We present a Newton-Krylov solver for a viscous-plastic sea-ice model. This constitutive relation is commonly used in climate models to describe the material properties of sea ice. Due to the strong nonlinearity introduced by the material law in the momentum equation, the development of fast, robust and scalable solvers is still a substantial challenge. In this paper, we propose a novel primal-dual Newton linearization for the implicitly-in-time discretized momentum equation. Compared to existing methods, it converges faster and more robustly with respect to mesh refinement, and thus enables numerically converged sea-ice simulations at high resolutions. Combined with an algebraic multigrid-preconditioned Krylov method for the linearized systems, which contain strongly varying coefficients, the resulting solver scales well and can be used in parallel. We present experiments for two challenging test problems and study solver performance for problems with up to 8.4 million spatial unknowns.
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The $3$SUM-Indexing problem was introduced as a data structure version of the $3$SUM problem, with the goal of proving strong conditional lower bounds for static data structures via reductions. Ideally, the conjectured hardness of $3$SUM-Indexing should be replaced by an unconditional lower bound. Unfortunately, we are far from proving this, with the strongest current lower bound being a logarithmic query time lower bound by Golovnev et al. from STOC'20. Moreover, their lower bound holds only for non-adaptive data structures and they explicitly asked for a lower bound for adaptive data structures. Our main contribution is precisely such a lower bound against adaptive data structures. As a secondary result, we also strengthen the non-adaptive lower bound of Golovnev et al. and prove strong lower bounds for $2$-bit-probe non-adaptive $3$SUM-Indexing data structures via a completely new approach that we find interesting in its own right.
We study recovery of amplitudes and nodes of a finite impulse train from a limited number of equispaced noisy frequency samples. This problem is known as super-resolution (SR) under sparsity constraints and has numerous applications, including direction of arrival and finite rate of innovation sampling. Prony's method is an algebraic technique which fully recovers the signal parameters in the absence of measurement noise. In the presence of noise, Prony's method may experience significant loss of accuracy, especially when the separation between Dirac pulses is smaller than the Nyquist-Shannon-Rayleigh (NSR) limit. In this work we combine Prony's method with a recently established decimation technique for analyzing the SR problem in the regime where the distance between two or more pulses is much smaller than the NSR limit. We show that our approach attains optimal asymptotic stability in the presence of noise. Our result challenges the conventional belief that Prony-type methods tend to be highly numerically unstable.
We consider a high-dimensional random constrained optimization problem in which a set of binary variables is subjected to a linear system of equations. The cost function is a simple linear cost, measuring the Hamming distance with respect to a reference configuration. Despite its apparent simplicity, this problem exhibits a rich phenomenology. We show that different situations arise depending on the random ensemble of linear systems. When each variable is involved in at most two linear constraints, we show that the problem can be partially solved analytically, in particular we show that upon convergence, the zero-temperature limit of the cavity equations returns the optimal solution. We then study the geometrical properties of more general random ensembles. In particular we observe a range in the density of constraints at which the systems enters a glassy phase where the cost function has many minima. Interestingly, the algorithmic performances are only sensitive to another phase transition affecting the structure of configurations allowed by the linear constraints. We also extend our results to variables belonging to $\text{GF}(q)$, the Galois Field of order $q$. We show that increasing the value of $q$ allows to achieve a better optimum, which is confirmed by the Replica Symmetric cavity method predictions.
Imposition methods of interface conditions for the second-order wave equation with non-conforming grids is considered. The spatial discretization is based on high order finite differences with summation-by-parts properties. Previously presented solution methods for this problem, based on the simultaneous approximation term (SAT) method, have shown to introduce significant stiffness. This can lead to highly inefficient schemes. Here, two new methods of imposing the interface conditions to avoid the stiffness problems are presented: 1) a projection method and 2) a hybrid between the projection method and the SAT method. Numerical experiments are performed using traditional and order-preserving interpolation operators. Both of the novel methods retain the accuracy and convergence behavior of the previously developed SAT method but are significantly less stiff.
The presence of outliers can significantly degrade the performance of ellipse fitting methods. We develop an ellipse fitting method that is robust to outliers based on the maximum correntropy criterion with variable center (MCC-VC), where a Laplacian kernel is used. For single ellipse fitting, we formulate a non-convex optimization problem to estimate the kernel bandwidth and center and divide it into two subproblems, each estimating one parameter. We design sufficiently accurate convex approximation to each subproblem such that computationally efficient closed-form solutions are obtained. The two subproblems are solved in an alternate manner until convergence is reached. We also investigate coupled ellipses fitting. While there exist multiple ellipses fitting methods that can be used for coupled ellipses fitting, we develop a couple ellipses fitting method by exploiting the special structure. Having unknown association between data points and ellipses, we introduce an association vector for each data point and formulate a non-convex mixed-integer optimization problem to estimate the data associations, which is approximately solved by relaxing it into a second-order cone program. Using the estimated data associations, we extend the proposed method to achieve the final coupled ellipses fitting. The proposed method is shown to have significantly better performance over the existing methods in both simulated data and real images.
We propose and analyze exact and inexact regularized Newton-type methods for finding a global saddle point of a \textit{convex-concave} unconstrained min-max optimization problem. Compared to their first-order counterparts, investigations of second-order methods for min-max optimization are relatively limited, as obtaining global rates of convergence with second-order information is much more involved. In this paper, we highlight how second-order information can be used to speed up the dynamics of dual extrapolation methods {despite inexactness}. Specifically, we show that the proposed algorithms generate iterates that remain within a bounded set and the averaged iterates converge to an $\epsilon$-saddle point within $O(\epsilon^{-2/3})$ iterations in terms of a gap function. Our algorithms match the theoretically established lower bound in this context and our analysis provides a simple and intuitive convergence analysis for second-order methods without requiring any compactness assumptions. Finally, we present a series of numerical experiments on synthetic and real data that demonstrate the efficiency of the proposed algorithms.
We present two (a decoupled and a coupled) integral-equation-based methods for the Morse-Ingard equations subject to Neumann boundary conditions on the exterior domain. Both methods are based on second-kind integral equation (SKIE) formulations. The coupled method is well-conditioned and can achieve high accuracy. The decoupled method has lower computational cost and more flexibility in dealing with the boundary layer; however, it is prone to the ill-conditioning of the decoupling transform and cannot achieve as high accuracy as the coupled method. We show numerical examples using a Nystr\"om method based on quadrature-by-expansion (QBX) with fast-multipole acceleration. We demonstrate the accuracy and efficiency of the solvers in both two and three dimensions with complex geometry.
Recent studies have shown that gradient descent (GD) can achieve improved generalization when its dynamics exhibits a chaotic behavior. However, to obtain the desired effect, the step-size should be chosen sufficiently large, a task which is problem dependent and can be difficult in practice. In this study, we incorporate a chaotic component to GD in a controlled manner, and introduce multiscale perturbed GD (MPGD), a novel optimization framework where the GD recursion is augmented with chaotic perturbations that evolve via an independent dynamical system. We analyze MPGD from three different angles: (i) By building up on recent advances in rough paths theory, we show that, under appropriate assumptions, as the step-size decreases, the MPGD recursion converges weakly to a stochastic differential equation (SDE) driven by a heavy-tailed L\'evy-stable process. (ii) By making connections to recently developed generalization bounds for heavy-tailed processes, we derive a generalization bound for the limiting SDE and relate the worst-case generalization error over the trajectories of the process to the parameters of MPGD. (iii) We analyze the implicit regularization effect brought by the dynamical regularization and show that, in the weak perturbation regime, MPGD introduces terms that penalize the Hessian of the loss function. Empirical results are provided to demonstrate the advantages of MPGD.
One common approach to detecting change-points is minimizing a cost function over possible numbers and locations of change-points. The framework includes several well-established procedures, such as the penalized likelihood and minimum description length. Such an approach requires finding the cost value repeatedly over different segments of the data set, which can be time-consuming when (i) the data sequence is long and (ii) obtaining the cost value involves solving a non-trivial optimization problem. This paper introduces a new sequential method (SE) that can be coupled with gradient descent (SeGD) and quasi-Newton's method (SeN) to find the cost value effectively. The core idea is to update the cost value using the information from previous steps without re-optimizing the objective function. The new method is applied to change-point detection in generalized linear models and penalized regression. Numerical studies show that the new approach can be orders of magnitude faster than the Pruned Exact Linear Time (PELT) method without sacrificing estimation accuracy.
When is heterogeneity in the composition of an autonomous robotic team beneficial and when is it detrimental? We investigate and answer this question in the context of a minimally viable model that examines the role of heterogeneous speeds in perimeter defense problems, where defenders share a total allocated speed budget. We consider two distinct problem settings and develop strategies based on dynamic programming and on local interaction rules. We present a theoretical analysis of both approaches and our results are extensively validated using simulations. Interestingly, our results demonstrate that the viability of heterogeneous teams depends on the amount of information available to the defenders. Moreover, our results suggest a universality property: across a wide range of problem parameters the optimal ratio of the speeds of the defenders remains nearly constant.
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