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First- and second-order optimality conditions in the sparse optimal control of Cahn--Hilliard systems
Full text linkThis paper deals with the sparse distributed control of viscous and nonviscous Cahn--Hilliard systems. We report on results concerning first-order necessary and second-order sufficient optimality conditions that have recently established by the authors. The analysis covers both the cases when the nonlinear double well potential governing the evolution is of either regular or logarithmic type. A major difficulty originates from the sparsity-enhancing term in the cost functional which typically is nondifferentiable
Hierarchical proximal Galerkin: A fast hp-FEM solver for variational problems with pointwise inequality constraints
Full text linkWe leverage the proximal Galerkin algorithm (Keith and Surowiec, Foundations of Computational Mathematics, 2024), a recently introduced mesh-independent algorithm, to obtain a high-order finite element solver for variational problems with pointwise inequality constraints. This is achieved by discretizing the saddle point systems, arising from the latent variable proximal point method, with the hierarchical p-finite element basis. This results in discretized sparse Newton systems that admit a simple and effective block preconditioner. The solver can handle both obstacle-type and gradient-type constraints. We apply the resulting algorithm to solve obstacle problems with hp-adaptivity, a gradient-type constrained problem, and the thermoforming problem, an example of an obstacle-type quasi-variational inequality. We observe hp-robustness in the number of Newton iterations and only mild growth in the number of inner Krylov iterations to solve the Newton systems. Crucially we also provide wall-clock timings that are faster than low-order discretization counterparts
Shape of polystyrene droplets on soft PDMS: Exploring the gap between theory and experiment at the three-phase contact line
Full text linkThe shapes of liquid polystyrene (PS) droplets on viscoelastic polydimethylsiloxane (PDMS) substrates are investigated experimentally using atomic force microscopy for a range of droplet sizes and substrate elasticities. These shapes, which comprise the PS-air, PS-PDMS, and PDMS-air interfaces as well as the three-phase contact line, are compared to theoretical predictions using axisymmetric sharp-interface models derived through energy minimization. We find that the polystyrene droplets are cloaked by a thin layer of uncrosslinked molecules migrating from the PDMS substrate. By incorporating the effects of cloaking into the surface energies in our theoretical model, we show that the global features of the experimental droplet shapes are in excellent quantitative agreement for all droplet sizes and substrate elasticities. However, our comparisons also reveal systematic discrepancies between the experimental results and the theoretical predictions in the vicinity of the three-phase contact line. Moreover, the relative importance of these discrepancies systematically increases for softer substrates and smaller droplets. We demonstrate that global variations in system parameters, such as surface tension and elastic shear moduli, cannot explain these differences but instead point to a locally larger elastocapillary length, whose possible origin is discussed thoroughly
Hyperbolic relaxation of the chemical potential in the viscous Cahn--Hilliard equation
No full textIn this paper, we study a hyperbolic relaxation of the viscous Cahn--Hilliard system with zero Neumann boundary conditions. In fact, we consider a relaxation term involving the second time derivative of the chemical potential in the first equation of the system. We develop a well-posedness, continuous dependence and regularity theory for the initial-boundary value problem. Moreover, we investigate the asymptotic behavior of the system as the relaxation parameter tends to 0 and prove the convergence to the viscous Cahn--Hilliard system
Multi-scale hybrid band simulation of (Al,Ga)N UV-C light emitting diodes
No full textAluminium gallium nitride alloys are used for developing light emitting diodes operating in the UV part of the electromagnetic spectrum. These devices suffer from a low efficiency. To gain insight to this question we develop a 3-D modified drift-diffusion model which takes into account both alloy disorder effects and valence band mixing, and investigate the device efficiency. Results show that the current injection efficiency is strongly influenced by the chosen doping profile
A sparse hierarchical hp-finite element method on disks and annuli
Full text linkWe develop a sparse hierarchical hp-finite element method (hp-FEM) for the Helmholtz equation with variable coefficients posed on a two-dimensional disk or annulus. The mesh is an inner disk cell (omitted if on an annulus domain) and concentric annuli cells. The discretization preserves the Fourier mode decoupling of rotationally invariant operators, such as the Laplacian, which manifests as block diagonal mass and stiffness matrices. Moreover, the matrices have a sparsity pattern independent of the order of the discretization and admit an optimal complexity factorization. The sparse hp-FEM can handle radial discontinuities in the right-hand side and in rotationally invariant Helmholtz coefficients. Rotationally anisotropic coefficients that are approximated by low-degree polynomials in Cartesian coordinates also result in sparse linear systems. We consider examples such as a high-frequency Helmholtz equation with radial discontinuities and rotationally anisotropic coefficients, singular source terms, the time-dependent Schrödinger equation, and an extension to a three-dimensional cylinder domain, with a quasi-optimal solve, via the Alternating Direction Implicit (ADI) algorithm
Minimal and maximal solution maps of elliptic QVIs of obstacle type: Lipschitz stability, differentiability, and optimal control
No full textQuasi-variational inequalities (QVIs) of obstacle type in many cases have multiple solutions that can be ordered. We study a multitude of properties of the operator mapping the source term to the minimal or maximal solution of such QVIs. We prove that the solution maps are locally Lipschitz continuous and directionally differentiable and show existence of optimal controls for problems that incorporate these maps as the control-to-state operator. We also consider a Moreau–Yosida-type penalisation for the QVI, wherein we show that it is possible to approximate the minimal and maximal solutions by sequences of minimal and maximal solutions (respectively) of certain PDEs, which have a simpler structure and offer a convenient characterisation in particular for computation. For solution mappings of these penalised problems, we prove a number of properties including Lipschitz and differential stability. Making use of the penalised equations, we derive (in the limit) C-stationarity conditions for the control problem, in addition to the Bouligand stationarity we get from the differentiability result
An Eulerian formulation for dissipative materials using Lie derivatives and GENERIC
No full textWe recall the systematic formulation of Eulerian mechanics in terms of Lie derivatives along the vector field of the material points. Using the abstract properties of Lie derivatives we show that the transport via Lie derivatives generates in a natural way a Poisson structure on the chosen phase space. The evolution equations for thermo-viscoelastic-viscoplastic materials in the Eulerian setting are formulated in the abstract framework of GENERIC (General Equations for Non-Equilibrium Reversible-Irreversible Coupling). The equations may not be new, but the systematic splitting between reversible Hamiltonian and dissipative effects allows us to see the equations in a new light that is especially useful for future generalizing of the system, e.g. for adding new effects like reactive species
LeAP-SSN: A semismooth Newton method with global convergence rates
Full text linkWe propose LeAP-SSN (Levenberg–Marquardt Adaptive Proximal SemismoothNewton method), a semismooth Newton-type method with a simple, parameter-free globalisation strategy that guarantees convergence from arbitrary starting points in nonconvex settings to stationary points, and under a Polyak–Łojasiewicz condition, to a global minimum, in Hilbert spaces. The method employs an adaptive Levenberg–Marquardt regularisation for the Newton steps, combined with backtracking, and does not require knowledge of problem-specific constants. We establish global nonasymptotic rates: O(1/k) for convex problems in terms of objective values, O(1/sqrt{k}) under nonconvexity in terms of subgradients, and linear convergence under a Polyak–Łojasiewicz condition. The algorithm achieves superlinear convergence under mild semismoothness and Dennis–Moré or partial smoothness conditions, even for non-isolated minimisers. By combining strong global guarantees with superlinear local rates in a fully parameter-agnostic framework, LeAP-SSN bridges the gap between globally convergent algorithms and the fast asymptotics of Newton's method. The practical efficiency of the method is illustrated on representative problems from imaging, contact mechanics, and machine learning