We present a generic framework for gradient reconstruction schemes on unstructured meshes using the notion of a dyadic sum-vector product. The proposed formulation reconstructs centroidal gradients of a scalar from its directional derivatives along specific directions in a suitably defined neighbourhood. We show that existing gradient reconstruction schemes can be encompassed within this framework by a suitable choice of the geometric vectors that define the dyadic sum tensor. The proposed framework also allows us to re-interpret certain hybrid schemes, which might not be derivable through traditional routes. Additionally, a generalization of flexible gradient schemes is proposed that can be employed to enhance the robustness of consistent gradient schemes without compromising on the accuracy of the computed gradients.
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We consider the equilibrium equations for a linearized Cosserat material. We identify their structure in terms of a differential complex, which is isomorphic to six copies of the de Rham complex through an algebraic isomorphism. Moreover, we show how the Cosserat materials can be analyzed by inheriting results from linearized elasticity. Both perspectives give rise to mixed finite element methods, which we refer to as strongly and weaky coupled, respectively. We prove convergence of both classes of methods, with particular attention to improved convergence rate estimates, and stability in the limit of vanishing Cosserat material parameters. The theoretical results are fully reflected in the actual performance of the methods, as shown by the numerical verifications.
We formulate a uniform tail bound for empirical processes indexed by a class of functions, in terms of the individual deviations of the functions rather than the worst-case deviation in the considered class. The tail bound is established by introducing an initial "deflation" step to the standard generic chaining argument. The resulting tail bound is the sum of the complexity of the "deflated function class" in terms of a generalization of Talagrand's $\gamma$ functional, and the deviation of the function instance, both of which are formulated based on the natural seminorm induced by the corresponding Cram\'{e}r functions. We also provide certain approximations for the mentioned seminorm when the function class lies in a given (exponential type) Orlicz space, that can be used to make the complexity term and the deviation term more explicit.
Homogeneous normalized random measures with independent increments (hNRMIs) represent a broad class of Bayesian nonparametric priors and thus are widely used. In this paper, we obtain the strong law of large numbers, the central limit theorem and the functional central limit theorem of hNRMIs when the concentration parameter $a$ approaches infinity. To quantify the convergence rate of the obtained central limit theorem, we further study the Berry-Esseen bound, which turns out to be of the form $O \left( \frac{1}{\sqrt{a}}\right)$. As an application of the central limit theorem, we present the functional delta method, which can be employed to obtain the limit of the quantile process of hNRMIs. As an illustration of the central limit theorems, we demonstrate the convergence numerically for the Dirichlet processes and the normalized inverse Gaussian processes with various choices of the concentration parameters.
The subject of this work is an adaptive stochastic Galerkin finite element method for parametric or random elliptic partial differential equations, which generates sparse product polynomial expansions with respect to the parametric variables of solutions. For the corresponding spatial approximations, an independently refined finite element mesh is used for each polynomial coefficient. The method relies on multilevel expansions of input random fields and achieves error reduction with uniform rate. In particular, the saturation property for the refinement process is ensured by the algorithm. The results are illustrated by numerical experiments, including cases with random fields of low regularity.
We propose a material design method via gradient-based optimization on compositions, overcoming the limitations of traditional methods: exhaustive database searches and conditional generation models. It optimizes inputs via backpropagation, aligning the model's output closely with the target property and facilitating the discovery of unlisted materials and precise property determination. Our method is also capable of adaptive optimization under new conditions without retraining. Applying to exploring high-Tc superconductors, we identified potential compositions beyond existing databases and discovered new hydrogen superconductors via conditional optimization. This method is versatile and significantly advances material design by enabling efficient, extensive searches and adaptability to new constraints.
Lattices are architected metamaterials whose properties strongly depend on their geometrical design. The analogy between lattices and graphs enables the use of graph neural networks (GNNs) as a faster surrogate model compared to traditional methods such as finite element modelling. In this work, we generate a big dataset of structure-property relationships for strut-based lattices. The dataset is made available to the community which can fuel the development of methods anchored in physical principles for the fitting of fourth-order tensors. In addition, we present a higher-order GNN model trained on this dataset. The key features of the model are (i) SE(3) equivariance, and (ii) consistency with the thermodynamic law of conservation of energy. We compare the model to non-equivariant models based on a number of error metrics and demonstrate its benefits in terms of predictive performance and reduced training requirements. Finally, we demonstrate an example application of the model to an architected material design task. The methods which we developed are applicable to fourth-order tensors beyond elasticity such as piezo-optical tensor etc.
We present the design and implementation of a tool for semi-automatic verification of functional specifications of operating system modules. Such verification tasks are traditionally done in interactive theorem provers, where the functionalities of the module are specified at abstract and concrete levels using data such as structures, algebraic datatypes, arrays, maps and so on. In this work, we provide encodings to SMT for these commonly occurring data types. This allows verification conditions to be reduced into a form suitable for SMT solvers. The use of SMT solvers combined with a tactic language allows semi-automatic verification of the specification. We apply the tool to verify functional specification for key parts of the uC-OS/II operating system, based on earlier work giving full verification of the system in Coq. We demonstrate a large reduction in the amount of human effort due to increased level of automation.
We present a novel framework for the development of fourth-order lattice Boltzmann schemes to tackle multidimensional nonlinear systems of conservation laws. Our numerical schemes preserve two fundamental characteristics inherent in classical lattice Boltzmann methods: a local relaxation phase and a transport phase composed of elementary shifts on a Cartesian grid. Achieving fourth-order accuracy is accomplished through the composition of second-order time-symmetric basic schemes utilizing rational weights. This enables the representation of the transport phase in terms of elementary shifts. Introducing local variations in the relaxation parameter during each stage of relaxation ensures the entropic nature of the schemes. This not only enhances stability in the long-time limit but also maintains fourth-order accuracy. To validate our approach, we conduct comprehensive testing on scalar equations and systems in both one and two spatial dimensions.
We present a finite element approach for diffusion problems with thermal fluctuations based on a fluctuating hydrodynamics model. The governing transport equations are stochastic partial differential equations with a fluctuating forcing term. We propose a discrete formulation of the stochastic forcing term that has the correct covariance matrix up to a standard discretization error. Furthermore, to obtain a numerical solution with spatial correlations that converge to those of the continuum equation, we derive a linear mapping to transform the finite element solution into an equivalent discrete solution that is free from the artificial correlations introduced by the spatial discretization. The method is validated by applying it to two diffusion problems: a second-order diffusion equation and a fourth-order diffusion equation. The theoretical (continuum) solution to the first case presents spatially decorrelated fluctuations, while the second case presents fluctuations correlated over a finite length. In both cases, the numerical solution presents a structure factor that approximates well the continuum one.
Incomplete factorizations have long been popular general-purpose algebraic preconditioners for solving large sparse linear systems of equations. Guaranteeing the factorization is breakdown free while computing a high quality preconditioner is challenging. A resurgence of interest in using low precision arithmetic makes the search for robustness more urgent and tougher. In this paper, we focus on symmetric positive definite problems and explore a number of approaches: a look-ahead strategy to anticipate break down as early as possible, the use of global shifts, and a modification of an idea developed in the field of numerical optimization for the complete Cholesky factorization of dense matrices. Our numerical simulations target highly ill-conditioned sparse linear systems with the goal of computing the factors in half precision arithmetic and then achieving double precision accuracy using mixed precision refinement.
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