamics of dislocations in elastic solids. The dual variational principle accounting for a specified set of initial and boundary conditions for a general class of PDE is also developed.

IETI-based Low-Rank method for PDE-constrained optimization
Alexandra B\"unger, Tom-Christian Riemer, Martin Stoll
https://arxiv.org/abs/2405.06458 ht…

Numerical simulations of a stochastic dynamics leading to cascades and loss of regularity: applications to fluid turbulence and generation of fractional Gaussian fields
Geoffrey Beck, Charles-Edouard Br\'ehier, Laurent Chevillard, Ricardo Grande, Wandrille Ruffenach
https://arxiv.org/abs/2403.05401

Numerical simulations of a stochastic dynamics leading to cascades and loss of regularity: applications to fluid turbulence and generation of fractional Gaussian fields
Motivated by the modeling of the spatial structure of the velocity field of three-dimensional turbulent flows, and the phenomenology of cascade phenomena, a linear dynamics has been recently proposed able to generate high velocity gradients from a smooth-in-space forcing term. It is based on a linear Partial Differential Equation (PDE) stirred by an additive random forcing term which is delta-correlated in time. The underlying proposed deterministic mechanism corresponds to a transport in Fouri…

Deep Backward and Galerkin Methods for the Finite State Master Equation
Asaf Cohen, Mathieu Lauri\`ere, Ethan Zell
https://arxiv.org/abs/2403.04975 https:/…

Deep Backward and Galerkin Methods for the Finite State Master Equation
This paper proposes and analyzes two neural network methods to solve the master equation for finite-state mean field games (MFGs). Solving MFGs provides approximate Nash equilibria for stochastic, differential games with finite but large populations of agents. The master equation is a partial differential equation (PDE) whose solution characterizes MFG equilibria for any possible initial distribution. The first method we propose relies on backward induction in a time component while the second …

Local sparse limit of a particle coalescence model under environmental noise
Franco Flandoli, Ruojun Huang
https://arxiv.org/abs/2404.05233 https://…

Local sparse limit of a particle coalescence model under environmental noise
A system of diffusions subject to common (environmental) noise is investigated. Due to the scaling limit assumptions on the common noise, a deterministic PDE limit is obtained. Particles interact by coalescence and the sparse regime of Hammond and Rezakhanlou is imposed, so that a nontrivial cell equation comes into play to determine the average coalescence rate. The corrector to the mean coalescence rate provided by that theory explains the role of turbulence in the enhancement of coalescence.…

IETI-based Low-Rank method for PDE-constrained optimization
Alexandra B\"unger, Tom-Christian Riemer, Martin Stoll
https://arxiv.org/abs/2405.06458 ht…

This https://arxiv.org/abs/2309.15326 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_qbi…

Structural identifiability analysis of linear reaction-advection-diffusion processes in mathematical biology
Effective application of mathematical models to interpret biological data and make accurate predictions often requires that model parameters are identifiable. Approaches to assess the so-called structural identifiability of models are well-established for ordinary differential equation models, yet there are no commonly adopted approaches that can be applied to assess the structural identifiability of the partial differential equation (PDE) models that are requisite to capture spatial features i…

This https://arxiv.org/abs/2311.03091 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_mat…

Abstract dissipative Hamiltonian differential-algebraic equations are everywhere
In this paper we study the representation of partial differential equations (PDEs) as abstract differential-algebraic equations (DAEs) with dissipative Hamiltonian structure (adHDAEs). We show that these systems not only arise when there are constraints coming from the underlying physics, but many standard PDE models can be seen as an adHDAE on an extended state space. This reflects the fact that models often include closure relations and structural properties. We present a unifying operator th…

This https://arxiv.org/abs/2202.09894 has been replaced.
link: https://scholar.google.com/scholar?q=a

Third-order affine-invariant (systems of) PDEs in two independent variables as vanishing of the Fubini-Pick invariant
If one imposes that a $3^{\mathrm{rd}}$ order PDE $\mathcal{E}:=\{F(x,y,u,u_x,\dots,u_{xyy},u_{yyy})=0\}$ in two independent variables $x,y$ and one unknown function $u$ be invariant with respect to the group of affine transformation $\mathrm{Aff}(3)$ of $\mathbb{R}^3=\{(x,y,u)\}$, then it turns out that there is (up to a nowhere zero factor) only one choice of $F$. In this paper we study more in depth the geometry behind the occurrence of such a PDE: after proving the relationship of $F$ with …

Generating self-similar membrane solutions
Jens Hoppe
https://arxiv.org/abs/2404.18856 https://arxiv.org/pdf/2404.18856

Generating self-similar membrane solutions
Several ways to reduce to a first order ODE the non-linear PDE's governing the relativistic motion of an axially symmetric membrane in 4 space time dimensions, as well as examples for a previously found non-trivial transformation generating solutions, are given.

This https://arxiv.org/abs/2108.06740 has been replaced.
link: https://scholar.google.com/scholar?q=a

A fast iterative PDE-based algorithm for feedback controls of nonsmooth mean-field control problems
We propose a PDE-based accelerated gradient algorithm for optimal feedback controls of McKean-Vlasov dynamics that involve mean-field interactions both in the state and action. The method exploits a forward-backward splitting approach and iteratively refines the approximate controls based on the gradients of smooth costs, the proximal maps of nonsmooth costs, and dynamically updated momentum parameters. At each step, the state dynamics is approximated via a particle system, and the required gra…

This https://arxiv.org/abs/2204.07497 has been replaced.
link: https://scholar.google.com/scholar?q=a

Helicity-conservative Physics-informed Neural Network Model for Navier-Stokes Equations
We design the helicity-conservative physics-informed neural network model for the Navier-Stokes equation in the ideal case. The key is to provide an appropriate PDE model as loss function so that its neural network solutions produce helicity conservation. Physics-informed neural network model is based on the strong form of PDE. We compare the proposed Physics-informed neural network model and a relevant helicity-conservative finite element method. We arrive at the conclusion that the strong for…

Generative downscaling of PDE solvers with physics-guided diffusion models
Yulong Lu, Wuzhe Xu
https://arxiv.org/abs/2404.05009 https://

Generative downscaling of PDE solvers with physics-guided diffusion models
Solving partial differential equations (PDEs) on fine spatio-temporal scales for high-fidelity solutions is critical for numerous scientific breakthroughs. Yet, this process can be prohibitively expensive, owing to the inherent complexities of the problems, including nonlinearity and multiscale phenomena. To speed up large-scale computations, a process known as downscaling is employed, which generates high-fidelity approximate solutions from their low-fidelity counterparts. In this paper, we pr…

Iterated INLA for State and Parameter Estimation in Nonlinear Dynamical Systems
Rafael Anderka, Marc Peter Deisenroth, So Takao
https://arxiv.org/abs/2402.17036

Iterated INLA for State and Parameter Estimation in Nonlinear Dynamical Systems
Data assimilation (DA) methods use priors arising from differential equations to robustly interpolate and extrapolate data. Popular techniques such as ensemble methods that handle high-dimensional, nonlinear PDE priors focus mostly on state estimation, however can have difficulty learning the parameters accurately. On the other hand, machine learning based approaches can naturally learn the state and parameters, but their applicability can be limited, or produce uncertainties that are hard to i…

The behaviour of the DC current density at the edge of electrodes
Spyros Alexakis
https://arxiv.org/abs/2404.04336 https://arxiv.org/…

The behaviour of the DC current density at the edge of electrodes
We study the complete electrode model boundary condition for second order elliptic PDE. A specific case of this is the PDE describing the electrostatic potential for a conductive body into which current is injected through electrodes that touch the boundary. We obtain the optimal description of the gradient of the electrostatic potential upon approach to the edge of the electrodes.

Generative downscaling of PDE solvers with physics-guided diffusion models
Yulong Lu, Wuzhe Xu
https://arxiv.org/abs/2404.05009 https://

Generative downscaling of PDE solvers with physics-guided diffusion models
Solving partial differential equations (PDEs) on fine spatio-temporal scales for high-fidelity solutions is critical for numerous scientific breakthroughs. Yet, this process can be prohibitively expensive, owing to the inherent complexities of the problems, including nonlinearity and multiscale phenomena. To speed up large-scale computations, a process known as downscaling is employed, which generates high-fidelity approximate solutions from their low-fidelity counterparts. In this paper, we pr…

Rockafellian Relaxation for PDE-Constrained Optimization with Distributional Uncertainty
Harbir Antil, Sean P. Carney, Hugo D\'iaz, Johannes O. Royset
https://arxiv.org/abs/2405.00176

Rockafellian Relaxation for PDE-Constrained Optimization with Distributional Uncertainty
Stochastic optimization problems are generally known to be ill-conditioned to the form of the underlying uncertainty. A framework is introduced for optimal control problems with partial differential equations as constraints that is robust to inaccuracies in the precise form of the problem uncertainty. The framework is based on problem relaxation and involves optimizing a bivariate, "Rockafellian" objective functional that features both a standard control variable and an additional perturbation …

Bathymetry reconstruction from experimental data using PDE-constrained optimisation
Judith Angel, J\"orn Behrens, Sebastian G\"otschel, Marten Hollm, Daniel Ruprecht, Robert Seifried
https://arxiv.org/abs/2404.05556

Bathymetry reconstruction from experimental data using PDE-constrained optimisation
Knowledge of the bottom topography, also called bathymetry, of rivers, seas or the ocean is important for many areas of maritime science and civil engineering. While direct measurements are possible, they are time consuming and expensive. Therefore, many approaches have been proposed how to infer the bathymetry from measurements of surface waves. Mathematically, this is an inverse problem where an unknown system state needs to be reconstructed from observations with a suitable model for the flo…

This https://arxiv.org/abs/2401.15661 has been replaced.
link: https://scholar.google.com/scholar?q=a

Brain-Inspired Physics-Informed Neural Networks: Bare-Minimum Neural Architectures for PDE Solvers
Physics-Informed Neural Networks (PINNs) have emerged as a powerful tool for solving partial differential equations~(PDEs) in various scientific and engineering domains. However, traditional PINN architectures typically rely on large, fully connected multilayer perceptrons~(MLPs), lacking the sparsity and modularity inherent in many traditional numerical solvers. An unsolved and critical question for PINN is: What is the minimum PINN complexity regarding nodes, layers, and connections needed to…

This https://arxiv.org/abs/2401.13962 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_mat…

An inf-sup Approach to $C_0$-Semigroup Generation for An Interactive Composite Structure-Stokes PDE Dynamics
In this work, we investigate the existence and uniqueness properties of a composite structure (multilayered) fluid interaction PDE system which arises in multi-physics problems, and particularly in biofluidic applications related to the mammalian blood transportation process. The PDE system under consideration consists of the interactive coupling of 3D Stokes flow and 3D elastic dynamics which gives rise to an additional 2D elastic equation on the boundary interface between these 3D PDE systems…

Bathymetry reconstruction from experimental data using PDE-constrained optimisation
Judith Angel, J\"orn Behrens, Sebastian G\"otschel, Marten Hollm, Daniel Ruprecht, Robert Seifried
https://arxiv.org/abs/2404.05556

Bathymetry reconstruction from experimental data using PDE-constrained optimisation
Knowledge of the bottom topography, also called bathymetry, of rivers, seas or the ocean is important for many areas of maritime science and civil engineering. While direct measurements are possible, they are time consuming and expensive. Therefore, many approaches have been proposed how to infer the bathymetry from measurements of surface waves. Mathematically, this is an inverse problem where an unknown system state needs to be reconstructed from observations with a suitable model for the flo…

This https://arxiv.org/abs/2403.01667 has been replaced.
link: https://scholar.google.com/scholar?q=a

Reaction-diffusion models of biological invasion: Open source computational tools, key concepts and analysis
This review provides open-access computational tools that support a range of mathematical approaches to analyse three related scalar reaction-diffusion models used to study biological invasion. Starting with the classic Fisher-Kolmogorov (Fisher-KPP) model, we illustrate how computational methods can be used to explore time-dependent partial differential equation (PDE) solutions in parallel with phase plane and regular perturbation techniques to explore invading travelling wave solutions moving…

A data-driven approach to PDE-constrained optimization in inverse problems
Tristan van Leeuwen, Yunan Yang
https://arxiv.org/abs/2403.15292 https://…

A data-driven approach to PDE-constrained optimization in inverse problems
Inverse problems are ubiquitous in science and engineering. Many of these are naturally formulated as a PDE-constrained optimization problem. These non-linear, large-scale, constrained optimization problems know many challenges, of which the inherent non-linearity of the problem is an important one. As an alternative to this physics-driven approach, data-driven methods have been proposed. These methods come with their own set of challenges, and it appears that, ideally, one would devise hybrid …

Designing Poisson Integrators Through Machine Learning
Miguel Vaquero, David Mart\'in de Diego, Jorge Cort\'es
https://arxiv.org/abs/2403.20139 htt…

Designing Poisson Integrators Through Machine Learning
This paper presents a general method to construct Poisson integrators, i.e., integrators that preserve the underlying Poisson geometry. We assume the Poisson manifold is integrable, meaning there is a known local symplectic groupoid for which the Poisson manifold serves as the set of units. Our constructions build upon the correspondence between Poisson diffeomorphisms and Lagrangian bisections, which allows us to reformulate the design of Poisson integrators as solutions to a certain PDE (Hami…

Colored Stochastic Multiplicative Processes with Additive Noise Unveil a Third-Order PDE, Defying Conventional FPE and Fick-Law Paradigms
Marco Bianucci, Mauro Bologna, Riccardo Mannella
https://arxiv.org/abs/2404.14229

Colored Stochastic Multiplicative Processes with Additive Noise Unveil a Third-Order PDE, Defying Conventional FPE and Fick-Law Paradigms
Research on stochastic differential equations (SDE) involving both additive and multiplicative noise has been extensive. In situations where the primary process is driven by a multiplicative stochastic process, additive white noise typically represents an intrinsic and unavoidable fast factor, including phenomena like thermal fluctuations, inherent uncertainties in measurement processes, or rapid wind forcing in ocean dynamics. This work focuses on a significant class of such systems, particula…

This https://arxiv.org/abs/2311.03091 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_mat…

Abstract dissipative Hamiltonian differential-algebraic equations are everywhere
In this paper we study the representation of partial differential equations (PDEs) as abstract differential-algebraic equations (DAEs) with dissipative Hamiltonian structure (adHDAEs). We show that these systems not only arise when there are constraints coming from the underlying physics, but many standard PDE models can be seen as an adHDAE on an extended state space. This reflects the fact that models often include closure relations and structural properties. We present a unifying operator th…

About semilinear low dimension Bessel PDEs
Alberto Ohashi (UnB), Francesco Russo (OC, ENSTA Paris), Alan Teixeira (ENSTA Paris)
https://arxiv.org/abs/2404.03243

About semilinear low dimension Bessel PDEs
We prove existence and uniqueness of solutions of a semilinear PDE driven by a Bessel type generator$L^δ$ with low dimension $0 < δ< 1$. $L^δ$ is a local operator, whose drift is thederivative of $x \mapsto \log (\vert x\vert)$:in particular it is a Schwartz distribution, whichis not the derivative of a continuous function.The solutions are intended in a duality (''weak'') sensewith respect to state space$L^2(\mathbb{R}_+, dμ),$ $μ$ being an invariant measure for the Bessel semigroup.

This https://arxiv.org/abs/2310.03201 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_mat…

A Hidden Convexity in Continuum Mechanics, with application to classical, continuous-time, rate-(in)dependent plasticity
A methodology for defining variational principles for a class of PDE models from continuum mechanics is demonstrated, and some of its features explored. The scheme is applied to quasi-static and dynamic models of rate-independent and rate-dependent, single crystal plasticity at finite deformation.

This https://arxiv.org/abs/2102.01432 has been replaced.
link: https://scholar.google.com/scholar?q=a

Bayesian data-driven discovery of partial differential equations with variable coefficients
The discovery of Partial Differential Equations (PDEs) is an essential task for applied science and engineering. However, data-driven discovery of PDEs is generally challenging, primarily stemming from the sensitivity of the discovered equation to noise and the complexities of model selection. In this work, we propose an advanced Bayesian sparse learning algorithm for PDE discovery with variable coefficients, predominantly when the coefficients are spatially or temporally dependent. Specificall…

Designing Poisson Integrators Through Machine Learning
Miguel Vaquero, David Mart\'in de Diego, Jorge Cort\'es
https://arxiv.org/abs/2403.20139 htt…

Designing Poisson Integrators Through Machine Learning
This paper presents a general method to construct Poisson integrators, i.e., integrators that preserve the underlying Poisson geometry. We assume the Poisson manifold is integrable, meaning there is a known local symplectic groupoid for which the Poisson manifold serves as the set of units. Our constructions build upon the correspondence between Poisson diffeomorphisms and Lagrangian bisections, which allows us to reformulate the design of Poisson integrators as solutions to a certain PDE (Hami…

Robust elastic full-waveform inversion using an alternating direction method of multipliers with reconstructed wavefields
Kamal Aghazade, Ali Gholami, Hossein S. Aghamiry, Hamid Reza Siahkoohi
https://arxiv.org/abs/2404.07927

Robust elastic full-waveform inversion using an alternating direction method of multipliers with reconstructed wavefields
Elastic full-waveform inversion (EFWI) is a process used to estimate subsurface properties by fitting seismic data while satisfying wave propagation physics. The problem is formulated as a least-squares data fitting minimization problem with two sets of constraints: Partial-differential equation (PDE) constraints governing elastic wave propagation and physical model constraints implementing prior information. The alternating direction method of multipliers is used to solve the problem, resultin…

Chemotaxis-inspired PDE model for airborne infectious disease transmission: analysis and simulations
Pierluigi Colli, Gabriela Marinoschi, Elisabetta Rocca, Alex Viguerie
https://arxiv.org/abs/2404.17506

Chemotaxis-inspired PDE model for airborne infectious disease transmission: analysis and simulations
Partial differential equation (PDE) models for infectious disease have received renewed interest in recent years. Most models of this type extend classical compartmental formulations with additional terms accounting for spatial dynamics, with Fickian diffusion being the most common such term. However, while diffusion may be appropriate for modeling vector-borne diseases, or diseases among plants or wildlife, the spatial propagation of airborne diseases in human populations is heavily dependent …

The heat equation with time-correlated random potential in d=2: Edwards-Wilkinson fluctuations
Sotirios Kotitsas
https://arxiv.org/abs/2405.01519 https://<…

The heat equation with time-correlated random potential in d=2: Edwards-Wilkinson fluctuations
We consider the stochastic PDE: $\partial_tu(t,x)=\frac{1}{2}Δu(t,x)+β{}u(t,x)V(t,x),$ in dimension $d=2$, where the potential V is the space and time mollification of the two-dimensional space-time white noise. We show that after renormalizing, the fluctuations of the solution converge to the Edwards-Wilkinson limit with an explicit effective variance and constant effective diffusivity. Our main tool is a Markov chain on the space of paths which we use to establish an extension of the Ka…

Perturbations in PDE-constrained optimal control decay exponentially in space
Simone G\"ottlich, Manuel Schaller, Karl Worthmann
https://arxiv.org/abs/2403.15056

Perturbations in PDE-constrained optimal control decay exponentially in space
For linear-quadratic optimal control problems (OCPs) governed by elliptic and parabolic partial differential equations (PDEs), we investigate the impact of perturbations on optimal solutions. Local perturbations may occur, e.g., due to discretization of the optimality system or disturbed problem data. Whereas these perturbations may exhibit global effects in the uncontrolled case, we prove that the ramifications are exponentially damped in space under stabilizability- and detectability-like con…

This https://arxiv.org/abs/2403.01667 has been replaced.
link: https://scholar.google.com/scholar?q=a

Reaction-diffusion models of biological invasion: Open source computational tools, key concepts and analysis
This review provides open-access computational tools that support a range of mathematical approaches to analyse three related scalar reaction-diffusion models used to study biological invasion. Starting with the classic Fisher-Kolmogorov (Fisher-KPP) model, we illustrate how computational methods can be used to explore time-dependent partial differential equation (PDE) solutions in parallel with phase plane and regular perturbation techniques to explore invading travelling wave solutions moving…

A New Hybrid Automaton Framework with Partial Differential Equation Dynamics
Tianshu Bao, Hengrong Du, Weiming Xiang, Taylor T. Johnson
https://arxiv.org/abs/2404.11900

A New Hybrid Automaton Framework with Partial Differential Equation Dynamics
This paper presents the syntax and semantics of a novel type of hybrid automaton (HA) with partial differential equation (PDE) dynamic, partial differential hybrid automata (PDHA). In PDHA, we add a spatial domain $X$ and harness a mathematic conception, partition, to help us formally define the spatial relations. While classically the dynamics of HA are described by ordinary differential equations (ODEs) and differential inclusions, PDHA is capable of describing the behavior of cyber-physical …

This https://arxiv.org/abs/2304.01294 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_mat…

Sparse Cholesky Factorization for Solving Nonlinear PDEs via Gaussian Processes
In recent years, there has been widespread adoption of machine learning-based approaches to automate the solving of partial differential equations (PDEs). Among these approaches, Gaussian processes (GPs) and kernel methods have garnered considerable interest due to their flexibility, robust theoretical guarantees, and close ties to traditional methods. They can transform the solving of general nonlinear PDEs into solving quadratic optimization problems with nonlinear, PDE-induced constraints. H…

Dynamics of Cayley Forms
Kirill Krasnov
https://arxiv.org/abs/2403.16661 https://arxiv.org/pdf/2403.16661

Dynamics of Cayley Forms
The most natural first-order PDE's to be imposed on a Cayley 4-form in eight dimensions is the condition that it is closed. As is known, this implies integrability of the Spin(7) structure defined by the Cayley form, as well as Ricci-flatness of the associated metric. We address the question as to what the most natural second-order in derivatives set of conditions is. We start by constructing the most general diffeomorphism invariant second order in derivatives Lagrangian that is quadratic in t…

This https://arxiv.org/abs/2306.05305 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_mat…

A Stochastic Analysis Approach to Tensor Field Theories
We present two different arguments using stochastic analysis to construct super-renormalizable tensor field theories, namely the $\mathrm{T}^4_3$ and $\mathrm{T}^4_4$ models. The first approach is the construction of a Langevin dynamic combined with a PDE energy estimate while the second is an application of the variational approach of Barashkov and Gubinelli. By leveraging the melonic structure of divergences, regularising properties of non-local products, and controlling certain random operat…

This https://arxiv.org/abs/2308.10430 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_mat…

Modeling of electronic dynamics in twisted bilayer graphene
We consider the problem of numerically computing the quantum dynamics of an electron in twisted bilayer graphene. The challenge is that atomic-scale models of the dynamics are aperiodic for generic twist angles because of the incommensurability of the layers. The Bistritzer-MacDonald PDE model, which is periodic with respect to the bilayer's moiré pattern, has recently been shown to rigorously describe these dynamics in a parameter regime. In this work, we first prove that the dynamics of the …

This https://arxiv.org/abs/2402.03133 has been replaced.
link: https://scholar.google.com/scholar?q=a

Revivals, or the Talbot effect, for the Airy equation
We study Dirichlet-type problems for the simplest third-order linear dispersive PDE, often referred to as the Airy equation. Such problems have not been extensively studied, perhaps due to the complexity of the spectral structure of the spatial operator. Our specific interest is to determine whether the peculiar phenomenon of revivals, also known as Talbot effect, is supported by these boundary conditions, which for third order problems are not reducible to periodic ones. We prove that this is …

This https://arxiv.org/abs/2311.00529 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_mat…

A Unified Framework for the Error Analysis of Physics-Informed Neural Networks
We prove a priori and a posteriori error estimates for physics-informed neural networks (PINNs) for linear PDEs. We analyze elliptic equations in primal and mixed form, elasticity, parabolic, hyperbolic and Stokes equations; and a PDE constrained optimization problem. For the analysis, we propose an abstract framework in the common language of bilinear forms, and we show that coercivity and continuity lead to error estimates. The obtained estimates are sharp and reveal that the $L^2$ penalty ap…

This https://arxiv.org/abs/2401.02393 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_mat…

A PDE approach for solving the characteristic function of the generalised signature process
The signature of a path, as a fundamental object in Rough path theory, serves as a generating function for non-commutative monomials on path space. It transforms the path into a grouplike element in the tensor algebra space, summarising the path faithfully up to a generalised form of re-parameterisation (a negligible equivalence class in this context). Our paper concerns stochastic processes and studies the characteristic function of the path signature of the stochastic process. In contrast to …

This https://arxiv.org/abs/2404.09323 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_mat…

Incremental data compression for PDE-constrained optimization with a data assimilation application
This paper proposes and analyzes an inexact gradient method based on incremental proper orthogonal decomposition (iPOD) to address the data storage difficulty in time-dependent PDE-constrained optimization, particularly for a data assimilation problem as a detailed demonstration for the key ideas. The proposed method is proved robust by rigorous analysis. We first derive a sharp data compression error estimate of the iPOD with the help of Hilbert-Schmidt operators. Then we demonstrate a numeric…

Efficient numerical computation of spiral spectra with exponentially-weighted preconditioners
Stephanie Dodson, Ryan Goh, Bjorn Sandstede
https://arxiv.org/abs/2405.05897

Efficient numerical computation of spiral spectra with exponentially-weighted preconditioners
The stability of nonlinear waves on spatially extended domains is commonly probed by computing the spectrum of the linearization of the underlying PDE about the wave profile. It is known that convective transport, whether driven by the nonlinear pattern itself or an underlying fluid flow, can cause exponential growth of the resolvent of the linearization as a function of the domain length. In particular, sparse eigenvalue algorithms may result in inaccurate and spurious spectra in the convectiv…

On a mathematical model for tissue regeneration
Shimi Chettiparambil Mohanan, Nishith Mohan, Christina Surulescu
https://arxiv.org/abs/2403.04516 https://<…

On a mathematical model for tissue regeneration
We propose a PDE-ODE model for tissue regeneration, obtained by parabolic upscaling from kinetic transport equations written for the mesoscopic densities of mesenchymal stem cells and chondrocytes which evolve in an artificial scaffold impregnated with hyaluron. Due to the simple chosen turning kernels, the effective equations obtained on the macroscopic level are of the usual reaction-diffusion-taxis type. We prove global existence of solutions to the coupled macroscopic system and perform a s…

Efficient numerical computation of spiral spectra with exponentially-weighted preconditioners
Stephanie Dodson, Ryan Goh, Bjorn Sandstede
https://arxiv.org/abs/2405.05897

Efficient numerical computation of spiral spectra with exponentially-weighted preconditioners
The stability of nonlinear waves on spatially extended domains is commonly probed by computing the spectrum of the linearization of the underlying PDE about the wave profile. It is known that convective transport, whether driven by the nonlinear pattern itself or an underlying fluid flow, can cause exponential growth of the resolvent of the linearization as a function of the domain length. In particular, sparse eigenvalue algorithms may result in inaccurate and spurious spectra in the convectiv…

This https://arxiv.org/abs/2208.02767 has been replaced.
link: https://scholar.google.com/scholar?q=a

Parabolic PDE-constrained optimal control under uncertainty with entropic risk measure using quasi-Monte Carlo integration
We study the application of a tailored quasi-Monte Carlo (QMC) method to a class of optimal control problems subject to parabolic partial differential equation (PDE) constraints under uncertainty: the state in our setting is the solution of a parabolic PDE with a random thermal diffusion coefficient, steered by a control function. To account for the presence of uncertainty in the optimal control problem, the objective function is composed with a risk measure. We focus on two risk measures, both…

This https://arxiv.org/abs/2212.04466 has been replaced.
link: https://scholar.google.com/scholar?q=a

On inclusion of time-varying source in the acoustic wave equation
Acoustic wave equation is a partial differential equation (PDE) which describes propagation of acoustic waves through a material. In general, the solution to this PDE is nonunique. Therefore, it is necessary to impose initial conditions in the form of Cauchy conditions for obtaining a unique solution. Theoretically, solving the wave equation is equivalent to representing the wavefield in terms of a radiation source which possesses finite energy over space and time.The radiation source is repres…

Globally Convergent Distributed Sequential Quadratic Programming with Overlapping Decomposition and Exact Augmented Lagrangian Merit Function
Runxin Ni, Sen Na, Sungho Shin, Mihai Anitescu
https://arxiv.org/abs/2402.17170

Globally Convergent Distributed Sequential Quadratic Programming with Overlapping Decomposition and Exact Augmented Lagrangian Merit Function
In this paper, we address the problem of solving large-scale graph-structured nonlinear programs (gsNLPs) in a scalable manner. GsNLPs are problems in which the objective and constraint functions are associated with a graph node, and they depend only on the variables of adjacent nodes. This graph-structured formulation encompasses various specific instances, such as dynamic optimization, PDE-constrained optimization, multi-stage stochastic optimization, and optimization over networks. We propos…

Concentration-compactness via profile decomposition for systems of coupled Schr\"{o}dinger equations of Hamiltonian type
Anderson Cardoso, Jo\~ao Marcos do \'O, Diego Ferraz
https://arxiv.org/abs/2403.03195

Concentration-compactness via profile decomposition for systems of coupled Schrödinger equations of Hamiltonian type
We analyse Hamiltonian-type systems of second-order elliptic PDE invariant under a non-compact group and, consequently, involve a lack of compactness of the Sobolev embedding. We show that the loss of compactness can be compensated by using a concentration-compactness principle via weak profile decomposition for bounded Palais-Smale sequences in Banach spaces. Our analysis to prove the existence of ground states involves a reduction by the inversion method of the system to a fourth-order equati…

Stabilization of a Class of Large-Scale Systems of Linear Hyperbolic PDEs via Continuum Approximation of Exact Backstepping Kernels
Jukka-Pekka Humaloja, Nikolaos Bekiaris-Liberis
https://arxiv.org/abs/2403.19455

Stabilization of a Class of Large-Scale Systems of Linear Hyperbolic PDEs via Continuum Approximation of Exact Backstepping Kernels
We establish that stabilization of a class of linear, hyperbolic partial differential equations (PDEs) with a large (nevertheless finite) number of components, can be achieved via employment of a backstepping-based control law, which is constructed for stabilization of a continuum version (i.e., as the number of components tends to infinity) of the PDE system. This is achieved by proving that the exact backstepping kernels, constructed for stabilization of the large-scale system, can be approxi…

Parameter identification in PDEs by the solution of monotone inclusion problems
Pankaj Gautam, Markus Grasmair
https://arxiv.org/abs/2403.04557 https://

Parameter identification in PDEs by the solution of monotone inclusion problems
In this paper we consider a parameter identification problem for a semilinear parabolic PDE. For the regularized solution of this problem, we introduce a total variation based regularization method requiring the solution of a monotone inclusion problem. We show well-posedness in the sense of inverse problems of the resulting regularization scheme. In addition, we introduce and analyze a numerical algorithm for the solution of this inclusion problem using a nested inertial primal dual method. We…

A Mathematical Framework for Spatio-Temporal Control in Industrial Drying
Lennon \'O N\'araigh
https://arxiv.org/abs/2404.16604 https://

A Mathematical Framework for Spatio-Temporal Control in Industrial Drying
We introduce two models of industrial drying - a simplified one-equation model, and a detailed three-equation model. The purpose of the simplified model is rigorous validation of numerical methods for PDE-constrained optimal control. The purpose of the detailed model is to be able to predict and control the behaviour of an industrial disk drier. For both models, we introduce a fully validated numerical method to compute the optimal source term to maintain the outlet temperature as close as poss…

Stationary non-radial localized patterns in the planar Swift-Hohenberg PDE: constructive proofs of existence
Matthieu Cadiot, Jean-Philippe Lessard, Jean-Christophe Nave
https://arxiv.org/abs/2403.10450

Stationary non-radial localized patterns in the planar Swift-Hohenberg PDE: constructive proofs of existence
In this paper, we present a methodology for establishing constructive proofs of existence of smooth, stationary, non-radial localized patterns in the planar Swift-Hohenberg equation. Specifically, given an approximate solution $u_0$, we construct an approximate inverse for the linearization around $u_0$, enabling the development of a Newton-Kantorovich approach. Consequently, we derive a sufficient condition for the existence of a unique localized pattern in the vicinity of $u_0$. The verificat…

This https://arxiv.org/abs/2302.05742 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_mat…

A single player and a mass of agents: a pursuit evasion-like game
We study a finite-horizon differential game of pursuit-evasion like, between a single player and a mass of agents. The player and the mass directly control their own evolution, which for the mass is given by a first order PDE of transport equation type. Using also an adapted concept of non-anticipating strategies, we derive an infinite dimensional Isaacs equation, and by dynamic programming techniques we prove that the value function is the unique viscosity solution on a suitable invariant subs…

Density estimation for elliptic PDE with random input by preintegration and quasi-Monte Carlo methods
Alexander D. Gilbert, Frances Y. Kuo, Abirami Srikumar
https://arxiv.org/abs/2402.11807

Density estimation for elliptic PDE with random input by preintegration and quasi-Monte Carlo methods
In this paper, we apply quasi-Monte Carlo (QMC) methods with an initial preintegration step to estimate cumulative distribution functions and probability density functions in uncertainty quantification (UQ). The distribution and density functions correspond to a quantity of interest involving the solution to an elliptic partial differential equation (PDE) with a lognormally distributed coefficient and a normally distributed source term. There is extensive previous work on using QMC to compute e…

This https://arxiv.org/abs/2403.19123 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_mat…

Schrödingerisation based computationally stable algorithms for ill-posed problems in partial differential equations
We introduce a simple and stable computational method for ill-posed partial differential equation (PDE) problems. The method is based on Schrödingerisation, introduced in [S. Jin, N. Liu and Y. Yu, Phys. Rev. A, 108 (2023), 032603], which maps all linear PDEs into Schrödinger-type equations in one higher dimension, for quantum simulations of these PDEs. Although the original problem is ill-posed, the Schrödingerized equations are Hamiltonian systems and time-reversible, allowing stable compu…

This https://arxiv.org/abs/2308.16910 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_mat…

Robust Variational Physics-Informed Neural Networks
We introduce a Robust version of the Variational Physics-Informed Neural Networks method (RVPINNs). As in VPINNs, we define the quadratic loss functional in terms of a Petrov-Galerkin-type variational formulation of the PDE problem: the trial space is a (Deep) Neural Network (DNN) manifold, while the test space is a finite-dimensional vector space. Whereas the VPINN's loss depends upon the selected basis functions of a given test space, herein, we minimize a loss based on the discrete dual norm…

A closure theorem for $\Gamma$-convergence and H-convergence with applications to non-periodic homogenization
Andrea Braides, Gianni Dal Maso, Claude Le Bris
https://arxiv.org/abs/2402.19031

A closure theorem for $Γ$-convergence and H-convergence with applications to non-periodic homogenization
In this work we examine the stability of some classes of integrals, and in particular with respect to homogenization. The prototypical case is the homogenization of quadratic energies with periodic coefficients perturbed by a term vanishing at infinity, which has been recently examined in the framework of elliptic PDE. We use localization techniques and higher-integrability Meyers-type results to provide a closure theorem by $Γ$-convergence within a large class of integral functionals. From su…

Quantitative homogenization for log-normal coefficients via Malliavin calculus: the one-dimensional case
Antoine Gloria, Siguang Qi
https://arxiv.org/abs/2402.19182

Quantitative homogenization for log-normal coefficients via Malliavin calculus: the one-dimensional case
The quantitative analysis of stochastic homogenization problems has been a very active field in the last fifteen years. Whereas the first results were motivated by applied questions (namely, the numerical approximation of homogenized coefficients), the more recent achievements in the field are much more analytically-driven and focus on the subtle interplay between PDE analysis (and in particular elliptic regularity theory) and probability (concentration, stochastic cancellations, scaling limits…

A fast cosine transformation accelerated method for predicting effective thermal conductivity
Changqing Ye, Shubin Fu, Eric T. Chung
https://arxiv.org/abs/2404.02433

A fast cosine transformation accelerated method for predicting effective thermal conductivity
Predicting effective thermal conductivity by solving a Partial Differential Equation (PDE) defined on a high-resolution Representative Volume Element (RVE) is a computationally intensive task. In this paper, we tackle the task by proposing an efficient and implementation-friendly computational method that can fully leverage the computing power offered by hardware accelerators, namely, graphical processing units (GPUs). We first employ the Two-Point Flux-Approximation scheme to discretize the PD…

Infinite dimensional Slow Manifolds for a Linear Fast-Reaction System
Christian Kuehn, Pascal Lehner, Jan-Eric Sulzbach
https://arxiv.org/abs/2404.17220 ht…

Infinite dimensional Slow Manifolds for a Linear Fast-Reaction System
The aim of this expository paper is twofold. We start with a concise overview of the theory of invariant slow manifolds for fast-slow dynamical systems starting with the work by Tikhonov and Fenichel to the most recent works on infinite-dimensional fast-slow systems. The main part focuses on a class of linear fast-reaction PDE, which are particular forms of fast-reaction systems. The first result shows the convergence of solutions of the linear system to the limit system as the time-scale param…

This https://arxiv.org/abs/2306.13174 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_mat…

Finite element approximation of time-dependent mean field games with nondifferentiable Hamiltonians
The standard formulation of the PDE system of Mean Field Games (MFG) requires the differentiability of the Hamiltonian. However in many cases, the structure of the underlying optimal problem leads to a convex but nondifferentiable Hamiltonian. For time-dependent MFG systems, we introduce a generalization of the problem as a Partial Differential Inclusion (PDI) by interpreting the derivative of the Hamiltonian in terms of the subdifferential set. In particular, we prove the existence and uniquen…

A multi-scale multi-lane model for traffic regulation via autonomous vehicles
Paola Goatin (ACUMES, UniCA, LJAD), Benedetto Piccoli
https://arxiv.org/abs/2404.17192

A multi-scale multi-lane model for traffic regulation via autonomous vehicles
We propose a new model for multi-lane traffic with moving bottlenecks, e.g., autonomous vehicles (AV). It consists of a system of balance laws for traffic in each lane, coupled in the source terms for lane changing, and fully coupled to ODEs for the AVs' trajectories.More precisely, each AV solves a controlled equation depending on the traffic density, while the PDE on the corresponding lane has a flux constraint at the AV's location. We prove existence of entropy weak solutions, and we charac…

On the viscosity linearization method without compactness
Anthony Salib
https://arxiv.org/abs/2403.18860 https://arxiv.org/pdf/2403.1…

On the viscosity linearization method without compactness
Savin's small perturbation approach has had far reaching applications in the theory of non-linear elliptic and parabolic PDE. In this short note, we revisit his seminal proof of De-Giorgi's improvement of flatness theorem for minimal surfaces and provide an approach based on the Harnack inequality that avoids the use of compactness arguments.

PDEformer: Towards a Foundation Model for One-Dimensional Partial Differential Equations
Zhanhong Ye, Xiang Huang, Leheng Chen, Hongsheng Liu, Zidong Wang, Bin Dong
https://arxiv.org/abs/2402.12652

PDEformer: Towards a Foundation Model for One-Dimensional Partial Differential Equations
This paper introduces PDEformer, a neural solver for partial differential equations (PDEs) capable of simultaneously addressing various types of PDEs. We advocate representing the PDE in the form of a computational graph, facilitating the seamless integration of both symbolic and numerical information inherent in a PDE. A graph Transformer and an implicit neural representation (INR) are employed to generate mesh-free predicted solutions. Following pretraining on data exhibiting a certain level …

Sixth-order parabolic equation on an interval: Eigenfunction expansion, Green's function, and intermediate asymptotics for a finite thin film with elastic resistance
Nectarios C. Papanicolaou, Ivan C. Christov
https://arxiv.org/abs/2402.18740

Sixth-order parabolic equation on an interval: Eigenfunction expansion, Green's function, and intermediate asymptotics for a finite thin film with elastic resistance
A linear sixth-order partial differential equation (PDE) of ``parabolic'' type describes the dynamics of thin liquid films beneath surfaces with elastic bending resistance when deflections from the equilibrium film height are small. On a finite domain, the associated sixth-order Sturm--Liouville eigenvalue value problem is self-adjoint for the boundary conditions corresponding to a thin film in a closed trough, and the eigenfunctions form a complete orthonormal set. Using these eigenfunctions, …