In this paper, we propose a dual-mixed formulation for stationary viscoplastic flows with yield, such as the Bingham or the Herschel-Bulkley flow. The approach is based on a Huber regularization of the viscosity term and a two-fold saddle point nonlinear operator equation for the resulting weak formulation. We provide the uniqueness of solutions for the continuous formulation and propose a discrete scheme based on Arnold-Falk-Winther finite elements. The discretization scheme yields a system of slantly differentiable nonlinear equations, for which a semismooth Newton algorithm is proposed and implemented. Local superlinear convergence of the method is also proved. Finally, we perform several numerical experiments in two and three dimensions to investigate the behavior and efficiency of the method.
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In this work, we consider fracture propagation in nearly incompressible and (fully) incompressible materials using a phase-field formulation. We use a mixed form of the elasticity equation to overcome volume locking effects and develop a robust, nonlinear and linear solver scheme and preconditioner for the resulting system. The coupled variational inequality system, which is solved monolithically, consists of three unknowns: displacements, pressure, and phase-field. Nonlinearities due to coupling, constitutive laws, and crack irreversibility are solved using a combined Newton algorithm for the nonlinearities in the partial differential equation and employing a primal-dual active set strategy for the crack irreverrsibility constraint. The linear system in each Newton step is solved iteratively with a flexible generalized minimal residual method (GMRES). The key contribution of this work is the development of a problem-specific preconditioner that leverages the saddle-point structure of the displacement and pressure variable. Four numerical examples in pure solids and pressure-driven fractures are conducted on uniformly and locally refined meshes to investigate the robustness of the solver concerning the Poisson ratio as well as the discretization and regularization parameters.
Kernel-based models such as kernel ridge regression and Gaussian processes are ubiquitous in machine learning applications for regression and optimization. It is well known that a serious downside for kernel-based models is the high computational cost; given a dataset of $n$ samples, the cost grows as $\mathcal{O}(n^3)$. Existing sparse approximation methods can yield a significant reduction in the computational cost, effectively reducing the real world cost down to as low as $\mathcal{O}(n)$ in certain cases. Despite this remarkable empirical success, significant gaps remain in the existing results for the analytical confidence bounds on the error due to approximation. In this work, we provide novel confidence intervals for the Nystr\"om method and the sparse variational Gaussian processes approximation method. Our confidence intervals lead to improved error bounds in both regression and optimization. We establish these confidence intervals using novel interpretations of the approximate (surrogate) posterior variance of the models.
A singularly perturbed parabolic problem of convection-diffusion type with a discontinuous initial condition is examined. An analytic function is identified which matches the discontinuity in the initial condition and also satisfies the homogenous parabolic differential equation associated with the problem. The difference between this analytical function and the solution of the parabolic problem is approximated numerically, using an upwind finite difference operator combined with an appropriate layer-adapted mesh. The numerical method is shown to be parameter-uniform. Numerical results are presented to illustrate the theoretical error bounds established in the paper.
In this work, a multirate in time approach resolving the different time scales of a convection-dominated transport and coupled fluid flow is developed and studied in view of goal-oriented error control by means of the Dual Weighted Residual (DWR) method. Key ingredients are an arbitrary degree discontinuous Galerkin time discretization of the underlying subproblems, an a posteriori error representation for the transport problem coupled with flow and its implementation using space-time tensor-product spaces. The error representation allows the separation of the temporal and spatial discretization error which serve as local error indicators for adaptive mesh refinement. The performance of the approach and its software implementation are studied by numerical convergence examples as well as an example of physical interest for convection-dominated transport.
We study ROUND-UFP and ROUND-SAP, two generalizations of the classical BIN PACKING problem that correspond to the unsplittable flow problem on a path (UFP) and the storage allocation problem (SAP), respectively. We are given a path with capacities on its edges and a set of tasks where for each task we are given a demand and a subpath. In ROUND-UFP, the goal is to find a packing of all tasks into a minimum number of copies (rounds) of the given path such that for each copy, the total demand of tasks on any edge does not exceed the capacity of the respective edge. In ROUND-SAP, the tasks are considered to be rectangles and the goal is to find a non-overlapping packing of these rectangles into a minimum number of rounds such that all rectangles lie completely below the capacity profile of the edges. We show that in contrast to BIN PACKING, both the problems do not admit an asymptotic polynomial-time approximation scheme (APTAS), even when all edge capacities are equal. However, for this setting, we obtain asymptotic $(2+\varepsilon)$-approximations for both problems. For the general case, we obtain an $O(\log\log n)$-approximation algorithm and an $O(\log\log\frac{1}{\delta})$-approximation under $(1+\delta)$-resource augmentation for both problems. For the intermediate setting of the no bottleneck assumption (i.e., the maximum task demand is at most the minimum edge capacity), we obtain absolute $12$- and asymptotic $(16+\varepsilon)$-approximation algorithms for ROUND-UFP and ROUND-SAP, respectively.
The main problem in the area of property testing is to understand which graph properties are \emph{testable}, which means that with constantly many queries to any input graph $G$, a tester can decide with good probability whether $G$ satisfies the property, or is far from satisfying the property. Testable properties are well understood in the dense model and in the bounded degree model, but little is known in sparse graph classes when graphs are allowed to have unbounded degree. This is the setting of the \emph{sparse model}. We prove that for any proper minor-closed class $\mathcal{G}$, any monotone property (i.e., any property that is closed under taking subgraphs) is testable for graphs from $\mathcal{G}$ in the sparse model. This extends a result of Czumaj and Sohler (FOCS'19), who proved it for monotone properties with finitely many obstructions. Our result implies for instance that for any integers $k$ and $t$, $k$-colorability of $K_t$-minor free graphs is testable in the sparse model. Elek recently proved that monotone properties of bounded degree graphs from minor-closed classes that are closed under disjoint union can be verified by an approximate proof labeling scheme in constant time. We show again that the assumption of bounded degree can be omitted in his result.
While many works exploiting an existing Lie group structure have been proposed for state estimation, in particular the Invariant Extended Kalman Filter (IEKF), few papers address the construction of a group structure that allows casting a given system into the framework of invariant filtering. In this paper we introduce a large class of systems encompassing most problems involving a navigating vehicle encountered in practice. For those systems we introduce a novel methodology that systematically provides a group structure for the state space, including vectors of the body frame such as biases. We use it to derive observers having properties akin to those of linear observers or filters. The proposed unifying and versatile framework encompasses all systems where IEKF has proved successful, improves state-of-the art "imperfect" IEKF for inertial navigation with sensor biases, and allows addressing novel examples, like GNSS antenna lever arm estimation.
In this paper, we characterize data-time tradeoffs of the proximal-gradient homotopy method used for solving linear inverse problems under sub-Gaussian measurements. Our results are sharp up to an absolute constant factor. We demonstrate that, in the absence of the strong convexity assumption, the proximal-gradient homotopy update can achieve a linear rate of convergence when the number of measurements is sufficiently large. Numerical simulations are provided to verify our theoretical results.
Minimax problems have gained tremendous attentions across the optimization and machine learning community recently. In this paper, we introduce a new quasi-Newton method for minimax problems, which we call $J$-symmetric quasi-Newton method. The method is obtained by exploiting the $J$-symmetric structure of the second-order derivative of the objective function in minimax problem. We show that the Hessian estimation (as well as its inverse) can be updated by a rank-2 operation, and it turns out that the update rule is a natural generalization of the classic Powell symmetric Broyden (PSB) method from minimization problems to minimax problems. In theory, we show that our proposed quasi-Newton algorithm enjoys local Q-superlinear convergence to a desirable solution under standard regularity conditions. Furthermore, we introduce a trust-region variant of the algorithm that enjoys global R-superlinear convergence. Finally, we present numerical experiments that verify our theory and show the effectiveness of our proposed algorithms compared to Broyden's method and the extragradient method on three classes of minimax problems.
Compact Approximate Taylor (CAT) methods for systems of conservation laws were introduced by Carrillo and Pares in 2019. These methods, based on a strategy that allows one to extend high-order Lax-Wendroff methods to nonlinear systems without using the Cauchy-Kovalevskaya procedure, have arbitrary even order of accuracy 2p and use (2p + 1)-point stencils, where p is an arbitrary positive integer. More recently in 2021 Carrillo, Macca, Pares, Russo and Zorio introduced a strategy to get rid of the spurious oscillations close to discontinuities produced by CAT methods. This strategy led to the so-called Adaptive CAT (ACAT) methods, in which the order of accuracy, and thus the width of the stencils, is adapted to the local smoothness of the solution. The goal of this paper is to extend CAT and ACAT methods to systems of balance laws. To do this, the source term is written as the derivative of its indefinite integral that is formally treated as a flux function. The well-balanced property of the methods is discussed and a variant that allows in principle to preserve any stationary solution is presented. The resulting methods are then applied to a number of systems going from a linear scalar conservation law to the 2D Euler equations with gravity, passing by the Burgers equations with source term and the 1D shallow water equations: the order and well-balanced properties are checked in several numerical tests.
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