We present a new method for constructing valid covariance functions of Gaussian processes over irregular nonconvex spatial domains such as water bodies, where the geodesic distance agrees with the Euclidean distance only for some pairs of points. Standard covariance functions based on geodesic distances are not positive definite on such domains. Using a visibility graph on the domain, we use the graphical method of "covariance selection" to propose a class of covariance functions that preserve Euclidean-based covariances between points that are connected through the domain. The proposed method preserves the partially Euclidean nature of the intrinsic geometry on the domain while maintaining validity (positive definiteness) and marginal stationarity over the entire parameter space, properties which are not always fulfilled by existing approaches to construct covariance functions on nonconvex domains. We provide useful approximations to improve computational efficiency, resulting in a scalable algorithm. We evaluate the performance of competing state-of-the-art methods using simulation studies on a contrived nonconvex domain. The method is applied to data regarding acidity levels in the Chesapeake Bay, showing its potential for ecological monitoring in real-world spatial applications on irregular domains.
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We consider a one-dimensional singularly perturbed 4th order problem with the additional feature of a shift term. An expansion into a smooth term, boundary layers and an inner layer yields a formal solution decomposition, and together with a stability result we have estimates for the subsequent numerical analysis. With classical layer adapted meshes we present a numerical method, that achieves supercloseness and optimal convergence orders in the associated energy norm. We also consider coarser meshes in view of the weak layers. Some numerical examples conclude the paper and support the theory.
Information inequalities appear in many database applications such as query output size bounds, query containment, and implication between data dependencies. Recently Khamis et al. proposed to study the algorithmic aspects of information inequalities, including the information inequality problem: decide whether a linear inequality over entropies of random variables is valid. While the decidability of this problem is a major open question, applications often involve only inequalities that adhere to specific syntactic forms linked to useful semantic invariance properties. This paper studies the information inequality problem in different syntactic and semantic scenarios that arise from database applications. Focusing on the boundary between tractability and intractability, we show that the information inequality problem is coNP-complete if restricted to normal polymatroids, and in polynomial time if relaxed to monotone functions. We also examine syntactic restrictions related to query output size bounds, and provide an alternative proof, through monotone functions, for the polynomial-time computability of the entropic bound over simple sets of degree constraints.
We give a short survey of recent results on sparse-grid linear algorithms of approximate recovery and integration of functions possessing a unweighted or weighted Sobolev mixed smoothness based on their sampled values at a certain finite set. Some of them are extended to more general cases.
The idea of decision-aware model learning, that models should be accurate where it matters for decision-making, has gained prominence in model-based reinforcement learning. While promising theoretical results have been established, the empirical performance of algorithms leveraging a decision-aware loss has been lacking, especially in continuous control problems. In this paper, we present a study on the necessary components for decision-aware reinforcement learning models and we showcase design choices that enable well-performing algorithms. To this end, we provide a theoretical and empirical investigation into prominent algorithmic ideas in the field. We highlight that empirical design decisions established in the MuZero line of works are vital to achieving good performance for related algorithms, and we showcase differences in behavior between different instantiations of value-aware algorithms in stochastic environments. Using these insights, we propose the Latent Model-Based Decision-Aware Actor-Critic framework ($\lambda$-AC) for decision-aware model-based reinforcement learning in continuous state-spaces and highlight important design choices in different environments.
Point source localisation is generally modelled as a Lasso-type problem on measures. However, optimisation methods in non-Hilbert spaces, such as the space of Radon measures, are much less developed than in Hilbert spaces. Most numerical algorithms for point source localisation are based on the Frank-Wolfe conditional gradient method, for which ad hoc convergence theory is developed. We develop extensions of proximal-type methods to spaces of measures. This includes forward-backward splitting, its inertial version, and primal-dual proximal splitting. Their convergence proofs follow standard patterns. We demonstrate their numerical efficacy.
We propose a method to modify a polygonal mesh in order to fit the zero-isoline of a level set function by extending a standard body-fitted strategy to a tessellation with arbitrarily-shaped elements. The novel level set-fitted approach, in combination with a Discontinuous Galerkin finite element approximation, provides an ideal setting to model physical problems characterized by embedded or evolving complex geometries, since it allows skipping any mesh post-processing in terms of grid quality. The proposed methodology is firstly assessed on the linear elasticity equation, by verifying the approximation capability of the level set-fitted approach when dealing with configurations with heterogeneous material properties. Successively, we combine the level set-fitted methodology with a minimum compliance topology optimization technique, in order to deliver optimized layouts exhibiting crisp boundaries and reliable mechanical performances. An extensive numerical test campaign confirms the effectiveness of the proposed method.
Ordinary state-based peridynamic (OSB-PD) models have an unparalleled capability to simulate crack propagation phenomena in solids with arbitrary Poisson's ratio. However, their non-locality also leads to prohibitively high computational cost. In this paper, a fast solution scheme for OSB-PD models based on matrix operation is introduced, with which, the graphics processing units (GPUs) are used to accelerate the computation. For the purpose of comparison and verification, a commonly used solution scheme based on loop operation is also presented. An in-house software is developed in MATLAB. Firstly, the vibration of a cantilever beam is solved for validating the loop- and matrix-based schemes by comparing the numerical solutions to those produced by a FEM software. Subsequently, two typical dynamic crack propagation problems are simulated to illustrate the effectiveness of the proposed schemes in solving dynamic fracture problems. Finally, the simulation of the Brokenshire torsion experiment is carried out by using the matrix-based scheme, and the similarity in the shapes of the experimental and numerical broken specimens further demonstrates the ability of the proposed approach to deal with 3D non-planar fracture problems. In addition, the speed-up of the matrix-based scheme with respect to the loop-based scheme and the performance of the GPU acceleration are investigated. The results emphasize the high computational efficiency of the matrix-based implementation scheme.
We examine a stochastic formulation for data-driven optimization wherein the decision-maker is not privy to the true distribution, but has knowledge that it lies in some hypothesis set and possesses a historical data set, from which information about it can be gleaned. We define a prescriptive solution as a decision rule mapping such a data set to decisions. As there does not exist prescriptive solutions that are generalizable over the entire hypothesis set, we define out-of-sample optimality as a local average over a neighbourhood of hypotheses, and averaged over the sampling distribution. We prove sufficient conditions for local out-of-sample optimality, which reduces to functions of the sufficient statistic of the hypothesis family. We present an optimization problem that would solve for such an out-of-sample optimal solution, and does so efficiently by a combination of sampling and bisection search algorithms. Finally, we illustrate our model on the newsvendor model, and find strong performance when compared against alternatives in the literature. There are potential implications of our research on end-to-end learning and Bayesian optimization.
Parametric mathematical models such as parameterizations of partial differential equations with random coefficients have received a lot of attention within the field of uncertainty quantification. The model uncertainties are often represented via a series expansion in terms of the parametric variables. In practice, this series expansion needs to be truncated to a finite number of terms, introducing a dimension truncation error to the numerical simulation of a parametric mathematical model. There have been several studies of the dimension truncation error corresponding to different models of the input random field in recent years, but many of these analyses have been carried out within the context of numerical integration. In this paper, we study the $L^2$ dimension truncation error of the parametric model problem. Estimates of this kind arise in the assessment of the dimension truncation error for function approximation in high dimensions. In addition, we show that the dimension truncation error rate is invariant with respect to certain transformations of the parametric variables. Numerical results are presented which showcase the sharpness of the theoretical results.
Quantum neural networks (QNNs) and quantum kernels stand as prominent figures in the realm of quantum machine learning, poised to leverage the nascent capabilities of near-term quantum computers to surmount classical machine learning challenges. Nonetheless, the training efficiency challenge poses a limitation on both QNNs and quantum kernels, curbing their efficacy when applied to extensive datasets. To confront this concern, we present a unified approach: coreset selection, aimed at expediting the training of QNNs and quantum kernels by distilling a judicious subset from the original training dataset. Furthermore, we analyze the generalization error bounds of QNNs and quantum kernels when trained on such coresets, unveiling the comparable performance with those training on the complete original dataset. Through systematic numerical simulations, we illuminate the potential of coreset selection in expediting tasks encompassing synthetic data classification, identification of quantum correlations, and quantum compiling. Our work offers a useful way to improve diverse quantum machine learning models with a theoretical guarantee while reducing the training cost.
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