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In this work, we bridge standard adaptive mesh refinement and coarsening on scalable octree background meshes and robust unfitted finite element formulations for the automatic and efficient solution of large-scale nonlinear solid mechanics problems posed on complex geometries, as an alternative to standard body-fitted formulations, unstructured mesh generation and graph partitioning strategies. We pay special attention to those aspects requiring a specialized treatment in the extension of the unfitted h-adaptive aggregated finite element method on parallel tree-based adaptive meshes, recently developed for linear scalar elliptic problems, to handle nonlinear problems in solid mechanics. In order to accurately and efficiently capture localized phenomena that frequently occur in nonlinear solid mechanics problems, we perform pseudo time-stepping in combination with h-adaptive dynamic mesh refinement and rebalancing driven by a-posteriori error estimators. The method is implemented considering both irreducible and mixed (u/p) formulations and thus it is able to robustly face problems involving incompressible materials. In the numerical experiments, both formulations are used to model the inelastic behavior of a wide range of compressible and incompressible materials. First, a selected set of benchmarks are reproduced as a verification step. Second, a set of experiments is presented with problems involving complex geometries. Among them, we model a cantilever beam problem with spherical hollows distributed in a Simple Cubic array. This test involves a discrete domain with up to 11.7M Degrees Of Freedom solved in less than two hours on 3072 cores of a parallel supercomputer.

We present differentiable predictive control (DPC), a method for learning constrained adaptive neural control policies and dynamical models of unknown linear systems. DPC presents an approximate data-driven solution approach to the explicit Model Predictive Control (MPC) problem as a scalable alternative to computationally expensive multiparametric programming solvers. DPC is formulated as a constrained deep learning problem whose architecture is inspired by the structure of classical MPC. The optimization of the neural control policy is based on automatic differentiation of the MPC-inspired loss function through a differentiable closed-loop system model. This novel solution approach can optimize adaptive neural control policies for time-varying references while obeying state and input constraints without the prior need of an MPC controller. We show that DPC can learn to stabilize constrained neural control policies for systems with unstable dynamics. Moreover, we provide sufficient conditions for asymptotic stability of generic closed-loop system dynamics with neural feedback policies. In simulation case studies, we assess the performance of the proposed DPC method in terms of reference tracking, robustness, and computational and memory footprints compared against classical model-based and data-driven control approaches. We demonstrate that DPC scales linearly with problem size, compared to exponential scalability of classical explicit MPC based on multiparametric programming.

The algebraic flux correction (AFC) schemes presented in this work constrain a standard continuous finite element discretization of a nonlinear hyperbolic problem to satisfy relevant maximum principles and entropy stability conditions. The desired properties are enforced by applying a limiter to antidiffusive fluxes that represent the difference between the high-order baseline scheme and a property-preserving approximation of Lax--Friedrichs type. In the first step of the limiting procedure, the given target fluxes are adjusted in a way that guarantees preservation of local and/or global bounds. In the second step, additional limiting is performed, if necessary,to ensure the validity of fully discrete and/or semi-discrete entropy inequalities. The limiter-based entropy fixes considered in this work are applicable to finite element discretizations of scalar hyperbolic equations and systems alike. The underlying inequality constraints are formulated using Tadmor's entropy stability theory. The proposed limiters impose entropy-conservative or entropy-dissipative bounds on the rate of entropy production by antidiffusive fluxes and Runge--Kutta (RK) time discretizations. Two versions of the fully discrete entropy fix are developed for this purpose. To motivate the use of limiter-based entropy fixes, we prove a finite element version of the Lax--Wendroff theorem and perform numerical studies for standard test problems. In our numerical experiments, entropy-dissipative schemes converge to correct weak solutions of scalar conservation laws, of the Euler equations, and of the shallow water equations.

Contact and related phenomena, such as friction, wear or elastohydrodynamic lubrication, remain as one of the most challenging problem classes in nonlinear solid and structural mechanics. In the context of their computational treatment with finite element methods (FEM) or isogeometric analysis (IGA), the inherent non-smoothness of contact conditions, the design of robust discretization approaches as well as the implementation of efficient solution schemes seem to provide a never ending source of hard nuts to crack. This is particularly true for the case of beam-to-solid interaction with its mixed-dimensional 1D-3D contact models. Therefore, this contribution gives an overview of current steps being taken, starting from state-of-the-art beam-to-beam (1D) and solid-to-solid (3D) contact algorithms, towards a truly general 1D-3D beam-to-solid contact formulation.

We consider an auto-scaling technique in a cloud system where virtual machines hosted on a physical node are turned on and off depending on the queue's occupation (or thresholds), in order to minimise a global cost integrating both energy consumption and performance. We propose several efficient optimisation methods to find threshold values minimising this global cost: local search heuristics coupled with aggregation of Markov chain and with queues approximation techniques to reduce the execution time and improve the accuracy. The second approach tackles the problem with a Markov Decision Process (MDP) for which we proceed to a theoretical study and provide theoretical comparison with the first approach. We also develop structured MDP algorithms integrating hysteresis properties. We show that MDP algorithms (value iteration, policy iteration) and especially structured MDP algorithms outperform the devised heuristics, in terms of time execution and accuracy. Finally, we propose a cost model for a real scenario of a cloud system to apply our optimisation algorithms and show their relevance.

We propose and analyze a seamless extended Discontinuous Galerkin (DG) discretization of advection-diffusion equations on semi-infinite domains. The semi-infinite half line is split into a finite subdomain where the model uses a standard polynomial basis, and a semi-unbounded subdomain where scaled Laguerre functions are employed as basis and test functions. Numerical fluxes enable the coupling at the interface between the two subdomains in the same way as standard single domain DG interelement fluxes. A novel linear analysis on the extended DG model yields unconditional stability with respect to the P\'eclet number. Errors due to the use of different sets of basis functions on different portions of the domain are negligible, as highlighted in numerical experiments with the linear advection-diffusion and viscous Burgers' equations. With an added damping term on the semi-infinite subdomain, the extended framework is able to efficiently simulate absorbing boundary conditions without additional conditions at the interface. A few modes in the semi-infinite subdomain are found to suffice to deal with outgoing single wave and wave train signals more accurately than standard approaches at a given computational cost, thus providing an appealing model for fluid flow simulations in unbounded regions.

In this paper, a novel optimization model for joint beamforming and power control in the downlink (DL) of a cell-free massive MIMO (CFmMIMO) system is presented. The objective of the proposed optimization model is to minimize the maximum user interference while satisfying quality of service (QoS) constraints and power consumption limits. The proposed min-max optimization model is formulated as a mixed-integer nonlinear program, that is directly tractable. Numerical results show that the proposed joint beamforming and power control scheme is effective and outperforms competing schemes in terms of data rate, power consumption, and energy efficiency.

Colorizing a given gray-level image is an important task in the media and advertising industry. Due to the ambiguity inherent to colorization (many shades are often plausible), recent approaches started to explicitly model diversity. However, one of the most obvious artifacts, structural inconsistency, is rarely considered by existing methods which predict chrominance independently for every pixel. To address this issue, we develop a conditional random field based variational auto-encoder formulation which is able to achieve diversity while taking into account structural consistency. Moreover, we introduce a controllability mecha- nism that can incorporate external constraints from diverse sources in- cluding a user interface. Compared to existing baselines, we demonstrate that our method obtains more diverse and globally consistent coloriza- tions on the LFW, LSUN-Church and ILSVRC-2015 datasets.

Dynamic programming (DP) solves a variety of structured combinatorial problems by iteratively breaking them down into smaller subproblems. In spite of their versatility, DP algorithms are usually non-differentiable, which hampers their use as a layer in neural networks trained by backpropagation. To address this issue, we propose to smooth the max operator in the dynamic programming recursion, using a strongly convex regularizer. This allows to relax both the optimal value and solution of the original combinatorial problem, and turns a broad class of DP algorithms into differentiable operators. Theoretically, we provide a new probabilistic perspective on backpropagating through these DP operators, and relate them to inference in graphical models. We derive two particular instantiations of our framework, a smoothed Viterbi algorithm for sequence prediction and a smoothed DTW algorithm for time-series alignment. We showcase these instantiations on two structured prediction tasks and on structured and sparse attention for neural machine translation.