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The purpose of this paper is to perform an error analysis of the variational integrators of mechanical systems subject to external forcing. Essentially, we prove that when a discretization of contact order $r$ of the Lagrangian and force are used, the integrator has the same contact order. Our analysis is performed first for discrete forced mechanical systems defined over $TQ$, where we study the existence of flows, the construction and properties of discrete exact systems and the contact order of the flows (variational integrators) in terms of the contact order of the original systems. Then we use those results to derive the corresponding analysis for the analogous forced systems defined over $Q\times Q$.

The incorporation of appropriate inductive bias plays a critical role in learning dynamics from data. A growing body of work has been exploring ways to enforce energy conservation in the learned dynamics by encoding Lagrangian or Hamiltonian dynamics into the neural network architecture. These existing approaches are based on differential equations, which do not allow discontinuity in the states and thereby limit the class of systems one can learn. However, in reality, most physical systems, such as legged robots and robotic manipulators, involve contacts and collisions, which introduce discontinuities in the states. In this paper, we introduce a differentiable contact model, which can capture contact mechanics: frictionless/frictional, as well as elastic/inelastic. This model can also accommodate inequality constraints, such as limits on the joint angles. The proposed contact model extends the scope of Lagrangian and Hamiltonian neural networks by allowing simultaneous learning of contact and system properties. We demonstrate this framework on a series of challenging 2D and 3D physical systems with different coefficients of restitution and friction. The learned dynamics can be used as a differentiable physics simulator for downstream gradient-based optimization tasks, such as planning and control.

We revisit the (block-angular) min-max resource sharing problem, which is a well-known generalization of fractional packing and the maximum concurrent flow problem. It consists of finding an $\ell_{\infty}$-minimal element in a Minkowski sum $\mathcal{X}= \sum_{C \in \mathcal{C}} X_C$ of non-empty closed convex sets $X_C \subseteq \mathbb{R}^{\mathcal{R}}_{\geq 0}$, where $\mathcal{C}$ and $\mathcal{R}$ are finite sets. We assume that an oracle for approximate linear minimization over $X_C$ is given. In this setting, the currently fastest known FPTAS is due to M\"uller, Radke, and Vygen. For $\delta \in (0,1]$, it computes a $\sigma(1+\delta)$-approximately optimal solution using $\mathcal{O}((|\mathcal{C}|+|\mathcal{R}|)\log |\mathcal{R}| (\delta^{-2} + \log \log |\mathcal{R}|))$ oracle calls, where $\sigma$ is the approximation ratio of the oracle. We describe an extension of their algorithm and improve on previous results in various ways. Our FPTAS, which, as previous approaches, is based on the multiplicative weight update method, computes close to optimal primal and dual solutions using $\mathcal{O}\left(\frac{|\mathcal{C}|+ |\mathcal{R}|}{\delta^2} \log |\mathcal{R}|\right)$ oracle calls. We prove that our running time is optimal under certain assumptions, implying that no warm-start analysis of the algorithm is possible. A major novelty of our analysis is the concept of local weak duality, which illustrates that the algorithm optimizes (close to) independent parts of the instance separately. Interestingly, this implies that the computed solution is not only approximately $\ell_{\infty}$-minimal, but among such solutions, also its second-highest entry is approximately minimal. We prove that this statement cannot be extended to the third-highest entry.

We introduce a new regularization model for incompressible fluid flow, which is a regularization of the EMAC formulation of the Navier-Stokes equations (NSE) that we call EMAC-Reg. The EMAC (energy, momentum, and angular momentum conserving) formulation has proved to be a useful formulation because it conserves energy, momentum and angular momentum even when the divergence constraint is only weakly enforced. However it is still a NSE formulation and so cannot resolve higher Reynolds number flows without very fine meshes. By carefully introducing regularization into the EMAC formulation, we create a model more suitable for coarser mesh computations but that still conserves the same quantities as EMAC, i.e., energy, momentum, and angular momentum. We show that EMAC-Reg, when semi-discretized with a finite element spatial discretization is well-posed and optimally accurate. Numerical results are provided that show EMAC-Reg is a robust coarse mesh model.

We consider sensitivity of a generic stochastic optimization problem to model uncertainty. We take a non-parametric approach and capture model uncertainty using Wasserstein balls around the postulated model. We provide explicit formulae for the first order correction to both the value function and the optimizer and further extend our results to optimization under linear constraints. We present applications to statistics, machine learning, mathematical finance and uncertainty quantification. In particular, we provide explicit first-order approximation for square-root LASSO regression coefficients and deduce coefficient shrinkage compared to the ordinary least squares regression. We consider robustness of call option pricing and deduce a new Black-Scholes sensitivity, a non-parametric version of the so-called Vega. We also compute sensitivities of optimized certainty equivalents in finance and propose measures to quantify robustness of neural networks to adversarial examples.

Reconfigurable intelligent surface (RIS) has become a promising technology to improve wireless communication in recent years. It steers the incident signals to create a favorable propagation environment by controlling the reconfigurable passive elements with less hardware cost and lower power consumption. In this paper, we consider a RIS-aided multiuser multiple-input single-output downlink communication system. We aim to maximize the weighted sum-rate of all users by joint optimizing the active beamforming at the access point and the passive beamforming vector of the RIS elements. Unlike most existing works, we consider the more practical situation with the discrete phase shifts and imperfect channel state information (CSI). Specifically, for the situation that the discrete phase shifts and perfect CSI are considered, we first develop a deep quantization neural network (DQNN) to simultaneously design the active and passive beamforming while most reported works design them alternatively. Then, we propose an improved structure (I-DQNN) based on DQNN to simplify the parameters decision process when the control bits of each RIS element are greater than 1 bit. Finally, we extend the two proposed DQNN-based algorithms to the case that the discrete phase shifts and imperfect CSI are considered simultaneously. Our simulation results show that the two DQNN-based algorithms have better performance than traditional algorithms in the perfect CSI case, and are also more robust in the imperfect CSI case.

We consider the energy complexity of the leader election problem in the single-hop radio network model, where each device has a unique identifier in $\{1, 2, \ldots, N\}$. Energy is a scarce resource for small battery-powered devices. For such devices, most of the energy is often spent on communication, not on computation. To approximate the actual energy cost, the energy complexity of an algorithm is defined as the maximum over all devices of the number of time slots where the device transmits or listens. Much progress has been made in understanding the energy complexity of leader election in radio networks, but very little is known about the trade-off between time and energy. $\textbf{Time-energy trade-off:}$ For any $k \geq \log \log N$, we show that a leader among at most $n$ devices can be elected deterministically in $O(k \cdot n^{1+\epsilon}) + O(k \cdot N^{1/k})$ time and $O(k)$ energy if each device can simultaneously transmit and listen, where $\epsilon > 0$ is any small constant. This improves upon the previous $O(N)$-time $O(\log \log N)$-energy algorithm by Chang et al. [STOC 2017]. We provide lower bounds to show that the time-energy trade-off of our algorithm is near-optimal. $\textbf{Dense instances:}$ For the dense instances where the number of devices is $n = \Theta(N)$, we design a deterministic leader election algorithm using only $O(1)$ energy. This improves upon the $O(\log^* N)$-energy algorithm by Jurdzi\'{n}ski et al. [PODC 2002] and the $O(\alpha(N))$-energy algorithm by Chang et al. [STOC 2017]. More specifically, we show that the optimal deterministic energy complexity of leader election is $\Theta\left(\max\left\{1, \log \frac{N}{n}\right\}\right)$ if the devices cannot simultaneously transmit and listen, and it is $\Theta\left(\max\left\{1, \log \log \frac{N}{n}\right\}\right)$ if they can.

Flexible sparsity regularization means stably approximating sparse solutions of operator equations by using coefficient-dependent penalizations. We propose and analyse a general nonconvex approach in this respect, from both theoretical and numerical perspectives. Namely, we show convergence of the regularization method and establish convergence properties of a couple of majorization approaches for the associated nonconvex problems. We also test a monotone algorithm for an academic example where the operator is an $M$ matrix, and on a time-dependent optimal control problem, pointing out the advantages of employing variable penalties over a fixed penalty.

We derive a priori error of the Godunov method for the multidimensional Euler system of gas dynamics. To this end we apply the relative energy principle and estimate the distance between the numerical solution and the strong solution. This yields also the estimates of the $L^2$-norm of errors in density, momentum and entropy. Under the assumption that the numerical density and energy are bounded, we obtain a convergence rate of $1/2$ for the relative energy in the $L^1$-norm. Further, under the assumption -- the total variation of numerical solution is bounded, we obtain the first order convergence rate for the relative energy in the $L^1$-norm. Consequently, numerical solutions (density, momentum and entropy) converge in the $L^2$-norm with the convergence rate of $1/2$. The numerical results presented for Riemann problems are consistent with our theoretical analysis.

In this paper, we study the optimal convergence rate for distributed convex optimization problems in networks. We model the communication restrictions imposed by the network as a set of affine constraints and provide optimal complexity bounds for four different setups, namely: the function $F(\xb) \triangleq \sum_{i=1}^{m}f_i(\xb)$ is strongly convex and smooth, either strongly convex or smooth or just convex. Our results show that Nesterov's accelerated gradient descent on the dual problem can be executed in a distributed manner and obtains the same optimal rates as in the centralized version of the problem (up to constant or logarithmic factors) with an additional cost related to the spectral gap of the interaction matrix. Finally, we discuss some extensions to the proposed setup such as proximal friendly functions, time-varying graphs, improvement of the condition numbers.