Smooth Approximation Research Articles

The 𝑊^{𝑠,𝑝}-boundedness of stationary wave operators for the Schrödinger operator with inverse-square potential

In this paper, we investigate the W s , p W^{s,p} -boundedness for stationary wave operators of the Schrödinger operator with inverse-square potential L a = − Δ + a | x | 2 , a ≥ − ( d − 2 ) 2 4 , \begin{equation*} \mathcal L_a=-\Delta +\tfrac {a}{|x|^2}, \quad a\geq -\tfrac {(d-2)^2}{4}, \end{equation*} in dimension d ≥ 2 d\geq 2 . We construct the stationary wave operators in terms of integrals of Bessel functions and spherical harmonics, and prove that they are W s , p W^{s,p} -bounded for certain p p and s s which depend on a a . As corollaries, we solve some open problems associated with the operator L a \mathcal L_a , which include the dispersive estimates and the local smoothing estimates in dimension d ≥ 2 d\geq 2 . We also generalize some known results such as the uniform Sobolev inequalities, the equivalence of Sobolev norms and the Mikhlin multiplier theorem, to a larger range of indices. These results are important in the description of linear and nonlinear dynamics for dispersive equations with inverse-square potential.

Алгоритм розкладу цілих чисел і гладкого наближення функцій

The problem of expansion in powers is generalized into decomposition of positive integers in the sequence of degrees of different orders, the con-ditions of decomposition are determined, and the algorithm for decomposi-tion is constructed. The algorithm is based on two procedures: 1) achieve-ment a minimum of residual at each algorithm step; 2) speeding of decom-position through expanding the local base by reducing decomposition in-dex, which ensures finiteness of algorithm. The algorithm has such effi-ciency factors as high rate of decomposition, ease of implementation, availability of different options for the decomposition of numbers as in ex-tended, narrowed, sparse bases, which protects the encoded information from external influences. The algorithm can be used to encode large amounts of digital information under basic systems of small dimensions. Decomposition of positive integers into a sequence of powers is opti-mal and correct. Optimality of decomposition follows from the condition that at each step of algorithm the minimum value of disjunction in the space of mixed parameters x∈N,y∈Ris achieved. Correctness of algo-rithm is due to the fact that when the disjunction is reduced, the algorithm expands the basis of decomposition by reducing the degree indicators by one. By switching from a discrete model to a continuous model by replac-ing the degrees with power functions, we obtain a smooth approximation of the ill-conditioned function in the neighborhood of decomposition. The construction of posinomial polynomials on the basis of smooth polynomi-als is one of the promising directions of integration of ill-conditioned non-differentiable functions and smooth replacement of variables in the catas-trophe theory.Posinomials (functions with a variable exponent) predict the step of splitting the integration interval into parts, since they determine the loga-rithmic rate of change of an arbitrary monotonic function. The method of decomposition of positive integers provides an optimal decomposition into the sum of powers, and therefore the transition from a discrete model to a continuous model in the neighborhood of decomposition by replacing powers with power functions as well as allows to achieve the high accura-cy of approximation.

TTRISK: Tensor train decomposition algorithm for risk averse optimization

AbstractThis article develops a new algorithm named TTRISK to solve high‐dimensional risk‐averse optimization problems governed by differential equations (ODEs and/or partial differential equations [PDEs]) under uncertainty. As an example, we focus on the so‐called Conditional Value at Risk (CVaR), but the approach is equally applicable to other coherent risk measures. Both the full and reduced space formulations are considered. The algorithm is based on low rank tensor approximations of random fields discretized using stochastic collocation. To avoid nonsmoothness of the objective function underpinning the CVaR, we propose an adaptive strategy to select the width parameter of the smoothed CVaR to balance the smoothing and tensor approximation errors. Moreover, unbiased Monte Carlo CVaR estimate can be computed by using the smoothed CVaR as a control variate. To accelerate the computations, we introduce an efficient preconditioner for the Karush–Kuhn–Tucker (KKT) system in the full space formulation.The numerical experiments demonstrate that the proposed method enables accurate CVaR optimization constrained by large‐scale discretized systems. In particular, the first example consists of an elliptic PDE with random coefficients as constraints. The second example is motivated by a realistic application to devise a lockdown plan for United Kingdom under COVID‐19. The results indicate that the risk‐averse framework is feasible with the tensor approximations under tens of random variables.

Joint Pilot Optimization, Target Detection and Channel Estimation for Integrated Sensing and Communication Systems

Radar sensing will be integrated into the 6G communication system to support various applications. In this integrated sensing and communication system, a radar target may also be a communication channel scatterer. In this case, the radar and communication channels exhibit certain joint burst sparsity. We propose a two-stage joint pilot optimization, target detection and channel estimation scheme to exploit such joint burst sparsity and pilot beamforming gain to enhance detection/estimation performance. In Stage 1, the base station (BS) sends downlink pilots (DP) for initial target search, and the user sends uplink pilots (UP) for channel estimation. Then the BS performs joint target detection and channel estimation. In Stage 2, the BS exploits the prior information obtained in Stage 1 to optimize the DP signal to further refine the performance. A Turbo Sparse Bayesian inference algorithm is proposed for joint target detection and channel estimation in both stages. The pilot optimization problem in Stage 2 is a semi-definite programming with rank-1 constraints. By replacing the rank-1 constraint with a tight and smooth approximation, we propose an efficient pilot optimization algorithm based on the majorization-minimization (MM) method. Simulations verify the advantages of the proposed scheme.

Deep Neural Network-Embedded Stochastic Nonlinear State-Space Models and Their Applications to Process Monitoring.

Process complexities are characterized by strong nonlinearities, dynamics, and uncertainties. Monitoring such a complex process requires a high-quality model describing the corresponding nonlinear dynamic behavior. The proposed model is constructed using deep neural networks (DNNs) to represent the state transition and observation generation, both of which constitute a stochastic nonlinear state-space model. A new bidirectional recurrent neural network (RNN), creating a connection of the hidden layer between a forward RNN and a backward RNN, is proposed to generate the filtering estimation and the smoothing estimation of process states which further generate observations with DNN-based process models. The smoothing estimator and the process model are first learned offline with all collected samples. Then the filtering estimator is fine-tuned by the learned smoother and process models to achieve real-time monitoring since the filter state is estimated based on the past and the current observations. Two indices are designed based on the learned model for monitoring the process anomaly. The proposed process monitoring model can deal with complex nonlinearities, process dynamics, and process uncertainties, all of which can be very challenging for the existing methods, such as kernel mapping and stacked auto-encoder. Two case studies validate that the effectiveness of the proposed method outperforms the other comparative methods by at least 10% when using the averaged fault detection rate in the industrial experimental data.

Developing an oligogenic risk score for Alzheimer’s disease (adORS)

AbstractBackgroundPolygenic risk score (PRS) has become a popular approach for estimating an individual’s genetic risk for disease, recently including Alzheimer’s disease. However, its limitations reveal the need to improve genetic modeling of the disease, i.e. pinpointing relevant genetic drivers for score calculation and improving interpretability and clinical utility. We propose a novel genetic risk score for Alzheimer’s disease (adORS), which—compared to PRS—combines a smaller set of more informative risk variants using known biomarkers to better predict those with high or low risk of disease, as well as mild cognitive impairment (MCI) conversion to AD.MethodGenetic variants of 1,704 patients (364 controls, 550 AD cases, 273 MCI converters [MCIc], and 517 MCI non‐converters [MCInc]) were collected from the Alzheimer’s Disease Neuroimaging Initiative (ADNI). Participants were classified based on diagnosis at their last visit. Variants were transformed into a gene‐based burden to amplify the variance held by each variable, enabling smooth model estimation and gene‐level interpretation. The risk score was estimated based on genes exhibiting high correlations with well‐known biomarkers, FDG and AV45. PRS were generated with clumping and thresholding using the PLINK data analysis toolset. We compared the performances between adORS and PRS on two tasks: controls vs AD classification and MCI conversion prediction.ResultTop 20 highly correlated genes including known AD‐relevant genes such as TOMM40 and APOC1 were used to calculate the AD risk score. Overall, adORS demonstrated better prediction performance than PRS (Table 1). adORS also showed better distinguishability between cases and controls and predicted lower risk scores in controls and MCInc patients versus higher risk scores in AD cases and MCIc. Additionally, adORS is well‐calibrated compared to PRS (Figure 1). The number of AD and MICc patients increases smoothly as the risk score increases.ConclusionWe proposed a new method to calculate an AD‐specific genetic risk score. The main advantage of adORS is to use only significant AD‐relevant genes for calculation, yielding a score that is a better predictor of AD status and progression. Experiments demonstrated that adORS has enhanced performance of prediction accuracy, calibration, and interpretability over PRS.

Are CEO characteristics associated with income smoothing?

ABSTRACT This study examines whether CEO characteristics are potentially linked to firms’ financial reporting behaviour, i.e. income smoothing behaviour. Using a unique hand-collected data set on CEO characteristics between 2008 and 2017 from U.K. firms, we find that firms with longer CEO tenure, run by CEOs with accounting or finance educational backgrounds, and those with female CEOs tend to engage in more income smoothing. Our results are robust to alternative measures of income smoothing and different estimation specifications. Our empirical evidence suggests that certain types of CEO characteristics play an important role in determining a firm’s income-smoothing activities. Our study sheds light on how CEO characteristics extend to other aspects of corporate behaviour, in this case, firms’ income smoothing behaviours.

Smooth-polylines vs. polynomials in balanced identification technology: solving inverse problems of thermal conductivity

Various parameters’ identification problems of one-dimensional nonlinear heat equation are considered. Their numerical study was carried out on the basis of balanced identification technology, which provides a compromise between the simplicity of the model (the curvature of the functions) and the proximity to experimental data. The problem of identifying functions whose arguments are model variables is considered. When approximating such a function, we had to use a polynomial function – the use of polylines (polygonal lines) in this case (superposition of functions) leads to nonsmooth mathematical programming problems (with discontinuous derivatives) with the solution not supported by standard solvers. An investigation of the use of special smooth approximations of polygonal curves (smooth-polylines) is presented.

Modeling, simulation, and optimization of multiproduct cryogenic air separation unit startup

AbstractThe startup of multiproduct cryogenic air separation plants takes several hours, during which time limited revenue is generated with high costs incurred due to the highly energy‐intensive nature of these operations. This motivates the development of high‐fidelity dynamic models to capture the complexity of the startup process to aid decision‐making. This article focuses on the development of a startup model for a multiproduct air separation unit (ASU), and its use in dynamic simulation and optimization. To accomplish this, a first‐principles based dynamic ASU model is extended by including various discontinuities using smooth approximations, adding dynamics to the primary heat exchanger, and extending the handling phase change within process streams. Dynamic simulations demonstrate plant response behavior during startup, including a failed startup resulting from an injudicious choice of input trajectory. In addition, improvement of startup operation is demonstrated through the incorporation of the model within a dynamic optimization framework.

Efficient smooth minorants for global optimization of univariate functions with the first derivative satisfying the interval Lipschitz condition

AbstractIn 1998, the paper Sergeyev (Math Program 81(1):127–146, 1998) has been published where a smooth piece-wise quadratic minorant has been proposed for multiextremal functions f(x) with the first derivative $$f'(x)$$ f ′ ( x ) satisfying the Lipschitz condition with a constant L, i.e., $$f'(x)$$ f ′ ( x ) cannot increase with the slope higher than L and decrease with the slope smaller than $$-L$$ - L . This minorant has been successfully applied in several efficient global optimization algorithms and used in engineering applications. In the present paper, it is supposed that the first derivative $$f'(x)$$ f ′ ( x ) cannot increase with the slope higher than a constant $$\beta $$ β and decrease with the slope smaller than $$\alpha $$ α . The interval $$[\alpha ,\beta ]$$ [ α , β ] is called the Lipschitz interval (clearly, in this case the Lipschitz constant $$L = \max \{|\alpha |, |\beta | \}$$ L = max { | α | , | β | } ). For this class of functions, smooth piece-wise estimators (minorants and majorants) have been proposed and applied in global optimization. Both theoretically and experimentally (on 200 randomly generated test problems) it has been shown that in cases where $$ |\alpha | \ne |\beta |$$ | α | ≠ | β | the new estimators can give a significant improvement w.r.t. those proposed in Sergeyev (Math Program 81(1):127–146, 1998), for example, in the framework of branch-and-bound global optimization methods.

A distribution-free smoothed combination method to improve discrimination accuracy in multi-category classification.

Results from multiple diagnostic tests are combined in many ways to improve the overall diagnostic accuracy. For binary classification, maximization of the empirical estimate of the area under the receiver operating characteristic curve has widely been used to produce an optimal linear combination of multiple biomarkers. However, in the presence of a large number of biomarkers, this method proves to be computationally expensive and difficult to implement since it involves maximization of a discontinuous, non-smooth function for which gradient-based methods cannot be used directly. The complexity of this problem further increases when the classification problem becomes multi-category. In this article, we develop a linear combination method that maximizes a smooth approximation of the empirical Hyper-volume Under Manifolds for the multi-category outcome. We approximate HUM by replacing the indicator function with the sigmoid function and normal cumulative distribution function. With such smooth approximations, efficient gradient-based algorithms are employed to obtain better solutions with less computing time. We show that under some regularity conditions, the proposed method yields consistent estimates of the coefficient parameters. We derive the asymptotic normality of the coefficient estimates. A simulation study is performed to study the effectiveness of our proposed method as compared to other existing methods. The method is illustrated using two real medical data sets.

Strichartz and Local Smoothing Estimates for Stochastic Dispersive Equations with Linear Multiplicative Noise

We study a quite general class of stochastic dispersive equations with linear multiplicative noise, including especially the Schrodinger and Airy equations. The pathwise Strichartz and local smoothing estimates are derived here in both the conservative and nonconservative case. In particular, we obtain the P-integrability of constants in these estimates, where P is the underlying probability measure. Several applications are given to nonlinear problems, including local well-posedness of stochastic nonlinear Schrodinger equations with variable coefficients and lower order perturbations, integrability of global solutions to stochastic nonlinear Schrodinger equations with constant coefficients. As another consequence, we prove as well the large deviation principle for the small noise asymptotics.

CDRNet: accurate cup-to-disc ratio measurement with tight bounding box supervision in fundus photography using deep learning

The cup-to-disc ratio (CDR) is one of the most significant indicator for glaucoma diagnosis. Different from the use of costly fully supervised learning formulation with pixel-wise annotations in the literature, this study investigates the feasibility of accurate CDR measurement in fundus images using only tight bounding box supervision. For this purpose, we develop a two-task network named as CDRNet for accurate CDR measurement, one for weakly supervised image segmentation, and the other for bounding-box regression. The weakly supervised image segmentation task is implemented based on generalized multiple instance learning formulation and smooth maximum approximation, and the bounding-box regression task outputs class-specific bounding box prediction in a single scale at the original image resolution. To get accurate bounding box prediction, a class-specific bounding-box normalizer and an expected intersection-over-union are proposed. In the experiments, the proposed approach was evaluated by a testing set with 1200 images using CDR error and $F_1$ score for CDR measurement and dice coefficient for image segmentation. A grader study was conducted to compare the performance of the proposed approach with those of individual graders. The experimental results indicate that the proposed approach outperforms the state-of-the-art performance obtained from the fully supervised image segmentation (FSIS) approach using pixel-wise annotation for CDR measurement. Its performance is also better than those of individual graders. In addition, the proposed approach gets performance close to the state-of-the-art obtained from FSIS and the performance of individual graders for optic cup and disc segmentation. The codes are available at \url{https://github.com/wangjuan313/CDRNet}.

On the asymptotic distribution of the maximum sample spectral coherence of Gaussian time series in the high dimensional regime

We investigate the asymptotic distribution of the maximum of a frequency smoothed estimate of the spectral coherence of a M-variate complex Gaussian time series with mutually independent components when the dimension M and the number of samples N both converge to infinity. If B denotes the smoothing span of the underlying smoothed periodogram estimator, a type I extreme value limiting distribution is obtained under the rate assumptions MN→0 and MB→c∈(0,+∞). This result is then exploited to build a statistic with controlled asymptotic level for testing independence between the M components of the observed time series. Numerical simulations support our results.

Regularizing effects of the entropy functional in optimal transport and planning problems

We analyze optimal transport problems with additional entropic cost evaluated along curves in the Wasserstein space which join two probability measures m0,m1. The effect of the additional entropy functional results into an elliptic regularization for the (so-called) Kantorovich potentials of the dual problem. Assuming the initial and terminal measures to be positive and smooth, we prove that the optimal curve remains smooth for all time. We focus on the case that the transport problem is set on a convex bounded domain in the d-dimensional Euclidean space (with no-flux condition on the boundary), but we also mention the case of Gaussian-like measures in the whole space. The approach follows ideas introduced by P.-L. Lions in the theory of mean-field games [27]. The result provides with a smooth approximation of minimizers in optimization problems with penalizing congestion terms, which appear in mean-field control or mean-field planning problems. This allows us to exploit new estimates for this kind of problems by using displacement convexity properties in the Eulerian approach.

Cloud-Edge-End Collaboration in Air–Ground Integrated Power IoT: A Semidistributed Learning Approach

The combination of air–ground integrated power Internet of Things (AGI-PIoT) and cloud-edge-end collaboration enables flexible coverage and real-time data processing. However, how to achieve intelligent cloud-edge-end collaboration in AGI-PIoT faces several challenges such as dynamics of aerial networks, coupling of resource allocation in multiple layers, timescales, and dimensions, incomplete information, and dimensionality curse. In this article, we propose a FEderated Deep rEinforcement leaRning-based multi-lAyer multi-Timescale multi-dImensional resOurce allocatioN algorithm (FEDERATION). The multilayer multitimescale multidimensional resource allocation problem is decomposed into three subproblems based on Lyapunov optimization. For the subproblem of joint task offloading and power control, a federated deep actor-critic-based semidistributed algorithm is developed. The subproblem of admission control is solved by quadratic programming. The third subproblem is addressed through smooth approximation and Lagrange dual decomposition. Simulation results indicate that FEDERATION outperforms existing algorithms in queuing delay, energy consumption, and convergence.

Optimal control approach for nonlinear chemical processes with uncertainty and application to a continuous stirred-tank reactor problem

Practical chemical process is usually a dynamic process including uncertainty. Stochastic constraints can be used to chemical process modeling, where constraints cannot be strictly satisfied or need not be fully satisfied. Thus, optimal control of nonlinear systems with stochastic constraints can be available to address practical nonlinear chemical process problems. This problem is hard to cope with due to the stochastic constraints. By introducing a novel smooth and differentiable approximation function, an approximation-based approach is proposed to address this issue, where the stochastic constraints are replaced by some deterministic ones. Following that, the stochastic constrained optimal control problem is converted into a deterministic parametric optimization problem. Convergence results show that the approximation function and the corresponding feasible set converge uniformly to that of the original problem. Then, the optimal solution of the deterministic parametric optimization problem is guaranteed to converge uniformly to the optimal solution of the original problem. Following that, a computation approach is proposed for solving the original problem. Numerical results, obtained from a nonlinear continuous stirred-tank reactor problem including stochastic constraints, show that the proposed approach is less conservative compared with the existing typical methods and can obtain a stable and robust performance when considering the small perturbations in initial system state.

Deep-Learning-Based Gas Leak Source Localization From Sparse Sensor Data

In this article, we address the problem of estimating the location of gas leak sources using sparse unreliable spatio-temporal chemical sensor data. We pose the task of estimating the underlying gas signal and predicting the source location as an inverse problem. For this purpose, we develop a novel deep-learning projection-based framework. We incorporate traditional projection-onto-convex-sets (POCS) iteration in the structure of the deep model to obtain a regularized solution that conforms to our prior knowledge of the spatio-temporal structure of the gas concentration distribution. We use discrete cosine transform (DCT) layers to model the smooth nature of the gas plume signal. In the DCT domain, we project the feature maps onto a low-pass region, whose boundary is determined during training using the backpropagation algorithm. This operation is equivalent to projecting onto a convex set. Furthermore, these projection operations are embedded in the nonlinear structure of a convolutional neural network. We address two different types of data: methane–propane leak from industrial plants and isopropyl alcohol (isopropanol) vapor leak in an indoor environment. Experimental results are presented. Our results show that we can obtain a smooth estimate of the underlying gas signal while obtaining a good source location prediction with high accuracy.

A presmoothing approach for estimation in the semiparametric Cox mixture cure model

A challenge when dealing with survival analysis data is accounting for a cure fraction, meaning that some subjects will never experience the event of interest. Mixture cure models have been frequently used to estimate both the probability of being cured and the time to event for the susceptible subjects, by usually assuming a parametric (logistic) form of the incidence. We propose a new estimation procedure for a parametric cure rate that relies on a preliminary smooth estimator and is independent of the model assumed for the latency. On a second stage one can assume a semiparametric model for the latency and estimate also the survival distribution of the uncured subject. For the particular case of the logistic/Cox model, we investigate the theoretical properties of the estimators and show through simulations that presmoothing leads to more accurate results compared to the maximum likelihood estimator. To illustrate the practical use, we apply the new estimation procedure to two studies of melanoma survival data.

Towards Differentiable Agent-Based Simulation

Simulation-based optimization using agent-based models is typically carried out under the assumption that the gradient describing the sensitivity of the simulation output to the input cannot be evaluated directly. To still apply gradient-based optimization methods, which efficiently steer the optimization towards a local optimum, gradient estimation methods can be employed. However, many simulation runs are needed to obtain accurate estimates if the input dimension is large. Automatic differentiation (AD) is a family of techniques to compute gradients of general programs directly. Here, we explore the use of AD in the context of time-driven agent-based simulations. By substituting common discrete model elements such as conditional branching with smooth approximations, we obtain gradient information across discontinuities in the model logic. On the examples of a synthetic grid-based model, an epidemics model, and a microscopic traffic model, we study the fidelity and overhead of the differentiable simulations as well as the convergence speed and solution quality achieved by gradient-based optimization compared with gradient-free methods. In traffic signal timing optimization problems with high input dimension, the gradient-based methods exhibit substantially superior performance. A further increase in optimization progress is achieved by combining gradient-free and gradient-based methods. We demonstrate that the approach enables gradient-based training of neural network-controlled simulation entities embedded in the model logic. Finally, we show that the performance overhead of differentiable agent-based simulations can be reduced substantially by exploiting sparsity in the model logic.