We propose a two-step Newton's method for refining an approximation of a singular zero whose deflation process terminates after one step, also known as a deflation-one singularity. Given an isolated singular zero of a square analytic system, our algorithm exploits an invertible linear operator obtained by combining the Jacobian and a projection of the Hessian in the direction of the kernel of the Jacobian. We prove the quadratic convergence of the two-step Newton method when it is applied to an approximation of a deflation-one singular zero. Also, the algorithm requires a smaller size of matrices than the existing methods, making it more efficient. We demonstrate examples and experiments to show the efficiency of the method.
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Peridynamics (PD), as a nonlocal theory, is well-suited for solving problems with discontinuities, such as cracks. However, the nonlocal effect of peridynamics makes it computationally expensive for dynamic fracture problems in large-scale engineering applications. As an alternative, this study proposes a multi-time-step (MTS) coupling model of PD and classical continuum mechanics (CCM) based on the Arlequin framework. Peridynamics is applied to the fracture domain of the structure, while continuum mechanics is applied to the rest of the structure. The MTS method enables the peridynamic model to be solved at a small time step and the continuum mechanical model is solved at a larger time step. Consequently, higher computational efficiency is achieved for the fracture domain of the structure while ensuring computational accuracy, and this coupling method can be easily applied to large-scale engineering fracture problems.
This work explores the dimension reduction problem for Bayesian nonparametric regression and density estimation. More precisely, we are interested in estimating a functional parameter $f$ over the unit ball in $\mathbb{R}^d$, which depends only on a $d_0$-dimensional subspace of $\mathbb{R}^d$, with $d_0 < d$.It is well-known that rescaled Gaussian process priors over the function space achieve smoothness adaptation and posterior contraction with near minimax-optimal rates. Moreover, hierarchical extensions of this approach, equipped with subspace projection, can also adapt to the intrinsic dimension $d_0$ (\cite{Tokdar2011DimensionAdapt}).When the ambient dimension $d$ does not vary with $n$, the minimax rate remains of the order $n^{-\beta/(2\beta +d_0)}$.%When $d$ does not vary with $n$, the order of the minimax rate remains the same regardless of the ambient dimension $d$. However, this is up to multiplicative constants that can become prohibitively large when $d$ grows. The dependences between the contraction rate and the ambient dimension have not been fully explored yet and this work provides a first insight: we let the dimension $d$ grow with $n$ and, by combining the arguments of \cite{Tokdar2011DimensionAdapt} and \cite{Jiang2021VariableSelection}, we derive a growth rate for $d$ that still leads to posterior consistency with minimax rate.The optimality of this growth rate is then discussed.Additionally, we provide a set of assumptions under which consistent estimation of $f$ leads to a correct estimation of the subspace projection, assuming that $d_0$ is known.
Non-verbal signals in speech are encoded by prosody and carry information that ranges from conversation action to attitude and emotion. Despite its importance, the principles that govern prosodic structure are not yet adequately understood. This paper offers an analytical schema and a technological proof-of-concept for the categorization of prosodic signals and their association with meaning. The schema interprets surface-representations of multi-layered prosodic events. As a first step towards implementation, we present a classification process that disentangles prosodic phenomena of three orders. It relies on fine-tuning a pre-trained speech recognition model, enabling the simultaneous multi-class/multi-label detection. It generalizes over a large variety of spontaneous data, performing on a par with, or superior to, human annotation. In addition to a standardized formalization of prosody, disentangling prosodic patterns can direct a theory of communication and speech organization. A welcome by-product is an interpretation of prosody that will enhance speech- and language-related technologies.
During the evolution of large models, performance evaluation is necessarily performed to assess their capabilities and ensure safety before practical application. However, current model evaluations mainly rely on specific tasks and datasets, lacking a united framework for assessing the multidimensional intelligence of large models. In this perspective, we advocate for a comprehensive framework of cognitive science-inspired artificial general intelligence (AGI) tests, aimed at fulfilling the testing needs of large models with enhanced capabilities. The cognitive science-inspired AGI tests encompass the full spectrum of intelligence facets, including crystallized intelligence, fluid intelligence, social intelligence, and embodied intelligence. To assess the multidimensional intelligence of large models, the AGI tests consist of a battery of well-designed cognitive tests adopted from human intelligence tests, and then naturally encapsulates into an immersive virtual community. We propose increasing the complexity of AGI testing tasks commensurate with advancements in large models and emphasizing the necessity for the interpretation of test results to avoid false negatives and false positives. We believe that cognitive science-inspired AGI tests will effectively guide the targeted improvement of large models in specific dimensions of intelligence and accelerate the integration of large models into human society.
As the development of formal proofs is a time-consuming task, it is important to devise ways of sharing the already written proofs to prevent wasting time redoing them. One of the challenges in this domain is to translate proofs written in proof assistants based on impredicative logics to proof assistants based on predicative logics, whenever impredicativity is not used in an essential way. In this paper we present a transformation for sharing proofs with a core predicative system supporting prenex universe polymorphism (like in Agda). It consists in trying to elaborate each term into a predicative universe polymorphic term as general as possible. The use of universe polymorphism is justified by the fact that mapping each universe to a fixed one in the target theory is not sufficient in most cases. During the elaboration, we need to solve unification problems in the equational theory of universe levels. In order to do this, we give a complete characterization of when a single equation admits a most general unifier. This characterization is then employed in a partial algorithm which uses a constraint-postponement strategy for trying to solve unification problems. The proposed translation is of course partial, but in practice allows one to translate many proofs that do not use impredicativity in an essential way. Indeed, it was implemented in the tool Predicativize and then used to translate semi-automatically many non-trivial developments from Matita's library to Agda, including proofs of Bertrand's Postulate and Fermat's Little Theorem, which (as far as we know) were not available in Agda yet.
It is well known that Newton's method, especially when applied to large problems such as the discretization of nonlinear partial differential equations (PDEs), can have trouble converging if the initial guess is too far from the solution. This work focuses on accelerating this convergence, in the context of the discretization of nonlinear elliptic PDEs. We first provide a quick review of existing methods, and justify our choice of learning an initial guess with a Fourier neural operator (FNO). This choice was motivated by the mesh-independence of such operators, whose training and evaluation can be performed on grids with different resolutions. The FNO is trained using a loss minimization over generated data, loss functions based on the PDE discretization. Numerical results, in one and two dimensions, show that the proposed initial guess accelerates the convergence of Newton's method by a large margin compared to a naive initial guess, especially for highly nonlinear or anisotropic problems.
For problems of time-harmonic scattering by rational polygonal obstacles, embedding formulae express the far-field pattern induced by any incident plane wave in terms of the far-field patterns for a relatively small (frequency-independent) set of canonical incident angles. Although these remarkable formulae are exact in theory, here we demonstrate that: (i) they are highly sensitive to numerical errors in practice, and (ii) direct calculation of the coefficients in these formulae may be impossible for particular sets of canonical incident angles, even in exact arithmetic. Only by overcoming these practical issues can embedding formulae provide a highly efficient approach to computing the far-field pattern induced by a large number of incident angles. Here we address challenges (i) and (ii), supporting our theory with numerical experiments. Challenge (i) is solved using techniques from computational complex analysis: we reformulate the embedding formula as a complex contour integral and prove that this is much less sensitive to numerical errors. In practice, this contour integral can be efficiently evaluated by residue calculus. Challenge (ii) is addressed using techniques from numerical linear algebra: we oversample, considering more canonical incident angles than are necessary, thus expanding the set of valid coefficient vectors. The coefficient vector can then be selected using either a least squares approach or column subset selection.
This work concerns the implementation of the hybridizable discontinuous Galerkin (HDG) method to solve the linear anisotropic elastic equation in the frequency domain. First-order formulation with the compliance tensor and Voigt notation are employed to provide a compact description of the discretized problem and flexibility with highly heterogeneous media. We further focus on the question of optimal choice of stabilization in the definition of HDG numerical traces. For this purpose, we construct a hybridized Godunov-upwind flux for anisotropic elasticity possessing three distinct wavespeeds. This stabilization removes the need to choose scaling factors, contrary to identity and Kelvin-Christoffel based stabilizations which are popular choices in literature. We carry out comparisons among these families for isotropic and anisotropic material, with constant background and highly heterogeneous ones, in two and three dimensions. They establish the optimality of the Godunov stabilization which can be used as a reference choice for generic material and different types of waves.
Computing the regularized solution of Bayesian linear inverse problems as well as the corresponding regularization parameter is highly desirable in many applications. This paper proposes a novel iterative method, termed the Projected Newton method (PNT), that can simultaneously update the regularization parameter and solution step by step without requiring any high-cost matrix inversions or decompositions. By reformulating the Tikhonov regularization as a constrained minimization problem and writing its Lagrangian function, a Newton-type method coupled with a Krylov subspace method, called the generalized Golub-Kahan bidiagonalization, is employed for the unconstrained Lagrangian function. The resulting PNT algorithm only needs solving a small-scale linear system to get a descent direction of a merit function at each iteration, thus significantly reducing computational overhead. Rigorous convergence results are proved, showing that PNT always converges to the unique regularized solution and the corresponding Lagrangian multiplier. Experimental results on both small and large-scale Bayesian inverse problems demonstrate its excellent convergence property, robustness and efficiency. Given that the most demanding computational tasks in PNT are primarily matrix-vector products, it is particularly well-suited for large-scale problems.
The fundamental computational issues in Bayesian inverse problems (BIP) governed by partial differential equations (PDEs) stem from the requirement of repeated forward model evaluations. A popular strategy to reduce such costs is to replace expensive model simulations with computationally efficient approximations using operator learning, motivated by recent progress in deep learning. However, using the approximated model directly may introduce a modeling error, exacerbating the already ill-posedness of inverse problems. Thus, balancing between accuracy and efficiency is essential for the effective implementation of such approaches. To this end, we develop an adaptive operator learning framework that can reduce modeling error gradually by forcing the surrogate to be accurate in local areas. This is accomplished by adaptively fine-tuning the pre-trained approximate model with train- ing points chosen by a greedy algorithm during the posterior computational process. To validate our approach, we use DeepOnet to construct the surrogate and unscented Kalman inversion (UKI) to approximate the BIP solution, respectively. Furthermore, we present a rigorous convergence guarantee in the linear case using the UKI framework. The approach is tested on a number of benchmarks, including the Darcy flow, the heat source inversion problem, and the reaction-diffusion problem. The numerical results show that our method can significantly reduce computational costs while maintaining inversion accuracy.
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