Nonlocal Nonlinear Diffusion: Variational Principles, Optimal Control, and Approximation

Schytt, Marcus Johan

4. term

Mathematics, Master

2023

2023-06-02

97

This thesis studies a nonlocal, nonlinear law for heat diffusion, inspired by peridynamics—a nonlocal reformulation of classical mechanics. The aim is to build a rigorous framework for steady (equilibrium) temperature states using the Dirichlet principle that systems minimize energy. We introduce nonlocal counterparts of the gradient and divergence to write the balance laws, and we prove that equilibrium states exist and are unique. They are characterized as weak solutions (generalized solutions) of a nonlocal p-Laplacian, a nonlinear diffusion operator. We also derive dual formulations based on Kelvin’s principle of minimum complementary energy, and show that these are well posed using the Ladyzhenskaya–Babuška–Brezzi condition and Fenchel–Rockafellar duality from convex analysis. A key link to the classical theory is established by proving that, as the interaction range (the nonlocal "horizon") shrinks to zero, nonlocal equilibria converge to their local counterparts, drawing on results by Bourgain–Brezis–Mironescu and Ponce. The scope then narrows to linear diffusion to study two application types: nonlocal optimal control (choosing inputs to achieve desired temperature profiles) and obstacle problems (solutions constrained by barriers). For linear–quadratic distributed control, the analysis closely mirrors the local case. For nonlinear control of conductivity, which is ill posed in the local setting, the nonlocal formulation yields practical solutions without extra regularization. We also analyze nonlocal obstacle problems under mild assumptions and argue that nonlocal models better accommodate discontinuous conductivities and obstacles, which are challenging for local models. Finally, we extend the finite element method to approximate nonlocal equilibrium states. Although this increases computational cost, we prove convergence of the numerical approximations and support the theory with numerical experiments. The nonlocal finite element approach is also applied to the optimal control and obstacle problems.

Afhandlingen undersøger en ikke-lokal, ikke-lineær lov for varmediffusion, inspireret af peridynamik—en ikke-lokal omformulering af klassisk mekanik. Målet er at opbygge en stringent ramme for stationære (ligevægts) temperaturtilstande via Dirichlets princip om minimal energi. Vi indfører ikke-lokale analoger til gradient og divergens til at formulere balanceligninger og beviser, at ligevægtstilstande findes og er entydige. De karakteriseres som svage løsninger (generaliserede løsninger) til en ikke-lokal p-Laplace-operator, en ikke-lineær diffusionsoperator. Vi udleder også duale formuleringer med udgangspunkt i Kelvins princip om minimal komplementær energi og viser, at de er velstillede ved hjælp af Ladyzhenskaya–Babuška–Brezzi-betingelsen og Fenchel–Rockafellar-dualitet fra konveks analyse. Et vigtigt bindeled til den klassiske teori er, at når interaktionshorisonten (den ikke-lokale "horisont") går mod nul, konvergerer de ikke-lokale ligevægte mod de lokale, støttet af resultater fra Bourgain–Brezis–Mironescu og Ponce. Dernæst indsnævrer vi fokus til lineær diffusion for at behandle to anvendelsestyper: ikke-lokal optimal styring (at vælge påvirkninger for at opnå ønskede temperaturprofiler) og hindringsproblemer (løsninger underlagt barrierer). For lineær-kvadratisk distribueret styring ligner analysen den lokale. For ikke-lineær styring af ledningsevne, som er dårligt stillet i den lokale ramme, giver den ikke-lokale formulering praktiske løsninger uden ekstra regularisering. Vi analyserer også ikke-lokale hindringsproblemer under milde antagelser og argumenterer for, at ikke-lokale modeller bedre håndterer diskontinuerlige ledningsevner og forhindringer, som er udfordrende for lokale modeller. Endelig udvider vi metoden med endelige elementer til at approksimere ikke-lokale ligevægtstilstande. Selvom det øger beregningsomkostningerne, beviser vi konvergens af de numeriske approksimationer og underbygger teorien med numeriske eksperimenter. Den ikke-lokale metode med endelige elementer anvendes også på styrings- og hindringsproblemerne.

[This apstract has been rewritten with the help of AI based on the project's original abstract]

Nonlocal modelling ; Diffusion ; PDEs ; Analysis of PDEs ; Nonlocal boundary value problems ; Nonlocal diffusion ; Optimal control ; Control in the coefficients ; Compliance minimization ; Nonlocal distributed control ; Fractional p-Laplacian ; p-Laplacian ; Nonlocal FEM ; Finite elements ; Finite element methods ; Nonlocality

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