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A General Framework for Distributed Partitioned Optimization
Get PDFDistributed optimization is widely used in large scale and privacy preserving machine learning and various distributed control and sensing systems. It is assumed that every agent in the network possesses a local objective function, and the nodes interact via a communication network. In the standard scenario, which is mostly studied in the literature, the local functions are dependent on a common set of variables, and, therefore, have to send the whole variable set at each communication round. In this work, we study a different problem statement, where each of the local functions held by the nodes depends only on some subset of the variables. Given a network, we build a general algorithm-independent framework for distributed partitioned optimization that allows to construct algorithms with reduced communication load using a generalization of Laplacian matrix. Moreover, our framework allows to obtain algorithms with non-asymptotic convergence rates with explicit dependence on the parameters of the network, including accelerated and optimal first-order methods. We illustrate the efficacy of our approach on a synthetic example
Parameter Space Study of Optimal Scale-Dependent Weights in TV Image Denoising
No full textWe investigate the choice of finite-dimensional parameter spaces within a bilevel optimization framework for selecting scale-dependent weights in total variation image denoising with non-uniform noise. Due to the pointwise box constraints on the parameter function, we prove existence of Lagrange multipliers in low regularity spaces, and derive a first-order optimality system. To cope with the difficulties related to the lack of regularity, a Moreau-Yosida regularization is introduced and convergence of the regularized optimal parameters towards the optimal weight for the original problem is verified. For each regularized bilevel problem a second-order quasi-Newton algorithm is proposed, together with a semismooth Newton scheme for solving the lower-level problem. Finally, several numerical tests are carried out to compare the different parameter space choices and draw some conclusions
Deterministic Homogenization under Optimal Moment Assumptions for Fast–Slow Systems. Part 2
No full textWe consider deterministic homogenization for discrete-time fast–slow systems of the form Xk+1=Xk+n−1an(Xk,Yk)+n−1/2bn(Xk,Yk),Yk+1=TnYkXk+1=Xk+n−1an(Xk,Yk)+n−1/2bn(Xk,Yk),Yk+1=TnYk and give conditions under which the dynamics of the slow equations converge weakly to an Itô diffusion X as n→∞n→∞. The drift and diffusion coefficients of the limiting stochastic differential equation satisfied by X are given explicitly. This extends the results of Kelly–Melbourne (J. Funct. Anal. 272 (2017) 4063–4102) from the continuous-time case to the discrete-time case. Moreover, our methods (p-variation rough paths) work under optimal moment assumptions. Combined with parallel developments on martingale approximations for families of nonuniformly expanding maps in Part 1 by Korepanov, Kosloff and Melbourne, we obtain optimal homogenization results when TnTn is such a family of maps. Nous étudions l’homogénéisation déterministe des systèmes lents-rapides en temps discret de la forme suivante Xk+1=Xk+n−1an(Xk,Yk)+n−1/2bn(Xk,Yk),Yk+1=TnYkXk+1=Xk+n−1an(Xk,Yk)+n−1/2bn(Xk,Yk),Yk+1=TnYk et donnons des conditions sous lesquelles la dynamique des équations lentes converge en loi vers une diffusion d’Itô X quand n→∞n→∞. Nous calculons explicitement la dérive et les coefficients de diffusion de l’équation différentielle stochastique vérifiée par X. Ceci étend les résultats de Kelly–Melbourne (J. Funct. Anal. 272 (2017) 4063–4102) du temps continu au temps discret. De plus, notre méthode (chemins rugueux en p-variation) fonctionne sous des conditions de moments optimales. Nous obtenons aussi des résultats optimaux d’homogénéisation quand TnTn est une famille de transformations non uniformément dilatantes. Ces résultats exploitent les développements parallèles dans la partie 1 (par Korepanov, Kosloff et Melbourne) sur l’approximation par des martingales pour ce type de transformations
Multiscale Simulations of Uni-Polar Hole Transport in (In,Ga)N Quantum Well Systems
No full textUnderstanding the impact of the alloy micro-structure on carrier transport becomes important when designing III-nitride-based light emitting diode (LED) structures. In this work, we study the impact of alloy fluctuations on the hole carrier transport in (In,Ga)N single and multi-quantum well systems. To disentangle hole transport from electron transport and carrier recombination processes, we focus our attention on uni-polar (p-i-p) systems. The calculations employ our recently established multi-scale simulation framework that connects atomistic tight-binding theory with a macroscale drift-diffusion model. In addition to alloy fluctuations, we pay special attention to the impact of quantum corrections on hole transport. Our calculations indicate that results from a virtual crystal approximation present an upper limit for the hole transport in a p-i-p structure in terms of the current-voltage characteristics. Thus we find that alloy fluctuations can have a detrimental effect on hole transport in (In,Ga)N quantum well systems, in contrast to uni-polar electron transport. However, our studies also reveal that the magnitude by which the random alloy results deviate from virtual crystal approximation data depends on several factors, e.g. how quantum corrections are treated in the transport calculations
Thermodynamic Models for a Concentration and Electric Field Dependent Susceptibility in Liquid Electrolytes
No full textThe dielectric susceptibility is an elementary quantity of the electrochemical double layer and the associated Poisson equation. While most often is treated as a material constant, its dependency on the salt concentration in liquid electrolytes is demonstrated by various bulk electrolyte experiments. This is usually referred to as dielectric decrement. Further, it is theoretically well accepted that the susceptibility declines for large electric fields. This effect is frequently termed dielectric saturation. We analyze the impact of a variable susceptibility in terms of species concentrations and electric fields based on non-equilibrium thermodynamics. This reveals some non-obvious generalizations compared to the case of a constant susceptibility. In particular the consistent coupling of the Poisson equation, the momentum balance and the chemical potentials functions are of ultimate importance. In a numerical study, we systematically analyze the effects of a concentration and field dependent susceptibility on the double layer of a planar electrode electrolyte interface. We compute the differential capacitance and the spatial structure of the electric potential, solvent concentration and ionic distribution for various non-constant models of
Research in Mathematics of Materials Science
No full textThis volume highlights contributions of women mathematicians in the study of complex materials and includes both original research papers and reviews. The featured topics and methods draw on the fields of Calculus of Variations, Partial Differential Equations, Functional Analysis, Differential Geometry and Topology, as well as Numerical Analysis and Mathematical Modelling. Areas of applications include foams, fluid-solid interactions, liquid crystals, shape-memory alloys, magnetic suspensions, failure in solids, plasticity, viscoelasticity, homogenization, crystallization, grain growth, and phase-field models
Generalized Nash equilibrium problems with partial differential operators: Theory, algorithms and risk aversion
No full textPDE-constrained (generalized) Nash equilibrium problems (GNEPs) are considered in a deterministic setting as well as under uncertainty. This includes a study of deterministic GNEPs with nonlinear and/or multivalued operator equations as forward problems and PDE-constrained GNEPs with uncertain data. The deterministic nonlinear problems are analyzed using the theory of generalized convexity for set-valued operators, and a variational approximation approach is proposed. The stochastic setting includes a detailed overview of the recently developed theory and algorithms for risk-averse PDE-constrained optimization problems. These new results open the way to a rigorous study of stochastic PDE-constrained GNEPs
Wick polynomials in non-commutative probability: A group-theoretical approach
No full textWick polynomials and Wick products are studied in the context of non-commutative probability theory. It is shown that free, boolean and conditionally free Wick polynomials can be defined and related through the action of the group of characters over a particular Hopf algebra. These results generalize our previous developments of a Hopf algebraic approach to cumulants and Wick products in classical probability theory
Oracle Complexity Separation in Convex Optimization
No full textMany convex optimization problems have structured objective functions written as a sum of functions with different oracle types (e.g., full gradient, coordinate derivative, stochastic gradient) and different arithmetic operations complexity of these oracles. In the strongly convex case, these functions also have different condition numbers that eventually define the iteration complexity of first-order methods and the number of oracle calls required to achieve a given accuracy. Motivated by the desire to call more expensive oracles fewer times, we consider the problem of minimizing the sum of two functions and propose a generic algorithmic framework to separate oracle complexities for each function. The latter means that the oracle for each function is called the number of times that coincide with the oracle complexity for the case when the second function is absent. Our general accelerated framework covers the setting of (strongly) convex objectives, the setting when both parts are given through full coordinate oracle, as well as when one of them is given by coordinate derivative oracle or has the finite-sum structure and is available through stochastic gradient oracle. In the latter two cases, we obtain accelerated random coordinate descent and accelerated variance reduced methods with oracle complexity separation
On the Convergence of Adaptive Stochastic Collocation for Elliptic Partial Differential Equations with Affine Diffusion
No full textConvergence of an adaptive collocation method for the stationary parametric diffusion equation with finite-dimensional affine coefficient is shown. The adaptive algorithm relies on a recently introduced residual-based reliable a posteriori error estimator. For the convergence proof, a strategy recently used for a stochastic Galerkin method with an hierarchical error estimator is transferred to the collocation setting