Optimal Sampling Problem Research Articles

Sampling for Remote Estimation of the Wiener Process over an Unreliable Channel

In this paper, we study a sampling problem where a source takes samples from a Wiener process and transmits them through a wireless channel to a remote estimator. Due to channel fading, interference, and potential collisions, the packet transmissions are unreliable and could take random time durations. Our objective is to devise an optimal causal sampling policy that minimizes the long-term average mean square estimation error. This optimal sampling problem is a recursive optimal stopping problem, which is generally quite difficult to solve. However, we prove that the optimal sampling strategy is, in fact, a simple threshold policy where a new sample is taken whenever the instantaneous estimation error exceeds a threshold. This threshold remains a constant value that does not vary over time. By exploring the structure properties of the recursive optimal stopping problem, a low-complexity iterative algorithm is developed to compute the optimal threshold. This work generalizes previous research by incorporating both transmission errors and random transmission times into remote estimation. Numerical simulations are provided to compare our optimal policy with the zero-wait and age-optimal policies.

Sampling and Remote Estimation for the Ornstein-Uhlenbeck Process Through Queues: Age of Information and Beyond

Recently, a connection between the age of information and remote estimation error was found in a sampling problem of Wiener processes: If the sampler has no knowledge of the signal being sampled, the optimal sampling strategy is to minimize the age of information; however, by exploiting causal knowledge of the signal values, it is possible to achieve a smaller estimation error. In this paper, we generalize the previous study by investigating a problem of sampling a stationary Gauss-Markov process named the Ornstein-Uhlenbeck (OU) process, where we aim to find useful insights for solving the problems of sampling more general signals. The optimal sampling problem is formulated as a constrained continuous-time Markov decision process (MDP) with an uncountable state space. We provide an exact solution to this MDP: The optimal sampling policy is a threshold policy on <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">instantaneous estimation error</i> and the threshold is found. Further, if the sampler has no knowledge of the OU process, the optimal sampling problem reduces to an MDP for minimizing a <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">nonlinear</i> age of information metric. The age-optimal sampling policy is a threshold policy on <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">expected estimation error</i> and the threshold is found. In both problems, the optimal sampling policies can be computed by low-complexity algorithms (e.g., bisection search and Newton’s method), and the curse of dimensionality is circumvented. These results hold for (i) general service time distributions of the queueing server and (ii) sampling problems both with and without a sampling rate constraint. Numerical results are provided to compare different sampling policies.

Asymptotic Optimality of the Triangular Lattice for a Class of Optimal Location Problems

We prove an asymptotic crystallization result in two dimensions for a class of nonlocal particle systems. To be precise, we consider the best approximation with respect to the 2-Wasserstein metric of a given absolutely continuous probability measure f mathrm {d}x by a discrete probability measure sum _i m_i delta _{z_i}, subject to a constraint on the particle sizes m_i. The locations z_i of the particles, their sizes m_i, and the number of particles are all unknowns of the problem. We study a one-parameter family of constraints. This is an example of an optimal location problem (or an optimal sampling or quantization problem) and it has applications in economics, signal compression, and numerical integration. We establish the asymptotic minimum value of the (rescaled) approximation error as the number of particles goes to infinity. In particular, we show that for the constrained best approximation of the Lebesgue measure by a discrete measure, the discrete measure whose support is a triangular lattice is asymptotically optimal. In addition, we prove an analogous result for a problem where the constraint is replaced by a penalization. These results can also be viewed as the asymptotic optimality of the hexagonal tiling for an optimal partitioning problem. They generalise the crystallization result of Bourne et al. (Commun Math Phys, 329: 117–140, 2014) from a single particle system to a class of particle systems, and prove a case of a conjecture by Bouchitté et al. (J Math Pures Appl, 95:382–419, 2011). Finally, we prove a crystallization result which states that optimal configurations with energy close to that of a triangular lattice are geometrically close to a triangular lattice.

Adaptive subset searching-based deep neural network method for structural reliability analysis

To deal with the problem of local optimal sampling in active learning methods for structural reliability analysis, an adaptive subset searching-based deep neural network (ASS-DNN) method is proposed. The proposed ASS-DNN method uses a global to local strategy to iteratively select important subsets close to the limit state surface from the Monte Carlo population to update a deep neural network (DNN) for calculating the structural failure probability. In each iteration, first a subset is selected from the previous subset of the Monte Carlo population to update the DNN of the limit state function. Second, the updated DNN is used to predict the failure probability and the conditional probability of the next subset in the current subset. Third, the convergence of the conditional probability is evaluated to evaluate the prediction accuracy of the DNN on the current subset. If the convergence condition is met, the sampling range of the next iteration is moved to a subset closer to the limit state surface, otherwise the sampling range remains in the current subset. Finally, the iteration is repeated until the required accuracy of the predicted failure probability is met. The efficiency and accuracy of the ASS-DNN method were demonstrated using three examples.

Optimal Sampling Pattern for the Free Final Time Linear Quadratic Regulator

The optimal sampling problem is the selection of the optimal sampling instants together with the optimal control actions such that a given cost function is minimized. In this article, we solve the optimal sampling problem for the free final time linear quadratic regulator. Each optimal sampling instant is computed as the minimization of a maximum eigenvalue problem that is formulated at each stage by previously applying dynamic programming. The solution provides the optimal sampling instants, control actions, and the optimal final time in a recursive and constructive way. Furthermore, the solution is optimal for any arbitrary number of control moves <formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex>$N \geq 1$</tex></formula> , as it is not based on asymptotic arguments. Two application examples show the feasibility of the approach.

Do Log Factors Matter? On Optimal Wavelet Approximation and the Foundations of Compressed Sensing

A signature result in compressed sensing is that Gaussian random sampling achieves stable and robust recovery of sparse vectors under optimal conditions on the number of measurements. However, in the context of image reconstruction, it has been extensively documented that sampling strategies based on Fourier measurements outperform this purportedly optimal approach. Motivated by this seeming paradox, we investigate the problem of optimal sampling for compressed sensing. Rigorously combining the theories of wavelet approximation and infinite-dimensional compressed sensing, our analysis leads to new error bounds in terms of the total number of measurements m for the approximation of piecewise \(\alpha \)-Hölder functions. Our theoretical findings suggest that Fourier sampling outperforms random Gaussian sampling when the Hölder exponent \(\alpha \) is large enough. Moreover, we establish a provably optimal sampling strategy. This work is an important first step towards the resolution of the claimed paradox and provides a clear theoretical justification for the practical success of compressed sensing techniques in imaging problems.

Optimal sampling pattern for free final time linear quadratic regulator: the scalar case

The optimal sampling problem is the selection of the optimal sampling instants together with the optimal control actions such that a given cost function is minimised. In this article, we solve the optimal sampling problem for free final time linear quadratic regulator with scalar dynamical system. The solution provides the optimal sampling instants, control actions and the optimal final time in a recursive and constructive way for any arbitrary number of samples , as it is not based on asymptotic arguments. An application example shows the feasibility of the approach.

Basic approaches to simulation of resist mask formation in computational lithography

Main currently used resist mask formation models and problems solved have been overviewed. Stages of "full physical simulation" have been briefly analyzed based on physicochemical principles for conventional diazonapthoquinone (DNQ) photoresists and chemically enhanced ones. We have considered the concepts of the main currently used compact models predicting resist mask contours for full-scale product topologies, i.e., VT5 (Variable Threshold 5) and CM1 (Compact Model 1). Computation examples have been provided for full and compact resist mask formation models. Full resist mask formation simulation has allowed us to optimize the lithographic stack for a new process. Optimal thickness ratios have been found for the binary anti-reflecting layers used in water immersion lithography. VT5 compact model calibration has allowed us to solve the problem of optimal calibration structure sampling for maximal coverage of optical image parameters space while employing the minimal number of structures. This problem has been solved using cluster analysis. Clustering has been implemented using the k-means method. The optimum sampling is 300 to 350 structures, the rms error being 1.4 nm which is slightly greater than the process noise for 100 nm structures. The use of SEM contours for VT5 model calibration allows us to reduce the rms error to 1.18 nm for 40 structures.

Speed, Accuracy, and the Optimal Timing of Choices

We model the joint distribution of choice probabilities and decision times in binary decisions as the solution to a problem of optimal sequential sampling, where the agent is uncertain of the utility of each action and pays a constant cost per unit time for gathering information. We show that choices are more likely to be correct when the agent chooses to decide quickly, provided the agent’s prior beliefs are correct. This better matches the observed correlation between decision time and choice probability than does the classical drift-diffusion model (DDM), where the agent knows the utility difference between the choices. (JEL C41, D11, D12, D83)

Optimal design of orders of DFrFTs for sparse representations

This study proposes an optimal design of the orders of the discrete fractional Fourier transforms (DFrFTs) and construct an overcomplete transform using the DFrFTs with these orders for performing the sparse representations. The design problem is formulated as an optimisation problem with an L 1 -norm non-convex objective function. To avoid all the orders of the DFrFTs to be the same, the exclusive OR of two constraints are imposed. The constrained optimisation problem is further reformulated to an optimal frequency sampling problem. A method based on solving the roots of a set of harmonic functions is employed for finding the optimal sampling frequencies. As the designed overcomplete transform can exploit the physical meanings of the signals in terms of representing the signals as the sums of the components in the time-frequency plane, the designed overcomplete transform can be applied to many applications.

Improved orientation sampling for indexing diffraction patterns of polycrystalline materials

Orientation mapping is a widely used technique for revealing the microstructure of a polycrystalline sample. The crystalline orientation at each point in the sample is determined by analysis of the diffraction pattern, a process known as pattern indexing. A recent development in pattern indexing is the use of a brute-force approach, whereby diffraction patterns are simulated for a large number of crystalline orientations and compared against the experimentally observed diffraction pattern in order to determine the most likely orientation. Whilst this method can robustly identify orientations in the presence of noise, it has very high computational requirements. In this article, the computational burden is reduced by developing a method for nearly optimal sampling of orientations. By using the quaternion representation of orientations, it is shown that the optimal sampling problem is equivalent to that of optimally distributing points on a four-dimensional sphere. In doing so, the number of orientation samples needed to achieve a desired indexing accuracy is significantly reduced. Orientation sets at a range of sizes are generated in this way for all Laue groups and are made available online for easy use.

Stochastic Choice and Optimal Sequential Sampling

We model the joint distribution of choice probabilities and decision times in binary choice tasks as the solution to a problem of optimal sequential sampling, where the agent is uncertain of the utility of each action and pays a constant cost per unit time for gathering information. In the resulting optimal policy, the agent's choices are more likely to be correct when the agent chooses to decide quickly, provided that the agent's prior beliefs are correct. For this reason it better matches the observed correlation between decision time and choice probability than does the classical drift-diffusion model, where the agent is uncertain which of two actions is best but knows the utility difference between them.

On Minimum Variance Optimal Controlled Sampling: A Simplified Approach

Controlled sampling, a technique of avoiding the nonpreferred samples in favor of preferred samples while maintaining the initial probabilities of each unit, is widely used due to its practical importance. With the introduction of linear programming methods, the time and labor involved in solving a controlled sampling problem have been reduced to a large extent. In this problem, we introduce a simple technique to solve the optimal controlled sampling problem with minimum variance, using the multiple objective linear programming approach. Some examples are considered to demonstrate the utility of the proposed procedure in comparison to the existing controlled selection procedures.

The Optimal Sampling Pattern for Linear Control Systems

In digital control systems, the state is sampled at given sampling instants and the input is kept constant between two consecutive instants. With the optimal sampling problem, we mean the selection of sampling instants and control inputs, such that a given function of the state and input is minimized. In this paper, we formulate the optimal sampling problem and we derive a necessary condition for the optimality of a set of sampling instants in the linear quadratic regulator problem. Since the numerical solution of the optimal sampling problem is very time consuming, we also propose a new quantization-based sampling strategy that is computationally tractable and capable of achieving near-optimal cost. Finally, and probably most interesting of all, we prove that the quantization-based sampling is optimal in first-order systems for a large number of samples. Experiments demonstrate that quantization-based sampling has near-optimal performance even when the system has a higher order. However, it is still an open question whether quantization-based sampling is asymptotically optimal in any case.

Optimal representation of non-stationary random fields with finite numbers of samples: A linear MMSE framework

In this article we consider the representation of a finite-energy non-stationary random field with a finite number of samples. We pose the problem as an optimal sampling problem where we seek the optimal sampling interval under the mean-square error criterion, for a given number of samples. We investigate the optimum sampling rates and the resulting trade-offs between the number of samples and the representation error. In our numerical experiments, we consider a parametric non-stationary field model, the Gaussian–Schell model, and present sampling schemes for varying noise levels and for sources with varying numbers of degrees of freedom. We discuss the dependence of the optimum sampling interval on the problem parameters. We also study the sensitivity of the error to the chosen sampling interval.

Adaptive Information Collection by Robotic Sensor Networks for Spatial Estimation

This work deals with trajectory optimization for a robotic sensor network sampling a spatio-temporal random field. We examine the optimal sampling problem of minimizing the maximum predictive variance of the estimator over the space of network trajectories. This is a high-dimensional, multi-modal, nonsmooth optimization problem, known to be NP-hard even for static fields and discrete design spaces. Under an asymptotic regime of near-independence between distinct sample locations, we show that the solutions to a novel generalized disk-covering problem are solutions to the optimal sampling problem. This result effectively transforms the search for the optimal trajectories into a geometric optimization problem. Constrained versions of the latter are also of interest as they can accommodate trajectories that satisfy a maximum velocity restriction on the robots. We characterize the solution for the unconstrained and constrained versions of the geometric optimization problem as generalized multicircumcenter trajectories, and provide algorithms which enable the network to find them in a distributed fashion. Several simulations illustrate our results.

Spatially Regularized Compressed Sensing for High Angular Resolution Diffusion Imaging

Despite the relative recency of its inception, the theory of compressive sampling (aka compressed sensing) (CS) has already revolutionized multiple areas of applied sciences, a particularly important instance of which is medical imaging. Specifically, the theory has provided a different perspective on the important problem of optimal sampling in magnetic resonance imaging (MRI), with an ever-increasing body of works reporting stable and accurate reconstruction of MRI scans from the number of spectral measurements which would have been deemed unacceptably small as recently as five years ago. In this paper, the theory of CS is employed to palliate the problem of long acquisition times, which is known to be a major impediment to the clinical application of high angular resolution diffusion imaging (HARDI). Specifically, we demonstrate that a substantial reduction in data acquisition times is possible through minimization of the number of diffusion encoding gradients required for reliable reconstruction of HARDI scans. The success of such a minimization is primarily due to the availability of spherical ridgelet transformation, which excels in sparsifying HARDI signals. What makes the resulting reconstruction procedure even more accurate is a combination of the sparsity constraints in the diffusion domain with additional constraints imposed on the estimated diffusion field in the spatial domain. Accordingly, the present paper describes an original way to combine the diffusion- and spatial-domain constraints to achieve a maximal reduction in the number of diffusion measurements, while sacrificing little in terms of reconstruction accuracy. Finally, details are provided on an efficient numerical scheme which can be used to solve the aforementioned reconstruction problem by means of standard and readily available estimation tools. The paper is concluded with experimental results which support the practical value of the proposed reconstruction methodology.

Optimal sub-Nyquist nonuniform sampling and reconstruction for multiband signals

We study the problem of optimal sub-Nyquist sampling for perfect reconstruction of multiband signals. The signals are assumed to have a known spectral support /spl Fscr/ that does not tile under translation. Such signals admit perfect reconstruction from periodic nonuniform sampling at rates approaching Landau's (1967) lower bound equal to the measure of /spl Fscr/. For signals with sparse /spl Fscr/, this rate can be much smaller than the Nyquist rate. Unfortunately the reduced sampling rates afforded by this scheme can be accompanied by increased error sensitivity. In a previous study, we derived bounds on the error due to mismodeling and sample additive noise. Adopting these bounds as performance measures, we consider the problems of optimizing the reconstruction sections of the system, choosing the optimal base sampling rate, and designing the nonuniform sampling pattern. We find that optimizing these parameters can improve system performance significantly. Furthermore, uniform sampling is optimal for signals with /spl Fscr/ that tiles under translation. For signals with nontiling /spl Fscr/, which are not amenable to efficient uniform sampling, the results reveal increased error sensitivities with sub-Nyquist sampling. However, these can be controlled by optimal design, demonstrating the potential for practical multifold reductions in sampling rate.