Weak Regularity Research Articles

A Data-Driven Model Reduction Method for Parabolic Inverse Source Problems and its Convergence Analysis

In this paper, we propose a data-driven model reduction method to solve parabolic inverse source problems efficiently. Our method consists of offline and online stages. In the off-line stage, we explore the low-dimensional structures in the solution space of the parabolic partial differential equations (PDEs) in the forward problem with a given class of source functions and construct a small number of proper orthogonal decomposition (POD) basis functions to achieve significant dimension reduction. Equipped with the POD basis functions, we can solve the forward problem extremely fast in the online stage. Thus, we develop a fast algorithm to solve the optimization problem in the parabolic inverse source problems, which is referred to as the POD algorithm in this paper. Under a weak regularity assumption on the solution of the parabolic PDEs, we prove the convergence of the POD algorithm in solving the forward parabolic PDEs. In addition, we obtain the error estimate of the POD algorithm for parabolic inverse source problems. Finally, we present numerical examples to demonstrate the accuracy and efficiency of the proposed method. Our numerical results show that the POD algorithm provides considerable computational savings over the finite element method.

Point forecasting and forecast evaluation with generalized Huber loss

Huber loss, its asymmetric variants and their associated functionals (here named Huber functionals) are studied in the context of point forecasting and forecast evaluation. The Huber functional of a distribution is the set of minimizers of the expected (asymmetric) Huber loss, is an intermediary between a quantile and corresponding expectile, and also arises in M-estimation. Each Huber functional is elicitable, generating the precise set of minimizers of an expected score, subject to weak regularity conditions on the class of probability distributions, and has a complete characterization of its consistent scoring functions. Such scoring functions admit a mixture representation as a weighted average of elementary scoring functions. Each elementary score can be interpreted as the relative economic loss of using a particular forecast for a class of investment decisions where profits and losses are capped. The relevance of this theory for comparative assessment of weather forecasts is also discussed.

Distributed least squares prediction for functional linear regression* *This work was partially supported by the National Natural Science Foundation of China (Grant No. 11871438).

To cope with the challenges of memory bottleneck and algorithmic scalability when massive data sets are involved, we propose a distributed least squares procedure in the framework of functional linear model and reproducing kernel Hilbert space. This approach divides the big data set into multiple subsets, applies regularized least squares regression on each of them, and then averages the individual outputs as a final prediction. We establish the non-asymptotic prediction error bounds for the proposed learning strategy under some regularity conditions. When the target function only has weak regularity, we also introduce some unlabelled data to construct a semi-supervised approach to enlarge the number of the partitioned subsets. Results in present paper provide a theoretical guarantee that the distributed algorithm can achieve the optimal rate of convergence while allowing the whole data set to be partitioned into a large number of subsets for parallel processing.

A Novel Short-Term Residential Electric Load Forecasting Method Based on Adaptive Load Aggregation and Deep Learning Algorithms

Short-term residential load forecasting is the precondition of the day-ahead and intra-day scheduling strategy of the household microgrid. Existing short-term electric load forecasting methods are mainly used to obtain regional power load for system-level power dispatch. Due to the high volatility, strong randomness, and weak regularity of the residential load of a single household, the mean absolute percentage error (MAPE) of the traditional methods forecasting results would be too big to be used for home energy management. With the increase in the total number of households, the aggregated load becomes more and more stable, and the cyclical pattern of the aggregated load becomes more and more distinct. In the meantime, the maximum daily load does not increase linearly with the increase in households in a small area. Therefore, in our proposed short-term residential load forecasting method, an optimal number of households would be selected adaptively, and the total aggregated residential load of the selected households is used for load prediction. In addition, ordering points to identify the clustering structure (OPTICS) algorithm are also selected to cluster households with similar power consumption patterns adaptively. It can be used to enhance the periodic regularity of the aggregated load in alternative. The aggregated residential load and encoded external factors are then used to predict the load in the next half an hour. The long short-term memory (LSTM) deep learning algorithm is used in the prediction because of its inherited ability to maintain historical data regularity in the forecasting process. The experimental data have verified the effectiveness and accuracy of our proposed method.

L 1 and L ∞ stability of transition densities of perturbed diffusions

Abstract In this paper, we derive a stability result for L 1 {L_{1}} and L ∞ {L_{\infty}} perturbations of diffusions under weak regularity conditions on the coefficients. In particular, the drift terms we consider can be unbounded with at most linear growth, and the estimates reflect the transport of the initial condition by the unbounded drift through the corresponding flow. Our approach is based on the study of the distance in L 1 {L_{1}} - L 1 {L_{1}} metric between the transition densities of a given diffusion and the perturbed one using the McKean–Singer parametrix expansion. In the second part, we generalize the well-known result on the stability of diffusions with bounded coefficients to the case of at most linearly growing drift.

A unified view of graph regularity via matrix decompositions

AbstractWe give a unified proof of algorithmic weak and Szemerédi regularity lemmas for several well‐studied classes of sparse graphs, for which only weak regularity lemmas were previously known. These include core‐dense graphs, low threshold rank graphs, and (a version of) upper regular graphs. More precisely, we define cut pseudorandom graphs, we prove our regularity lemmas for these graphs, and then we show that cut pseudorandomness captures all of the above graph classes as special cases. The core of our approach is an abstracted matrix decomposition, which can be computed by a simple algorithm by Charikar. Using work of Oveis Gharan and Trevisan, it also implies new PTASes for MAX‐CUT, MAX‐BISECTION, MIN‐BISECTION for a significantly expanded class of input graphs. (It is NP Hard to get PTASes for these graphs in general.)

Non-uniform L1/DG method for one-dimensional time-fractional convection equation

In this paper, we present an efficient numerical method to solve a one-dimensional time-fractional convection equation whose solution has a certain weak regularity at the starting time, where the time fractional derivative in the Caputo sense with the order in (0, 1) is discretized by the L1 finite difference method on non-uniform meshes and the spatial derivative by the discontinuous Galerkin (DG) finite element method. The stability and convergence of the method are analyzed. Numerical experiments are provided to confirm the theoretical results.

Rank-based estimation under asymptotic dependence and independence, with applications to spatial extremes

Multivariate extreme value theory is concerned with modeling the joint tail behavior of several random variables. Existing work mostly focuses on asymptotic dependence, where the probability of observing a large value in one of the variables is of the same order as observing a large value in all variables simultaneously. However, there is growing evidence that asymptotic independence is equally important in real world applications. Available statistical methodology in the latter setting is scarce and not well understood theoretically. We revisit non-parametric estimation and introduce rank-based M-estimators for parametric models that simultaneously work under asymptotic dependence and asymptotic independence, without requiring prior knowledge on which of the two regimes applies. Asymptotic normality of the proposed estimators is established under weak regularity conditions. We further show how bivariate estimators can be leveraged to obtain parametric estimators in spatial tail models, and again provide a thorough theoretical justification for our approach.

Transversality conditions for the existence of solutions of first-order discontinuous functional differential equations

We are concerned with the existence of extremal solutions to a large class of first-order functional differential problems under weak regularity assumptions. Our technique involves multivalued analysis and the method of lower and upper solutions in order to obtain a new existence result to a scalar Cauchy problem. As a consequence of this result and a monotone iterative method for discontinuous operators, we derive our main existence result which is illustrated by several examples concerning well-known models: a generalized logistic equation or second-order problems in the presence of dry friction.

Architecture-Driven Digital Image Correlation Technique (ADDICT) for the measurement of sub-cellular kinematic fields in speckle-free cellular materials

Measuring displacement and strain fields at low observable scales of complex microstructures still remains a challenge in experimental mechanics often because of the combination of low definition images with poor texture at this scale. This is the case for cellular materials, for which complex local phenomena can occur. The aim of this paper is to design and validate numerically and experimentally a Digital Image Correlation (DIC) technique for the measurement of local displacement fields of samples with complex cellular geometries (i.e. samples presenting multiple random holes). It consists of a DIC method assisted with a physically sound weak regularization using an elastic B-spline image-based model. This technique introduces a separation of scales above which DIC is dominant and below which it is assisted with image-based modeling. Several in silico experimentations are performed in order to finely analyze the influence of the introduced regularization lengths for different input mechanical behaviors (elastic, elasto-plastic and geometrically non-linear) and in comparison with true error quantification. We show that the method can estimate complex local displacement and strain fields with speckle-free low definition images, even in non-linear regimes such as local buckling or plasticity. Finally, an experimental validation is performed in 2D-DIC to allow for the comparison of the proposed method on low resolution speckle-free images with a classic DIC on speckled high resolution images.

Bergman projection induced by radial weight

We establish characterizations of the radial weights ω on the unit disc such that the Bergman projection Pω, induced by ω, is bounded and/or acts surjectively from L∞ to the Bloch space B, or the dual of the weighted Bergman space Aω1 is isomorphic to the Bloch space under the Aω2-pairing. We also solve the problem posed by Dostanić in 2004 of describing the radial weights ω such that Pω is bounded on the Lebesgue space Lωp, under a weak regularity hypothesis on the weight involved. With regard to Littlewood-Paley estimates, we characterize the radial weights ω such that the norm of any function in Aωp is comparable to the norm in Lωp of its derivative times the distance from the boundary. This last-mentioned result solves another well-known problem on the area. All characterizations can be given in terms of doubling conditions on moments and/or tail integrals ∫r1ω(t)dt of ω, and are therefore easy to interpret.

A Hybrid Scan Gibbs Sampler for Bayesian Models with Latent Variables

Gibbs sampling is a widely popular Markov chain Monte Carlo algorithm that can be used to analyze intractable posterior distributions associated with Bayesian hierarchical models. There are two standard versions of the Gibbs sampler: The systematic scan (SS) version, where all variables are updated at each iteration, and the random scan (RS) version, where a single, randomly selected variable is updated at each iteration. The literature comparing the theoretical properties of SS and RS Gibbs samplers is reviewed, and an alternative hybrid scan Gibbs sampler is introduced, which is particularly well suited to Bayesian models with latent variables. The word “hybrid” reflects the fact that the scan used within this algorithm has both systematic and random elements. Indeed, at each iteration, one updates the entire set of latent variables, along with a randomly chosen block of the remaining variables. The hybrid scan (HS) Gibbs sampler has important advantages over the two standard scan Gibbs samplers. First, the HS algorithm is often easier to analyze from a theoretical standpoint. In particular, it can be much easier to establish the geometric ergodicity of a HS Gibbs Markov chain than to do the same for the corresponding SS and RS versions. Second, the sandwich methodology developed in (Ann. Statist. 36 (2008) 532–554), which is also reviewed, can be applied to the HS Gibbs algorithm (but not to the standard scan Gibbs samplers). It is shown that, under weak regularity conditions, adding sandwich steps to the HS Gibbs sampler always results in a theoretically superior algorithm. Three specific Bayesian hierarchical models of varying complexity are used to illustrate the results. One is a simple location-scale model for data from the Student’s t distribution, which is used as a pedagogical tool. The other two are sophisticated, yet practical Bayesian regression models.

Numerical Methods for the Time Fractional Convection-Diffusion-Reaction Equation

In this article, efficient methods are derived for seeking numerical solution to the time fractional convection-diffusion-reaction equation whose solution very likely exhibits a weak regularity at the starting time. Here, the time fractional derivative in the Caputo sense with order in (0, 1) is discretized by the finite difference methods with uniform and non-uniform meshes and the spatial derivative by the local discontinuous Galerkin finite element method. The derived numerical schemes are stable and convergent. Finally, some numerical experiments are presented to verify the theoretical analysis.

Adaptive bridge estimator for Cox model with a diverging number of parameters

This article considers the adaptive bridge estimator for Cox proportional hazards model with a diverging number of parameters. Specifically, we assume that the number of covariates of the Cox model is not fixed, and use the adaptive bridge penalty method to select variables and estimate the parameters of the Cox model. We show that under some weak regularity conditions, our proposed method has oracle properties and can select non-zero coefficients with a probability converging to 1. We also present an algorithm based on the Minorization-Maximization method to solve the adaptive bridge estimation. In addition, simulation studies show that the proposed method is effective and the well-known primary biliary cirrhosis data is provided to illustrate the application of the proposed method in survival problems.

Multi-stage stochastic planning of regional integrated energy system based on scenario tree path optimization under long-term multiple uncertainties

Planning a regional energy system based on the advantages of different types of energies and building an economic and efficient regional integrated energy system (RIES) are hot research subjects in the energy planning field. The call for using clean energy and attaining carbon neutrality worldwide has also attracted research attention in the field of low-carbon and clean planning. Because of the growth of the energy system construction cycle together with the utilization of various types of energies, multi-stage planning considering the uncertainty and construction time sequence in a long time scale has gradually become the primary research direction for planning RIES. The uncertainty of multi-energy load growth and the energy price fluctuation in a long time scale considerably affects the construction and economy of the final planning scheme. Aiming to solve the problems of incomplete load data, low accuracy of single load forecasting, and weak regularity of the energy price fluctuation, this study proposes a multi-stage scenario tree generation method based on the conditional generative adversarial network–random forest–Markov chain. And a method of energy price upper and lower boundary determination is based on the multi-index artificial neural network method for analyzing and solving energy price fluctuation. A multi-stage stochastic planning model of RIES is built herein to minimize the comprehensive cost, including the construction investment and regional operation costs, energy hub income, and carbon emission reduction. Finally, the Beichen district of Tianjin, China is considered a case for verifying the effectiveness and applicability of the proposed model and method. Results show that the construction time sequence of the energy hub substantially affects the planning scheme. Energy hub construction in advance benefits the energy hub income and carbon emission reduction; however, the initial stage requires more investment, which affects the planning and construction of renewable energy. Compared with a single-stage deterministic planning scheme, the optimized multi-stage stochastic planning scheme reduces the number of idle facilities and benefits cost recovery, energy hub income, coverage scenarios, and carbon emission reduction.

Existence of a fundamental solution of partial differential equations associated to Asian options

We prove the existence and uniqueness of the fundamental solution for Kolmogorov operators associated to some stochastic processes, that arise in the Black & Scholes setting for the pricing problem relevant to path dependent options. We improve previous results in that we provide a closed form expression for the solution of the Cauchy problem under weak regularity assumptions on the coefficients of the differential operator. Our method is based on a limiting procedure, whose convergence relies on some barrier arguments and uniform a priori estimates recently discovered.

Bayesian nonparametrics for directional statistics

We introduce a density basis of the trigonometric polynomials that is suitable to mixture modelling. Statistical and geometric properties are derived, suggesting it as a circular analogue to the Bernstein polynomial densities. Nonparametric priors are constructed using this basis and a simulation study shows that the use of the resulting Bayes estimator may provide gains over comparable circular density estimators previously suggested in the literature.From a theoretical point of view, we propose a general prior specification framework for density estimation on compact metric space using sieve priors. This is tailored to density bases such as the one considered herein and may also be used to exploit their particular shape-preserving properties. Furthermore, strong posterior consistency is shown to hold under notably weak regularity assumptions and adaptive convergence rates are obtained in terms of the approximation properties of positive linear operators generating our models.

Convergence of Numerical Schemes for a Conservation Equation with Convection and Degenerate Diffusion

The approximation of problems with linear convection and degenerate nonlinear diffusion, which arise in the framework of the transport of energy in porous media with thermodynamic transitions, is done using a θ-scheme based on the centred gradient discretisation method. The convergence of the numerical scheme is proved, although the test functions which can be chosen are restricted by the weak regularity hypotheses on the convection field, owing to the application of a discrete Gronwall lemma and a general result for the time translate in the gradient discretisation setting. Some numerical examples, using both the Control Volume Finite Element method and the Vertex Approximate Gradient scheme, show the role of θ for stabilising the scheme.

Learning Graphons via Structured Gromov-Wasserstein Barycenters

We propose a novel and principled method to learn a nonparametric graph model called graphon, which is defined in an infinite-dimensional space and represents arbitrary-size graphs. Based on the weak regularity lemma from the theory of graphons, we leverage a step function to approximate a graphon. We show that the cut distance of graphons can be relaxed to the Gromov-Wasserstein distance of their step functions. Accordingly, given a set of graphs generated by an underlying graphon, we learn the corresponding step function as the Gromov-Wasserstein barycenter of the given graphs. Furthermore, we develop several enhancements and extensions of the basic algorithm, e.g., the smoothed Gromov-Wasserstein barycenter for guaranteeing the continuity of the learned graphons and the mixed Gromov-Wasserstein barycenters for learning multiple structured graphons. The proposed approach overcomes drawbacks of prior state-of-the-art methods, and outperforms them on both synthetic and real-world data. The code is available at https://github.com/HongtengXu/SGWB-Graphon.

784 Naturalistic Measurement of Sleep/Wake Disturbance in Adults Receiving Methadone Treatment for Opioid Use Disorder

Abstract Introduction Individuals with opioid use disorder (OUD) report significant sleep/wake disturbance, which continues even after stabilization on medication-assisted treatment (MAT). However, the nature of sleep/wake disturbance in this population has not been well documented objectively. Here we analyze naturalistic wrist actigraphy recordings in individuals with OUD receiving methadone-based MAT. Methods Seven adults undergoing methadone treatment for OUD (ages 26–50; 4 women) wore a wrist actigraph (Actiwatch-2, Philips Respironics) continuously for 7 days. They were asked to adhere to their normal sleep schedule in order to obtain naturalistic observations. Reference data were collected in a separate study of healthy controls, in which 14 hospital nurses (ages 20–60; 13 women) wore a wrist actigraph continuously for 7 days. In this reference group, 7 participants had a day shift schedule (07:00–19:00) and 7 had a night shift schedule (19:00–07:00), with six 12h shifts in a 2-week period. Actigraphic data were collected in 1min epochs, and the sleep/wake status for each epoch was estimated using Actiware 6.0.9 (Philips Respironics). The estimated sleep/wake patterns were subjected to cosinor analysis to assess 24h rhythmicity and analysis of the distribution of inactive periods to assess sleep continuity. Results For the reference group, nurses working day shifts displayed strong 24h rhythmicity, whereas nurses working night shifts showed blunted 24h rhythmicity (F[1,12]=66.11, p<0.001). However, both day and night nurses exhibited high sleep continuity (KS test, p=0.82). By contrast, for the group with OUD receiving methadone, the strength of 24h rhythmicity was reduced to between that of the day and night shift nurses in the reference group, indicating weak regularity of sleep/wake patterns (F[2,18]=33.79, p<0.001). Furthermore, individuals with OUD receiving methadone experienced low sleep continuity compared to the reference group (KS test, p=0.030). Conclusion These naturalistic observations confirm the presence of sleep/wake disturbance, resulting from both irregular sleep/wake patterns and low sleep continuity, in individuals receiving methadone-based MAT for OUD. Sleep/wake disturbance may interfere with the ability to achieve OUD recovery goals, and comparing sleep disturbance in MAT populations to reference data highlights the need to consider sleep in these populations as a clinical priority. Support (if any) State of Washington Initiative Measure No. 171