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Low-regularity and high oscillations: numerical analysis and computation of dispersive evolution equations

Descripción del proyecto

Baja regularidad y altas oscilaciones: cálculo de ecuaciones dispersivas (LAHACODE)

Las ecuaciones diferenciales parciales (EDP) desempeñan un papel esencial en las matemáticas, donde nos permiten describir fenómenos físicos que van desde los átomos ultrafríos a la materia ultracaliente y desde los algoritmos de aprendizaje a los fluidos del cerebro humano. Para comprender la naturaleza, necesitamos conocer su comportamiento cualitativo y calcular su aproximación numérica de forma fiable. Si bien los problemas lineales con soluciones suaves son bien conocidos en la actualidad, la descripción fiable de fenómenos «no suaves» sigue siendo un problema complicado y abierto. El objetivo general del proyecto LAHACODE, financiado con fondos del CEI, es avanzar de forma decisiva para solucionar esta carencia incluyendo profundamente la estructura subyacente de las resonancias en la discretización numérica. Esto nos permitirá vincular la discretización dimensional finita con potentes resultados de existencia para EDP no lineales con poca regularidad.

Objetivo

Partial differential equations (PDEs) play a central role in mathematics, allowing us to describe physical phenomena ranging from ultra-cold atoms (Bose–Einstein condensation) up to ultra-hot matter (nuclear fusion), from learning algorithms to fluids in the human brain. To understand nature we have to understand their qualitative behavior: existence and long time behavior of solutions, their geometric and dynamical properties – as well as to compute reliably their numerical solution. While linear problems and smooth solutions are nowadays well understood, a reliable description of ‘non-smooth’ phenomena remains one of the most challenging open problems in computational mathematics since the underlying PDEs have very complicated solutions exhibiting high oscillations and loss of regularity. This leads to huge errors, massive computational costs and ultimately provokes the failure of classical schemes. Nevertheless, ‘non-smooth phenomena’ play a fundamental role in modern physical modeling (e.g. blow-up phenomena, turbulences, high frequencies, low dispersion limits, etc.) which makes it an essential task to develop suitable numerical schemes. The overall ambition of LAHACODE is to make a crucial step towards closing this gap – addressing the fundamental question: How and to what extent can we reproduce the qualitative behavior of differential equations in a finite (discretized) world? LAHACODE is situated at the challenging frontiers of analysis and numerics. The main objective is to develop a novel class of numerical schemes for nonlinear PDEs with strong geometric structure at low regularity and high oscillations. The key idea in the construction of the new schemes is to tackle and deeply embed the underlying structure of resonances in the numerical discretizations. As in the continuous case, these terms are central to structure preservation, and provide the new schemes with remarkable properties – allowing reliable approximations where classical schemes fail.

Régimen de financiación

ERC-STG - Starting Grant

Institución de acogida

SORBONNE UNIVERSITE
Aportación neta de la UEn
€ 1 499 905,00
Dirección
21 RUE DE L'ECOLE DE MEDECINE
75006 Paris
Francia

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Región
Ile-de-France Ile-de-France Paris
Tipo de actividad
Higher or Secondary Education Establishments
Enlaces
Coste total
€ 1 499 905,00

Beneficiarios (2)