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Corrigendum and Addendum: Newton Differentiability of Convex Functions in Normed Spaces and of a Class of Operators
No full textAs it is formulated, Proposition 3.12 of [M. Brokate and M. Ulbrich, SIAM J. Optim., 32 (2022), pp. 1265–1287] contains an error. But this can be corrected in the way described below. The results of [M. Brokate and M. Ulbrich, SIAM J. Optim., 32 (2022), pp. 1265–1287] based on Proposition 3.12 are not affected. We also use the opportunity to add a further illustrating example and to rectify some inaccuracies which may be confusing
A Bilinear Flory Equation
No full textWe consider coagulation equations of Flory type where particles are represented by finite dimensional vectors and the coagulation rate between two particles of types x and y is given by a bilinear form y⋅Ax, generalising the multiplicative kernel. For these coagulation rates, a gelation transition occurs at a finite time tg∈(0,∞), which can be given exactly in terms of an eigenvalue problem in finite dimensions. We prove a hydrodynamic limit for the corresponding stochastic coagulant, including the phase transition for the largest particle, and exploit a coupling to random graphs to extend analysis of the limiting process beyond the gelation time. On considère des équations de coagulation de type Flory où les particules sont représentées par des vecteurs de dimension finie et le taux de coagulation entre deux particules de types x et y est donné par une forme bilinéaire y⋅Ax, généralisant le noyau multiplicatif. Pour ces taux de coagulation, une transition de gélification se produit à un temps fini tg∈(0,∞), qui peut être donné exactement en termes d’un problème aux valeurs propres en dimensions finies. Nous prouvons une limite hydrodynamique pour le coagulant stochastique correspondant, y compris la transition de phase pour la plus grande particule, et exploitons un couplage avec des graphes aléatoires pour étendre l’analyse du processus limite au-delà du temps de gélification
An Energy-Based Finite-Strain Model for 3D Heterostructured Materials and its Validation by Curvature Analysis
No full textThis paper presents a comprehensive study of the intrinsic strain response of 3D het- erostructures arising from lattice mismatch. Combining materials with different lattice constants induces strain, leading to the bending of these heterostructures. We propose a model for nonlinear elastic heterostructures such as bimetallic beams or nanowires that takes into account local prestrain within each distinct material region. The resulting system of partial differential equations (PDEs) in Lagrangian coordinates incorporates a nonlinear strain and a linear stress-strain relationship governed by Hooke?s law. To validate our model, we apply it to bimetallic beams and hexagonal hetero-nanowires and perform numerical simulations using finite element methods (FEM). Our simulations ex- amine how these structures undergo bending under varying material compositions and cross-sectional geometries. In order to assess the fidelity of the model and the accuracy of simulations, we compare the calculated curvature with analytically derived formula- tions. We derive these analytical expressions through an energy-based approach as well as a kinetic framework, adeptly accounting for the lattice constant mismatch present at each compound material of the heterostructures. The outcomes of our study yield valuable insights into the behavior of strained bent heterostructures. This is particularly significant as the strain has the potential to influence the electronic band structure, piezoelectricity, and the dynamics of charge carriers
Reliably Calibrating X-ray Images Required for Preoperative Planning of THA using a Device-Adapted Magnification Factor
No full textBackground and aim Calibrated pelvic X-ray images are needed in the preoperative planning of total hip arthroplasty (THA) to predict component sizes. Errors and mismatch in the size of one or more components are reported, which can lead to clinically relevant complications. Our aim is to investigate whether we can solve the fundamental problem of X-ray calibration and whether traditional X-ray still has a place in preoperative planning despite improved radiological alternatives. Methods Based on geometric and radiographic principles, we estimate that the magnification factor is adapted to the X-ray device and depends strongly on the source-image distance of the device. We analyse the errors of the various calibration methods and investigate which narrow range can be expected to show that the center of rotation is sufficiently accurate. Based on the results of several CT-scans we defined an adapted magnification factor and validated the degree of measurement accuracy. Results The true magnification of objects on X-ray images depends mainly on the device settings. Stem size prediction is possible to a limited extent, with an error margin of 4.3%. Components can be predicted with a safety margin of one size up and down as with CT or 3D images. The prerequisite is that the source-image distance is greater than or equal to 120 cm, the table-image distance is known, and the object-image distance is estimated according to the patient's BMI. We defined a device-adapted magnification factor that simplifies the templating routine and can be used to obtain the most reliable preoperative dimensional measurements that can be expected from X-ray images. We found the error margin of the magnification factor with the highest degrees of prediction and precision. Conclusion Preoperative planning is reliable and reproducible using X-ray images if calibration is performed with the device-adapted magnification factor suggested in this paper
On a Drift‐Diffusion Model for Perovskite Solar Cells
No full textWe introduce a vacancy-assisted charge transport model for perovskite solar cells. This instationary drift-diffusion system describes the motion of electrons, holes, and ionic vacancies and takes into account Fermi–Dirac statistics for electrons and holes and the Fermi–Dirac integral of order -1 for the mobile ionic vacancies in the perovskite. The free energy functional we work with corresponds to that choice of the statistical relations. To verify the existence of weak solutions, we consider a problem with regularized state equations and reaction terms on any arbitrarily chosen finite time interval. We motivate its solvability by time discretization and passage to the time-continuous limit. A priori estimates for the chemical potentials that are independent of the regularization level ensure the existence of solutions to the original problem. These types of estimates rely on Moser iteration techniques and can also be obtained for solutions to the original problem
Analysis of Kernel Mirror Prox for Measure Optimization
No full textBy choosing a suitable function space as the dual to the non-negative measure cone, we study in a unified framework a class of functional saddle-point optimization problems, which we term the Mixed Functional Nash Equilibrium (MFNE), that underlies several existing machine learning algorithms, such as implicit generative models, distributionally robust optimization (DRO), and Wasserstein barycenters. We model the saddle-point optimization dynamics as an interacting Fisher-Rao-RKHS gradient flow when the function space is chosen as a reproducing kernel Hilbert space (RKHS). As a discrete time counterpart, we propose a primal-dual kernel mirror prox (KMP) algorithm, which uses a dual step in the RKHS, and a primal entropic mirror prox step. We then provide a unified convergence analysis of KMP in an infinite-dimensional setting for this class of MFNE problems, which establishes a convergence rate of O(1/N) in the deterministic case and O(1/N−−√) in the stochastic case, where N is the iteration counter. As a case study, we apply our analysis to DRO, providing algorithmic guarantees for DRO robustness and convergence
Neutral delay differential equation model of an optically injected Kerr cavity
No full textA neutral delay differential equation (NDDE) model of a Kerr cavity with external coherent injection is developed that can be considered as a generalization of the Ikeda map with second and higher order dispersions being taken into account. It is shown that this model has solutions in the form of dissipative solitons both in the limit, where the model can be reduced to the Lugiato--Lefever equation (LLE), and beyond this limit, where the soliton is eventually destroyed by the Cherenkov radiation. Unlike the standard LLE the NDDE model is able to describe the overlap of multiple resonances associated with different cavity modes
Gradient-Robust Hybrid DG Discretizations for the Compressible Stokes Equations
No full textThis paper studies two hybrid discontinuous Galerkin (HDG) discretizations for the velocity-density formulation of the compressible Stokes equations with respect to several desired structural properties, namely provable convergence, the preservation of non-negativity and mass constraints for the density, and gradient-robustness. The later property dramatically enhances the accuracy in well-balanced situations, such as the hydrostatic balance where the pressure gradient balances the gravity force. One of the studied schemes employs an H(div)-conforming velocity ansatz space which ensures all mentioned properties, while a fully discontinuous method is shown to satisfy all properties but the gradient-robustness. Also higher-order schemes for both variants are presented and compared in three numerical benchmark problems. The final example shows the importance also for non-hydrostatic well-balanced states for the compressible Navier-Stokes equations
Near-Optimal Tensor Methods for Minimizing the Gradient Norm of Convex Functions and Accelerated Primal-Dual Tensor Methods
No full textMotivated, in particular, by the entropy-regularized optimal transport problem, we consider convex optimization problems with linear equality constraints, where the dual objective has Lipschitz -th order derivatives, and develop two approaches for solving such problems. The first approach is based on the minimization of the norm of the gradient in the dual problem and then the reconstruction of an approximate primal solution. Recently, Grapiglia and Nesterov in their work showed lower complexity bounds for the problem of minimizing the gradient norm of the function with Lipschitz -th order derivatives. Still, the question of optimal or near-optimal methods remained open as the algorithms presented in the paper achieve suboptimal bounds only. We close this gap by proposing two near-optimal (up to logarithmic factors) methods with complexity bounds and with respect to the initial objective residual and the distance between the starting point and solution respectively. We then apply these results (having independent interest) to our primal-dual setting. As the second approach, we propose a direct accelerated primal-dual tensor method for convex problems with linear equality constraints, where the dual objective has Lipschitz -th order derivatives. For this algorithm, we prove complexity in terms of the duality gap and the residual in the constraints. We illustrate the practical performance of the proposed algorithms in experiments on logistic regression, entropy-regularized optimal transport problem, and the minimal mutual information problem
High-Probability Complexity Bounds for Non-smooth Stochastic Convex Optimization with Heavy-Tailed Noise
No full textStochastic first-order methods are standard for training large-scale machine learning models. Random behavior may cause a particular run of an algorithm to result in a highly suboptimal objective value, whereas theoretical guarantees are usually proved for the expectation of the objective value. Thus, it is essential to theoretically guarantee that algorithms provide small objective residual with high probability. Existing methods for non-smooth stochastic convex optimization have complexity bounds with the dependence on the confidence level that is either negative-power or logarithmic but under an additional assumption of sub-Gaussian (light-tailed) noise distribution that may not hold in practice. In our paper, we resolve this issue and derive the first high-probability convergence results with logarithmic dependence on the confidence level for non-smooth convex stochastic optimization problems with non-sub-Gaussian (heavy-tailed) noise. To derive our results, we propose novel stepsize rules for two stochastic methods with gradient clipping. Moreover, our analysis works for generalized smooth objectives with Hölder-continuous gradients, and for both methods, we provide an extension for strongly convex problems. Finally, our results imply that the first (accelerated) method we consider also has optimal iteration and oracle complexity in all the regimes, and the second one is optimal in the non-smooth setting