We propose a novel variant of the Localized Orthogonal Decomposition (LOD) method for time-harmonic scattering problems of Helmholtz type with high wavenumber $\kappa$. On a coarse mesh of width $H$, the proposed method identifies local finite element source terms that yield rapidly decaying responses under the solution operator. They can be constructed to high accuracy from independent local snapshot solutions on patches of width $\ell H$ and are used as problem-adapted basis functions in the method. In contrast to the classical LOD and other state-of-the-art multi-scale methods, the localization error decays super-exponentially as the oversampling parameter $\ell$ is increased. This implies that optimal convergence is observed under the substantially relaxed oversampling condition $\ell \gtrsim (\log \tfrac{\kappa}{H})^{(d-1)/d}$ with $d$ denoting the spatial dimension. Numerical experiments demonstrate the significantly improved offline and online performance of the method also in the case of heterogeneous media and perfectly matched layers.
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We introduce the Conic Blackwell Algorithm$^+$ (CBA$^+$) regret minimizer, a new parameter- and scale-free regret minimizer for general convex sets. CBA$^+$ is based on Blackwell approachability and attains $O(\sqrt{T})$ regret. We show how to efficiently instantiate CBA$^+$ for many decision sets of interest, including the simplex, $\ell_{p}$ norm balls, and ellipsoidal confidence regions in the simplex. Based on CBA$^+$, we introduce SP-CBA$^+$, a new parameter-free algorithm for solving convex-concave saddle-point problems, which achieves a $O(1/\sqrt{T})$ ergodic rate of convergence. In our simulations, we demonstrate the wide applicability of SP-CBA$^+$ on several standard saddle-point problems, including matrix games, extensive-form games, distributionally robust logistic regression, and Markov decision processes. In each setting, SP-CBA$^+$ achieves state-of-the-art numerical performance, and outperforms classical methods, without the need for any choice of step sizes or other algorithmic parameters.
The aim of this paper is to propose an efficient adaptive finite element method for eigenvalue problems based on the multilevel correction scheme and inverse power method. This method involves solving associated boundary value problems on each adaptive partitions and very low dimensional eigenvalue problems on some special meshes which are controlled by the proposed algorithm. Since we Hence the efficiency of solving eigenvalue problems can be improved to be similar to the adaptive finite element method for the associated boundary value problems. The convergence and optimal complexity is theoretically verified and numerically demonstrated.
Accurate solutions to inverse supersonic compressible flow problems are often required for designing specialized aerospace vehicles. In particular, we consider the problem where we have data available for density gradients from Schlieren photography as well as data at the inflow and part of wall boundaries. These inverse problems are notoriously difficult and traditional methods may not be adequate to solve such ill-posed inverse problems. To this end, we employ the physics-informed neural networks (PINNs) and its extended version, extended PINNs (XPINNs), where domain decomposition allows deploying locally powerful neural networks in each subdomain, which can provide additional expressivity in subdomains, where a complex solution is expected. Apart from the governing compressible Euler equations, we also enforce the entropy conditions in order to obtain viscosity solutions. Moreover, we enforce positivity conditions on density and pressure. We consider inverse problems involving two-dimensional expansion waves, two-dimensional oblique and bow shock waves. We compare solutions obtained by PINNs and XPINNs and invoke some theoretical results that can be used to decide on the generalization errors of the two methods.
Deploying reconfigurable intelligent surface (RIS) to enhance wireless transmission is a promising approach. In this paper, we investigate large-scale multi-RIS-assisted multi-cell systems, where multiple RISs are deployed in each cell. Different from the full-buffer scenario, the mutual interference in our system is not known a priori, and for this reason we apply the load coupling model to analyze this system. The objective is to minimize the total resource consumption subject to user demand requirement by optimizing the reflection coefficients in the cells. The cells are highly coupled and the overall problem is non-convex. To tackle this, we first investigate the single-cell case with given interference, and propose a low-complexity algorithm based on the Majorization-Minimization method to obtain a locally optimal solution. Then, we embed this algorithm into an algorithmic framework for the overall multi-cell problem, and prove its feasibility and convergence to a solution that is at least locally optimal. Simulation results demonstrate the benefit of RIS in time-frequency resource utilization in the multi-cell system.
We study Bayesian data assimilation (filtering) for time-evolution PDEs, for which the underlying forward problem may be very unstable or ill-posed. Such PDEs, which include the Navier-Stokes equations of fluid dynamics, are characterized by a high sensitivity of solutions to perturbations of the initial data, a lack of rigorous global well-posedness results as well as possible non-convergence of numerical approximations. Under very mild and readily verifiable general hypotheses on the forward solution operator of such PDEs, we prove that the posterior measure expressing the solution of the Bayesian filtering problem is stable with respect to perturbations of the noisy measurements, and we provide quantitative estimates on the convergence of approximate Bayesian filtering distributions computed from numerical approximations. For the Navier-Stokes equations, our results imply uniform stability of the filtering problem even at arbitrarily small viscosity, when the underlying forward problem may become ill-posed, as well as the compactness of numerical approximants in a suitable metric on time-parametrized probability measures.
The analysis of structure-preserving numerical methods for the Poisson--Nernst--Planck (PNP) system has attracted growing interests in recent years. In this work, we provide an optimal rate convergence analysis and error estimate for finite difference schemes based on the Slotboom reformulation. Different options of mobility average at the staggered mesh points are considered in the finite-difference spatial discretization, such as the harmonic mean, geometric mean, arithmetic mean, and entropic mean. A semi-implicit temporal discretization is applied, which in turn results in a non-constant coefficient, positive-definite linear system at each time step. A higher order asymptotic expansion is applied in the consistency analysis, and such a higher order consistency estimate is necessary to control the discrete maximum norm of the concentration variables. In convergence estimate, the harmonic mean for the mobility average, which turns out to bring lots of convenience in the theoretical analysis, is taken for simplicity, while other options of mobility average would also lead to the desired error estimate, with more technical details involved. As a result, an optimal rate convergence analysis on concentrations, electric potential, and ionic fluxes is derived, which is the first such results for the structure-preserving numerical schemes based on the Slotboom reformulation. It is remarked that the convergence analysis leads to a theoretical justification of the conditional energy dissipation analysis, which relies on the maximum norm bounds of the concentration and the gradient of the electric potential. Some numerical results are also presented to demonstrate the accuracy and structure-preserving performance of the associated schemes.
Group synchronization asks to recover group elements from their pairwise measurements. It has found numerous applications across various scientific disciplines. In this work, we focus on orthogonal and permutation group synchronization which are widely used in computer vision such as object matching and structure from motion. Among many available approaches, the spectral methods have enjoyed great popularity due to their efficiency and convenience. We will study the performance guarantees of the spectral methods in solving these two synchronization problems by investigating how well the computed eigenvectors approximate each group element individually. We establish our theory by applying the recent popular~\emph{leave-one-out} technique and derive a~\emph{block-wise} performance bound for the recovery of each group element via eigenvectors. In particular, for orthogonal group synchronization, we obtain a near-optimal performance bound for the group recovery in presence of additive Gaussian noise. For permutation group synchronization under random corruption, we show that the widely-used two-step procedure (spectral method plus rounding) can recover all the group elements exactly if the SNR (signal-to-noise ratio) is close to the information theoretical limit. Our numerical experiments confirm our theory and indicate a sharp phase transition for the exact group recovery.
We introduce the problem of determining if the mode of the output distribution of a quantum circuit (given as a black-box) is larger than a given threshold, named HighDist, and a similar problem based on the absolute values of the amplitudes, named HighAmp. We design quantum algorithms for promised versions of these problems whose space complexities are logarithmic in the size of the domain of the distribution, but the query complexities are independent. Using these, we further design algorithms to estimate the largest probability and the largest amplitude among the output distribution of a quantum black-box. All of these allow us to improve the query complexity of a few recently studied problems, namely, $k$-distinctness and its gapped version, estimating the largest frequency in an array, estimating the min-entropy of a distribution, and the non-linearity of a Boolean function, in the $\tilde{O}(1)$-qubits scenario. The time-complexities of almost all of our algorithms have a small overhead over their query complexities making them efficiently implementable on currently available quantum backends.
Graph neural network (GNN) has shown superior performance in dealing with graphs, which has attracted considerable research attention recently. However, most of the existing GNN models are primarily designed for graphs in Euclidean spaces. Recent research has proven that the graph data exhibits non-Euclidean latent anatomy. Unfortunately, there was rarely study of GNN in non-Euclidean settings so far. To bridge this gap, in this paper, we study the GNN with attention mechanism in hyperbolic spaces at the first attempt. The research of hyperbolic GNN has some unique challenges: since the hyperbolic spaces are not vector spaces, the vector operations (e.g., vector addition, subtraction, and scalar multiplication) cannot be carried. To tackle this problem, we employ the gyrovector spaces, which provide an elegant algebraic formalism for hyperbolic geometry, to transform the features in a graph; and then we propose the hyperbolic proximity based attention mechanism to aggregate the features. Moreover, as mathematical operations in hyperbolic spaces could be more complicated than those in Euclidean spaces, we further devise a novel acceleration strategy using logarithmic and exponential mappings to improve the efficiency of our proposed model. The comprehensive experimental results on four real-world datasets demonstrate the performance of our proposed hyperbolic graph attention network model, by comparisons with other state-of-the-art baseline methods.
Stochastic gradient Markov chain Monte Carlo (SGMCMC) has become a popular method for scalable Bayesian inference. These methods are based on sampling a discrete-time approximation to a continuous time process, such as the Langevin diffusion. When applied to distributions defined on a constrained space, such as the simplex, the time-discretisation error can dominate when we are near the boundary of the space. We demonstrate that while current SGMCMC methods for the simplex perform well in certain cases, they struggle with sparse simplex spaces; when many of the components are close to zero. However, most popular large-scale applications of Bayesian inference on simplex spaces, such as network or topic models, are sparse. We argue that this poor performance is due to the biases of SGMCMC caused by the discretization error. To get around this, we propose the stochastic CIR process, which removes all discretization error and we prove that samples from the stochastic CIR process are asymptotically unbiased. Use of the stochastic CIR process within a SGMCMC algorithm is shown to give substantially better performance for a topic model and a Dirichlet process mixture model than existing SGMCMC approaches.
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