Constrained optimization in classes of analytic functions with prescribed pointwise values

We consider an overdetermined problem for Laplace equation on a disk with partial boundary data where additional pointwise data inside the disk have to be taken into account. After reformulation, this ill-posed problem reduces to a bounded extremal problem of best norm-constrained approximation of partial L2 boundary data by traces of holomorphic functions which satisfy given pointwise interpolation conditions. The problem of best norm-constrained approximation of a given L2 function on a subset of the circle by the trace of a H2 function has been considered in [Baratchart \& Leblond, 1998]. In the present work, we extend such a formulation to the case where the additional interpolation conditions are imposed. We also obtain some new results that can be applied to the original problem: we carry out stability analysis and propose a novel method of evaluation of the approximation and blow-up rates of the solution in terms of a Lagrange parameter leading to a highly-efficient computational algorithm for solving the problem

Constrained extremal problems in the Hardy space H2 and Carleman's formulas

We study some approximation problems on a strict subset of the circle by analytic functions of the Hardy space H2 of the unit disk (in C), whose modulus satisfy a pointwise constraint on the complentary part of the circle. Existence and uniqueness results, as well as pointwise saturation of the constraint, are established. We also derive a critical point equation which gives rise to a dual formulation of the problem. We further compute directional derivatives for this functional as a computational means to approach the issue. We then consider a finite-dimensional polynomial version of the bounded extremal problem

Uniqueness results for inverse Robin problems with bounded coefficient

In this paper we address the uniqueness issue in the classical Robin inverse problem on a Lipschitz domain \Omega\subset\RR^n, with $L^\infty$ Robin coefficient, $L^2$ Neumann data and isotropic conductivity of class $W^{1,r}(\Omega)$, r\textgreater{}n. We show that uniqueness of the Robin coefficient on a subpart of the boundary given Cauchy data on the complementary part, does hold in dimension $n=2$ but needs not hold in higher dimension. We also raise on open issue on harmonic gradients which is of interest in this context

Minimax density estimation in the adversarial framework under local differential privacy

We consider the problem of nonparametric density estimation under privacy constraints in an adversarial framework. To this end, we study minimax rates under local differential privacy over Sobolev spaces. We first obtain a lower bound which allows us to quantify the impact of privacy compared with the classical framework. Next, we introduce a new Coordinate block privacy mechanism that guarantees local differential privacy, which, coupled with a projection estimator, achieves the minimax optimal rates

State-driven Implicit Modeling for Sparsity and Robustness in Neural Networks

Implicit models are a general class of learning models that forgo the hierarchical layer structure typical in neural networks and instead define the internal states based on an ``equilibrium'' equation, offering competitive performance and reduced memory consumption. However, training such models usually relies on expensive implicit differentiation for backward propagation. In this work, we present a new approach to training implicit models, called State-driven Implicit Modeling (SIM), where we constrain the internal states and outputs to match that of a baseline model, circumventing costly backward computations. The training problem becomes convex by construction and can be solved in a parallel fashion, thanks to its decomposable structure. We demonstrate how the SIM approach can be applied to significantly improve sparsity (parameter reduction) and robustness of baseline models trained on FashionMNIST and CIFAR-100 datasets

Tele-expertise assessment of chronic wounds by advanced practice dermatology nurses

BackgroundThe prevalence of chronic wounds continues to increase with the lengthening of life expectancy. The decreasing medical demography exacerbates a delay in care.ObjectiveThis study investigated the feasibility of evaluating chronic wounds through tele-expertise (TLE) by an advanced nurse practitioner (ANP) in dermatology.Methods We conducted a single-centre, observational study from November 2020 to May 2021. The main objective was to assess the ability of an Advanced Nurse Practitioner (ANP) to manage the orientation of care (medical or nursing) for requests for specialised advice on the treatment of chronic wounds through tele-expertise (TLE). The relevance of the advice provided by the ANP was evaluated by a dermatologist (D1). Simultaneously, a second dermatologist (D2) blindly determined what would have been the most appropriate care for the situation. His decisions were compared to those of the ANP using a Chi-2 test.ResultsOut of the 55 requests for teleconsultation to treat chronic wounds, the ANP considered that 43 (78.1%) cases required medical advice, while 12 (21.8%) could be managed within the scope of their expertise. D1 confirmed 72% of the ANP's decisions regarding cases deemed medical and 75% of cases deemed nursing by the ANP. Compared to the ANP's recommendations, D2 considered that 14 (25.5%) cases could be handled solely by the ANP, and 41 (74.5%) cases required a physician's intervention.ConclusionIn teleconsultations for chronic wounds, a dermatologist confirms the orientation and care decisions made by a specialised ANP in approximately 75% of cases. This high level of agreement positions the ANP as a reliable partner in the wound care pathway in dermatology.<br/