18 papers · ranked by Valyu relevance
Antika Yadav, Prasad Vilas Chanekar
— In this paper we study the control co-design (CCD) synthesis problem for a class of systems with parabolic partial differential equation (PDE) dynamics. We formulate CCD problem and finally derive an approximate CCD problem with matrix algebraic constraint. We then solve this approximate problem with gradient-based…
Verena Bögelein, Frank Duzaar, Giulia Treu
We establish the existence of Lipschitz-continuous solutions to the Cauchy-Dirichlet problem for a class of evolutionary partial differential equations of the form \begin{aligned}\partial _tu-{{\,\textrm{div}\,}}_x \nabla _\xi f\nabla u=0 \end{aligned} in a space-time cylinder $\Omega _T=\Omega \times 0,T$, subject to…
Wenzhong Zhang, Zhenyuan Hu, Wei Cai, George EM Karniadakis
The numerical solution of high-dimensional partial differential equations (PDEs) is severely constrained by the curse of dimensionality (CoD), rendering classical grid-based methods impractical beyond a few dimensions. In recent years, deep neural networks have emerged as a promising mesh-free alternative, enabling the…
Alberto Cialdea, Carmine Sebastiano Mare
Let {vα} be a system of polynomial solutions of the parabolic equation ahk∂xhx k u−∂tu = 0 in a bounded C 1 -cylinder Ω T contained in R n+1. Here ahk∂xhx k is an elliptic operator with real constant coefficients. We prove that {vα} is complete in L p (Σ′ ), where Σ ′ is the parabolic boundary of Ω T . Similar results…
Antika Yadav, Prasad Vilas Chanekar
In this paper, we study the control co-design (CCD) synthesis problem for a class of systems with parabolic partial differential equation (PDE) dynamics. We first derive a sufficient stability condition for the PDE. By spatially discretizing the PDE and using the sufficient stability condition, we propose a…
Wenrui Hao, Lili Ju, Yuejin Xu
In this paper, we study the convergence behavior of the diffuse domain method (DDM) for solving a class of second-order parabolic partial differential equations with Neumann boundary condition posed on general irregular domains. The DDM employs a phase-field function to extend the original parabolic problem to a…
Martin Dindoš, Linhan Li, Jill Pipher
In this paper, we resolve the question of whether the Neumann problem for the parabolic PDE $-\partial_tu + \mathrm{div}(A\nabla u)=0$ on a Lipschitz cylinder $\mathcal O\times\mathbb R$ is solvable for some $p\in (1,\infty)$ under the assumption that the matrix $A$ is elliptic with bounded and measurable coefficients…
Amir Hossein Salehi Shayegan
In this work, we present a solution to the critical limitation of qubit capacity in near-term quantum hardware by giving a hybrid framework that integrates the spectral element method (SEM) with distributed quantum computing. Using domain decomposition techniques, the additive and multiplicative Schwarz methods, the…
Le Trong Thanh Bui, Thi Kim Loan Huynh, Bao Quoc Tang, Bao-Ngoc Tran
Singular limits for the following indirect signalling chemotaxis system \begin{aligned}\left{\begin{array}{lllllll}\partial _t n = \Delta n - \nabla \cdot n \nabla c & \text{in}\Omega \times 0,\infty , \ \varepsilon \partial _t c = \Delta c - c + w & \text{in}\Omega \times 0,\infty , \ \varepsilon \partial _t w = \tau…
Klaus Deckelnick, Hans-Christoph Grunau, Robert Nürnberg, Glen Wheeler + 1 more
The free boundary free elastic flow is the steepest descent gradient flow for the elastic energy of curves meeting parallel lines perpendicularly. In this article we prove that the straight line has, measured in Euler's scale-invariant bending energy, a basin of attraction at least to the level 1.9615 π. We show that…
Ray Zirui Zhang, Christopher E. Miles, Xiaohui Xie, John S. Lowengrub
We propose a new neural network based method for solving inverse problems for partial differential equations (PDEs) by formulating the PDE inverse problem as a bilevel optimization problem. At the upper level, we minimize the data loss with respect to the PDE parameters. At the lower level, we train a neural network to…
Nikos I. Kavallaris, Farrukh Javed
We introduce a mechanistic, nonlocal tumour-growth model designed specifically to capture explosive dynamics that are not adequately explained by standard logistic reaction–diffusion descriptions. The motivation is empirical: the universal scaling law reported in [1] provides compelling cross-sectional evidence of…
Andreas Chatziafratis, Sergey A. Rukolaine, Elias C. Aifantis
We obtain novel integral representations, expressed as contour integrals in the complex Fourier plane, for the solution of fully nonhomogeneous initial-boundary-value as well as interface problems for the Barenblatt-Zheltov-Kochina pseudo-parabolic equation of Sobolev-Galpern type formulated on the real line, half-line…
Ray Zirui Zhang, Christopher E. Miles, Xiaohui Xie, John S. Lowengrub
Uncertainty quantification in PDE inverse problems is essential in many applications. Scientific machine learning and AI enable data-driven learning of model components while preserving physical structure, and provide the scalability and adaptability needed for emerging imaging technologies and clinical insights. We…
Vanja Nikolić, Teresa Rauscher
Harmonic generation plays a crucial role in contrast-enhanced ultrasound, both for imaging and therapeutic applications. However, accurately capturing these nonlinear effects is computationally demanding when using traditional time-domain approaches. To address this issue, we develop algorithms based on a time…
Antonio Agresti, Max Sauerbrey
We consider strictly positive solutions to a class of fourth-order conservative quasilinear SPDEs on the d-dimensional torus modeled after the stochastic thin-film equation. We prove local Lipschitz estimates in Bessel potential spaces under minimal assumptions on the parameters and corresponding stochastic maximal…
Authors not listed
The quantum theory of coupled ion-electron transfer (CIET) unifies phenomenological Butler-Volmer kinetics with the Marcus theory of electron transfer in a single, thermodynamically consistent modeling framework for Faradaic reaction rates. Here, we extend CIET theory to explicitly incorporate the electronic properties…
Jie Deng, Xinyu Zhang, Xuchang Zhang, Xing Yang
Coupled diffusion–reaction partial differential equations (PDEs) describe biochemical network dynamics but are difficult to solve for realistic multi-species systems without combining mechanism and data. We present a multi-stage physics-informed neural network (PINN) for multi-species diffusion–reaction PDEs and apply…