Pointwise Estimates Research Articles

Hyperspheric Integral Reliability Method for efficient reliability analysis of geotechnical ultimate limit states

This paper introduces the Hyperspheric Integral Reliability Method (HINT) for efficient reliability analysis of geotechnical ultimate limit states. The method is motivated by the mechanism of the Shear Strength Reduction Method (SSRM), which is often employed to calculate the factor of safety for geotechnical ultimate limit states. HINT exploits the observation that a factor of safety, computed by the SSRM, is not only a pointwise estimate of safety, but also a measure of distance to the failure limit along a radial direction in the space of the shear strength parameters. This observation is utilized to transform the reliability integral in the hyperspheric coordinate system and develop an efficient estimator of failure probability for geotechnical ultimate limit states. HINT was examined on several benchmarking problems, demonstrating stable and highly efficient performance on low to medium dimensional reliability problems. HINT can be also applied to general reliability problems if a radial search mechanism is implemented as in the SSRM. Given that the SSRM is already available in most commercial geotechnical software packages, HINT is perfectly suited to advance geotechnical reliability analyses and make them accessible to a wider set of use-cases.

A Spatial-Frequency Approach to Point-Wise Frequency Response Function Estimation with Digital Image Correlation

The use of digital image correlation for modal analysis is becoming an appealing option thanks to its non-contact and full-field measurement process. However, frequency response function (FRF) estimation can be challenging due to the limited number of time domain data and heavy measurement noise. Thereby, the present work aims to propose a method which improves the estimation accuracy of point-wise FRFs. Firstly, a Gaussian-process-based spatial-frequency model is proposed, which makes use of the intrinsic properties of the FRF and the local spatial information of field measurements. Then, a Bayesian solution is developed, which is enforced by a stable and efficient numerical procedure. Finally, the effectiveness of the proposed method is verified by making a comparison with the spectral estimator through the use of simulated data, and it is further validated based on an experimental application.

Reduced-dimensional modelling for nonlinear convection-dominated flow in cylindric domains

The aim of the paper is to construct and justify asymptotic approximations for solutions to quasilinear convection–diffusion problems with a predominance of nonlinear convective flow in a thin cylinder, where an inhomogeneous nonlinear Robin-type boundary condition involving convective and diffusive fluxes is imposed on the lateral surface. The limit problem for vanishing diffusion and the cylinder shrinking to an interval is a nonlinear first-order conservation law. For a time span that allows for a classical solution of this limit problem corresponding uniform pointwise and energy estimates are proven. They provide precise model error estimates with respect to the small parameter that controls the double viscosity-geometric limit. In addition, other problems with more higher Péclet numbers are also considered.

Change-point detection in anomalous-diffusion trajectories utilising machine-learning-based uncertainty estimates

Abstract When recording the movement of individual animals, cells or molecules one will often observe changes in their diffusive behaviour at certain points in time along their trajectory. In order to capture the different diffusive modes assembled in such heterogeneous trajectories it becomes necessary to segment them by determining these change-points. Such a change-point detection can be challenging for conventional statistical methods, especially when the changes are subtle. We here apply Bayesian Deep Learning to obtain point-wise estimates of not only the anomalous diffusion exponent but also the uncertainties in these predictions from a single anomalous diffusion trajectory generated according to four theoretical models of anomalous diffusion. We show that we are able to achieve an accuracy similar to single-mode (without change-points) predictions as well as a well calibrated uncertainty predictions of this accuracy. Additionally, we find that the predicted uncertainties feature interesting behaviour at the change-points leading us to examine the capabilities of these predictions for change-point detection. While the series of predicted uncertainties on their own are not sufficient to improve change-point detection, they do lead to a performance boost when applied in combination with the predicted anomalous diffusion exponents.

Gaussian estimates vs. elliptic regularity on open sets

AbstractGiven an elliptic operator $$L= - {{\,\textrm{div}\,}}(A \nabla \cdot )$$ L = - div ( A ∇ · ) subject to mixed boundary conditions on an open subset of $$\mathbb {R}^d$$ R d , we study the relation between Gaussian pointwise estimates for the kernel of the associated heat semigroup, Hölder continuity of L-harmonic functions and the growth of the Dirichlet energy. To this end, we generalize an equivalence theorem of Auscher and Tchamitchian to the case of mixed boundary conditions and to open sets far beyond Lipschitz domains. Yet, we prove the consistency of our abstract result by encompassing operators with real-valued coefficients and their small complex perturbations into one of the aforementioned equivalent properties. The resulting kernel bounds open the door for developing a harmonic analysis for the associated semigroups on rough open sets.

Sharp maximal function estimates for linear and multilinear pseudo-differential operators

In this paper, we study pointwise estimates for linear and multilinear pseudo-differential operators with exotic symbols in terms of the Fefferman-Stein sharp maximal function and Hardy-Littlewood type maximal function. Especially in the multilinear case, we use a multi-sublinear variant of the classical Hardy-Littlewood maximal function introduced by Lerner, Ombrosi, Pérez, Torres, and Trujillo-González [16], which provides more elaborate and natural weighted estimates in the multilinear setting.

Estimation of random functions proxying for unobservables

This article considers the model Y = M ( X , U ) where U is an unobservable continuously distributed scalar and M is monotonic with respect to U. It is assumed there is an observable scalar W satisfying the restriction T F W | X , U = T F W | U almost surely where T is a known functional and F A|B denotes the distribution of A|B. This article shows that M can be identified under a mild monotonicity condition. This result requires neither statistical independence between X and U nor X to be continuously distributed. The estimation problem is treated when T F A | B ≡ E [ A | B ] . The proposed pointwise estimator of M is asymptotically normally distributed under weak technical conditions. Furthermore, the rate of convergence in probability is equal to n − r / ( 2 r + d + 1 ) where d denotes the dimension of the continuously distributed components of X and r is a positive integer which relates to the smoothness of certain functions. A Monte Carlo experiment is conducted and reveals the benefits of the estimator in the presence of endogeneity. We apply our estimator to estimate the returns to schooling.

Global classical solutions to a multidimensional radiation hydrodynamics model with symmetry and large initial data

AbstractAs a first stage to study the global large solutions of the radiation hydrodynamics model with viscosity and thermal conductivity in the high‐dimensional space, we study the problems in high dimensions with some symmetry, such as the spherically or cylindrically symmetric solutions. Specifically, we will study the global classical large solutions to the radiation hydrodynamics model with spherically or cylindrically symmetric initial data. The key point is to obtain the strict positive lower and upper bounds of the density and the lower bound of the temperature . Compared with the Navier–Stokes equations, these estimates in the present paper are more complicated due to the influence of the radiation. To overcome the difficulties caused by the radiation, we construct a pointwise estimate between the radiative heat flux and the temperature by studying the boundary value problem of the corresponding ordinary differential equation. And we consider a general heat conductivity: if ; if . This can be viewed as the first result about the global classical large solutions of the radiation hydrodynamics model with some symmetry in the high‐dimensional space.

Reduced Element for Adaptive Finite Element Analysis of First-Order IVP with Built-in Error Estimator in Maximum Norm

This paper proposes a novel yet simple approach to the adaptive finite element (FE) analysis of the first-order Initial Value Problems (IVPs) in the maximum norm by introducing the reduced element technique. In the present approach, the FE solution uh of the conventional Galerkin element of degree m + 1 is decomposed into two parts: a reduced solution uRh from the reduced element of degree m obtained by ignoring the highest degree term of uh, and a built-in point-wise error estimator εRh directly given by the ignored term. Since the end node solutions of the reduced element are inherited from the full order element, it gains O(h2m+2) accuracy and achieves a nodal/element accuracy ratio as high as two, which greatly enhances its adaptive capability regarding solving IVPs on long time domains. The related error analysis is addressed and a complete adaptivity algorithm is given. Typical numerical examples of both linear and nonlinear IVPs of both single and systems of equations are presented to show the validity and effectiveness of the proposed approach.

Pointwise estimates on the fundamental solution to nonlocal diffusion equation

In this article we study the pointwise behavior of the fundamental solution S ( t , x ) to the nonlocal diffusion model u t = D ( J ∗ u − u ) on R , S ( t , x ) can be written as e − Dt δ ( x ) + S 0 ( t , x ) , where S 0 ( t , x ) is called the regularizing part. Under some conditions on J and its Fourier transform J ^ , we obtain some new pointwise estimates on S 0 ( t , x ) . Then we apply our results to study the spatial dynamics of the nonlocal diffusion population model in a shifting environment considered in Li et al. [Spatial dynamics of a nonlocal dispersal population model in a shifting environment. J Nonlinear Sci. 2018;28:1189–1219]. In our study the convolution kernel J can be assumed to be more general, i.e. not necessarily of compact support, also the proof is greatly simplified. It is our believe that these pointwise estimates possess their own interest in further studying nonlocal diffusion models.

Equivalence of Sobolev Norms with Respect to Weighted Gaussian Measures

AbstractWe consider the spaces $${\text {L}}^p(X,\nu ;V)$$ L p ( X , ν ; V ) , where X is a separable Banach space, $$\mu $$ μ is a centred non-degenerate Gaussian measure, $$\nu :=Ke^{-U}\mu $$ ν : = K e - U μ with normalizing factor K and V is a separable Hilbert space. In this paper we prove a vector-valued Poincaré inequality for functions $$F\in W^{1,p}(X,\nu ;V)$$ F ∈ W 1 , p ( X , ν ; V ) , which allows us to show that for every $$p\in (1,\infty )$$ p ∈ ( 1 , ∞ ) and every $$k\in \mathbb {N}$$ k ∈ N the norm in $$W^{k,p}(X,\nu )$$ W k , p ( X , ν ) is equivalent to the graph norm of $$D_H^{k}$$ D H k (the k-th Malliavin derivative) in $${\text {L}}^p(X,\nu )$$ L p ( X , ν ) . To conclude, we show exponential decay estimates for the V-valued perturbed Ornstein-Uhlenbeck semigroup $$(T^V(t))_{t\ge 0}$$ ( T V ( t ) ) t ≥ 0 , defined in Section 2.6, as t goes to infinity. Useful tools are the study of the asymptotic behaviour of the scalar perturbed Ornstein-Uhlenbeck $$(T(t))_{t\ge 0}$$ ( T ( t ) ) t ≥ 0 , and pointwise estimates for $$|D_HT(t)f|_H^p$$ | D H T ( t ) f | H p by means of both $$T(t)|D_Hf|^p_H$$ T ( t ) | D H f | H p and $$T(t)|f|^p$$ T ( t ) | f | p .

Minimizing under relaxed symmetry constraints: Triple and N-junctions

We consider a nonnegative potential $W:\mathbb{R}^2\rightarrow\mathbb{R}$ invariant under the action of the rotation group $C_N$ of the regular polygon with $N$ sides, $N\geq 3$. We assume that $W$ has $N$ nondegenerate zeros and prove the existence of a $N$-junction solution to the vector Allen-Cahn equation. The proof is variational and is based on sharp lower and upper bounds for the energy and on a new pointwise estimate for vector minimizers.

Lower bounds for the trade-off between bias and mean absolute deviation

In nonparametric statistics, rate-optimal estimators typically balance bias and stochastic error. The recent work on overparametrization raises the question whether rate-optimal estimators exist that do not obey this trade-off. In this work we consider pointwise estimation in the Gaussian white noise model with regression function f in a class of β-Hölder smooth functions. Let ’worst-case’ refer to the supremum over all functions f in the Hölder class. It is shown that any estimator with worst-case bias ≲n−β/(2β+1)≕ψn must necessarily also have a worst-case mean absolute deviation that is lower bounded by ≳ψn. To derive the result, we establish abstract inequalities relating the change of expectation for two probability measures to the mean absolute deviation.

The forward self-similar solution of fractional incompressible Navier-Stokes system: The critical case

In this paper, we study the regularity and pointwise estimates of forward self-similar solutions of fractional Navier-Stokes system under the critical case. By employing a Caffarelli,Kohn and Nirenberg-type iteration, L∞ estimates of the self-similar solution's profile are established, which is a key ingredient to ensure that the global weighted energy estimate procedure used in [28] is performed under the critical case. As a product, its natural pointwise bounds are recovered. Moreover, to obtain the optimal spatial decay estimate of self-similar solution's profile, a new technique is required due to lack of the related regularity theory.

Decay estimates for defocusing energy-critical Hartree equation

Abstract In this paper, we are devoted to establishing the point-wise decay estimates for solution to the 5D defocusing energy-critical Hartree equation with an initial data in H 2 ( R 5 ) ∩ L 1 ( R 5 ) ${H}^{2}\left({\mathbb{R}}^{5}\right)\cap {L}^{1}\left({\mathbb{R}}^{5}\right)$ . We show that the nonlinear solution has the same time decay rate as the linear one. The main new ingredient is that we used the theories of Lorentz spaces to overcome the low power of nonlinearity.

Point-wise time-space estimates for a class of oscillatory integrals and their applications

Point-wise time-space estimates for a class of oscillatory integrals and their applications

Asymptotic expansion for convection-dominated transport in a thin graph-like junction

We consider for a small parameter [Formula: see text] a parabolic convection–diffusion problem with Péclet number of order [Formula: see text] in a three-dimensional graph-like junction consisting of thin curvilinear cylinders with radii of order [Formula: see text] connected through a domain (node) of diameter [Formula: see text] Inhomogeneous Robin type boundary conditions, that involve convective and diffusive fluxes, are prescribed both on the lateral surfaces of the thin cylinders and the boundary of the node. The asymptotic behavior of the solution is studied as [Formula: see text] i.e. when the diffusion coefficients are eliminated and the thin junction is shrunk into a three-part graph connected in a single vertex. A rigorous procedure for the construction of the complete asymptotic expansion of the solution is developed and the corresponding energy and uniform pointwise estimates are proved. Depending on the directions of the limit convective fluxes, the corresponding limit problems [Formula: see text] are derived in the form of first-order hyperbolic differential equations on the one-dimensional branches with novel gluing conditions at the vertex. These generalize the classical Kirchhoff transmission conditions and might require the solution of a three-dimensional cell-like problem associated with the vertex to account for the local geometric inhomogeneity of the node and the physical processes in the node. The asymptotic ansatz consists of three parts, namely, the regular part, node-layer part, and boundary-layer one. Their coefficients are classical solutions to mixed-dimensional limit problems. The existence and other properties of those solutions are analyzed.

Pointwise Error Estimates of Numerical Solutions to Linear Quadratic Optimal Control Problems

An auxiliary optimal control problem is formulated that provides with its unique solution, a continuous representation of the global error of a numerical approximation to the solution of a linear quadratic optimal control problem. The resulting error functions are characterized as the unique solutions to an optimality system that is reformulated as a boundary value problem. With this formulation, reliable pointwise error estimates can be generated utilizing well-established techniques of defect control. A novel algorithm based on defect correction and defect control is presented that generates pointwise approximations to the global error of numerical optimal control solutions on a uniform grid. It is proven and numerically validated that this algorithm can generate pointwise estimates that approximate the true global error with a prescribed accuracy.

Puiseux asymptotic expansions for convection-dominated transport problems in thin graph-like networks: Strong boundary interactions

This article completes the study of the influence of the intensity parameter α in the boundary condition ε ∂ ν ε u ε − u ε V ε → · ν ε = ε α φ ε given on the boundary of a thin three-dimensional graph-like network consisting of thin cylinders that are interconnected by small domains (nodes) with diameters of order O ( ε ). Inside of the thin network a time-dependent convection-diffusion equation with high Péclet number of order O ( ε − 1 ) is considered. The novelty of this article is the case of α < 1, which indicates a strong intensity of physical processes on the boundary, described by the inhomogeneity φ ε (the cases α = 1 and α > 1 were previously studied by the same authors). A complete Puiseux asymptotic expansion is constructed for the solution u ε as ε → 0, i.e., when the diffusion coefficients are eliminated and the thin network shrinks into a graph. Furthermore, the corresponding uniform pointwise and energy estimates are proved, which provide an approximation of the solution with a given accuracy in terms of the parameter ε.

Quasilinear Lane–Emden type systems with sub-natural growth terms

Global pointwise estimates are obtained for quasilinear Lane–Emden-type systems involving measures in the “sublinear growth” rate. We give necessary and sufficient conditions for existence expressed in terms of Wolff’s potential. Our approach is based on recent advances due to Kilpeläinen and Malý in the potential theory. This method enables us to treat several problems, such as equations involving general quasilinear operators and fractional Laplacian.