Casting nonlocal problems in variational form and discretizing them with the finite element (FE) method facilitates the use of nonlocal vector calculus to prove well-posedeness, convergence, and stability of such schemes. Employing an FE method also facilitates meshing of complicated domain geometries and coupling with FE methods for local problems. However, nonlocal weak problems involve the computation of a double-integral, which is computationally expensive and presents several challenges. In particular, the inner integral of the variational form associated with the stiffness matrix is defined over the intersections of FE mesh elements with a ball of radius $\delta$, where $\delta$ is the range of nonlocal interaction. Identifying and parameterizing these intersections is a nontrivial computational geometry problem. In this work, we propose a quadrature technique where the inner integration is performed using quadrature points distributed over the full ball, without regard for how it intersects elements, and weights are computed based on the generalized moving least squares method. Thus, as opposed to all previously employed methods, our technique does not require element-by-element integration and fully circumvents the computation of element-ball intersections. This paper considers one- and two-dimensional implementations of piecewise linear continuous FE approximations, focusing on the case where the element size h and the nonlocal radius $\delta$ are proportional, as is typical of practical computations. When boundary conditions are treated carefully and the outer integral of the variational form is computed accurately, the proposed method is asymptotically compatible in the limit of $h \sim \delta \to 0$, featuring at least first-order convergence in L^2 for all dimensions, using both uniform and nonuniform grids.
相關內容
Integration：Integration, the VLSI Journal。
Explanation：集成，VLSI雜志。
Publisher：Elsevier。
SIT：
We introduce and analyze various Regularized Combined Field Integral Equations (CFIER) formulations of time-harmonic Navier equations in media with piece-wise constant material properties. These formulations can be derived systematically starting from suitable coercive approximations of Dirichlet-to-Neumann operators (DtN), and we present a periodic pseudodifferential calculus framework within which the well posedness of CIER formulations can be established. We also use the DtN approximations to derive and analyze Optimized Schwarz (OS) methods for the solution of elastodynamics transmission problems. The pseudodifferential calculus we develop in this paper relies on careful singularity splittings of the kernels of Navier boundary integral operators which is also the basis of high-order Nystr\"om quadratures for their discretizations. Based on these high-order discretizations we investigate the rate of convergence of iterative solvers applied to CFIER and OS formulations of scattering and transmission problems. We present a variety of numerical results that illustrate that the CFIER methodology leads to important computational savings over the classical CFIE one, whenever iterative solvers are used for the solution of the ensuing discretized boundary integral equations. Finally, we show that the OS methods are competitive in the high-frequency high-contrast regime.
We employ kernel-based approaches that use samples from a probability distribution to approximate a Kolmogorov operator on a manifold. The self-tuning variable-bandwidth kernel method [Berry & Harlim, Appl. Comput. Harmon. Anal., 40(1):68--96, 2016] computes a large, sparse matrix that approximates the differential operator. Here, we use the eigendecomposition of the discretization to (i) invert the operator, solving a differential equation, and (ii) represent gradient vector fields on the manifold. These methods only require samples from the underlying distribution and, therefore, can be applied in high dimensions or on geometrically complex manifolds when spatial discretizations are not available. We also employ an efficient $k$-$d$ tree algorithm to compute the sparse kernel matrix, which is a computational bottleneck.
Vision Transformers (ViT) serve as powerful vision models. Unlike convolutional neural networks, which dominated vision research in previous years, vision transformers enjoy the ability to capture long-range dependencies in the data. Nonetheless, an integral part of any transformer architecture, the self-attention mechanism, suffers from high latency and inefficient memory utilization, making it less suitable for high-resolution input images. To alleviate these shortcomings, hierarchical vision models locally employ self-attention on non-interleaving windows. This relaxation reduces the complexity to be linear in the input size; however, it limits the cross-window interaction, hurting the model performance. In this paper, we propose a new shift-invariant local attention layer, called query and attend (QnA), that aggregates the input locally in an overlapping manner, much like convolutions. The key idea behind QnA is to introduce learned queries, which allow fast and efficient implementation. We verify the effectiveness of our layer by incorporating it into a hierarchical vision transformer model. We show improvements in speed and memory complexity while achieving comparable accuracy with state-of-the-art models. Finally, our layer scales especially well with window size, requiring up-to x10 less memory while being up-to x5 faster than existing methods. The code is publicly available at \url{//github.com/moabarar/qna}.
Continuous-time measurements are instrumental for a multitude of tasks in quantum engineering and quantum control, including the estimation of dynamical parameters of open quantum systems monitored through the environment. However, such measurements do not extract the maximum amount of information available in the output state, so finding alternative optimal measurement strategies is a major open problem. In this paper we solve this problem in the setting of discrete-time input-output quantum Markov chains. We present an efficient algorithm for optimal estimation of one-dimensional dynamical parameters which consists of an iterative procedure for updating a `measurement filter' operator and determining successive measurement bases for the output units. A key ingredient of the scheme is the use of a coherent quantum absorber as a way to post-process the output after the interaction with the system. This is designed adaptively such that the joint system and absorber stationary state is pure at a reference parameter value. The scheme offers an exciting prospect for optimal continuous-time adaptive measurements, but more work is needed to find realistic practical implementations.
Machine learning and computational intelligence technologies gain more and more popularity as possible solution for issues related to the power grid. One of these issues, the power flow calculation, is an iterative method to compute the voltage magnitudes of the power grid's buses from power values. Machine learning and, especially, artificial neural networks were successfully used as surrogates for the power flow calculation. Artificial neural networks highly rely on the quality and size of the training data, but this aspect of the process is apparently often neglected in the works we found. However, since the availability of high quality historical data for power grids is limited, we propose the Correlation Sampling algorithm. We show that this approach is able to cover a larger area of the sampling space compared to different random sampling algorithms from the literature and a copula-based approach, while at the same time inter-dependencies of the inputs are taken into account, which, from the other algorithms, only the copula-based approach does.
In this paper we get error bounds for fully discrete approximations of infinite horizon problems via the dynamic programming approach. It is well known that considering a time discretization with a positive step size $h$ an error bound of size $h$ can be proved for the difference between the value function (viscosity solution of the Hamilton-Jacobi-Bellman equation corresponding to the infinite horizon) and the value function of the discrete time problem. However, including also a spatial discretization based on elements of size $k$ an error bound of size $O(k/h)$ can be found in the literature for the error between the value functions of the continuous problem and the fully discrete problem. In this paper we revise the error bound of the fully discrete method and prove, under similar assumptions to those of the time discrete case, that the error of the fully discrete case is in fact $O(h+k)$ which gives first order in time and space for the method. This error bound matches the numerical experiments of many papers in the literature in which the behaviour $1/h$ from the bound $O(k/h)$ have not been observed.
Given a set $P$ of $n$ points in the plane, the $k$-center problem is to find $k$ congruent disks of minimum possible radius such that their union covers all the points in $P$. The $2$-center problem is a special case of the $k$-center problem that has been extensively studied in the recent past \cite{CAHN,HT,SH}. In this paper, we consider a generalized version of the $2$-center problem called \textit{proximity connected} $2$-center (PCTC) problem. In this problem, we are also given a parameter $\delta\geq 0$ and we have the additional constraint that the distance between the centers of the disks should be at most $\delta$. Note that when $\delta=0$, the PCTC problem is reduced to the $1$-center(minimum enclosing disk) problem and when $\delta$ tends to infinity, it is reduced to the $2$-center problem. The PCTC problem first appeared in the context of wireless networks in 1992 \cite{ACN0}, but obtaining a nontrivial deterministic algorithm for the problem remained open. In this paper, we resolve this open problem by providing a deterministic $O(n^2\log n)$ time algorithm for the problem.
We introduce a filtering technique for Discontinuous Galerkin approximations of hyperbolic problems. Following an approach already proposed for the Hamilton-Jacobi equations by other authors, we aim at reducing the spurious oscillations that arise in presence of discontinuities when high order spatial discretizations are employed. This goal is achieved using a filter function that keeps the high order scheme when the solution is regular and switches to a monotone low order approximation if it is not. The method has been implemented in the framework of the $deal.II$ numerical library, whose mesh adaptation capabilities are also used to reduce the region in which the low order approximation is used. A number of numerical experiments demonstrate the potential of the proposed filtering technique.
In the storied Colonel Blotto game, two colonels allocate $a$ and $b$ troops, respectively, to $k$ distinct battlefields. A colonel wins a battle if they assign more troops to that particular battle, and each colonel seeks to maximize their total number of victories. Despite the problem's formulation in 1921, the first polynomial-time algorithm to compute Nash equilibrium (NE) strategies for this game was discovered only quite recently. In 2016, \citep{ahmadinejad_dehghani_hajiaghayi_lucier_mahini_seddighin_2019} formulated a breakthrough algorithm to compute NE strategies for the Colonel Blotto game\footnote{To the best of our knowledge, the algorithm from \citep{ahmadinejad_dehghani_hajiaghayi_lucier_mahini_seddighin_2019} has computational complexity $O(k^{14}\max\{a,b\}^{13})$}, receiving substantial media coverage (e.g. \citep{Insider}, \citep{NSF}, \citep{ScienceDaily}). In this work, we present the first known $\epsilon$-approximation algorithm to compute NE strategies in the two-player Colonel Blotto game in runtime $\widetilde{O}(\epsilon^{-4} k^8 \max\{a,b\}^2)$ for arbitrary settings of these parameters. Moreover, this algorithm computes approximate coarse correlated equilibrium strategies in the multiplayer (continuous and discrete) Colonel Blotto game (when there are $\ell > 2$ colonels) with runtime $\widetilde{O}(\ell \epsilon^{-4} k^8 n^2 + \ell^2 \epsilon^{-2} k^3 n (n+k))$, where $n$ is the maximum troop count. Before this work, no polynomial-time algorithm was known to compute exact or approximate equilibrium (in any sense) strategies for multiplayer Colonel Blotto with arbitrary parameters. Our algorithm computes these approximate equilibria by a novel (to the author's knowledge) sampling technique with which we implicitly perform multiplicative weights update over the exponentially many strategies available to each player.
The numerical solution of singular eigenvalue problems is complicated by the fact that small perturbations of the coefficients may have an arbitrarily bad effect on eigenvalue accuracy. However, it has been known for a long time that such perturbations are exceptional and standard eigenvalue solvers, such as the QZ algorithm, tend to yield good accuracy despite the inevitable presence of roundoff error. Recently, Lotz and Noferini quantified this phenomenon by introducing the concept of $\delta$-weak eigenvalue condition numbers. In this work, we consider singular quadratic eigenvalue problems and two popular linearizations. Our results show that a correctly chosen linearization increases $\delta$-weak eigenvalue condition numbers only marginally, justifying the use of these linearizations in numerical solvers also in the singular case. We propose a very simple but often effective algorithm for computing well-conditioned eigenvalues of a singular quadratic eigenvalue problems by adding small random perturbations to the coefficients. We prove that the eigenvalue condition number is, with high probability, a reliable criterion for detecting and excluding spurious eigenvalues created from the singular part.
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