Simultaneous Approximation Research Articles

Approximation by polynomials in Sobolev spaces associated with classical moment functionals

Let textbf{u} be a moment functional associated with the Hermite, Laguerre, or Jacobi classical orthogonal polynomials. We study approximation by polynomials in H^r(textbf{u}), the Sobolev space consisting of functions whose derivatives of consecutive orders up to r belong to the L^2 space associated with textbf{u}. This requires the simultaneous approximation of a function f and its consecutive derivatives up to order Nleqslant r. We explicitly construct orthogonal polynomials that achieve such simultaneous approximation and provide error estimates in terms of E_n(f^{(r)}), the error of best approximation of f^{(r)} in L^{2}(textbf{u}).

Partially observed distributed optimization under unknown-but-bounded disturbances

In this paper, we consider non-stationary distributed optimization with partially observed parameters with acceleration based on the estimate sequence proposed by Y. Nesterov. We formulate this partial observability as time-varying communication matrix defined for each parameter separately. We propose the new distributed algorithm combining the accelerated Simultaneous Perturbation Stochastic Approximation (SPSA) and the described communication scheme as well as show its theoretical properties. The simulation validates the proposed algorithm in multi-sensor multi-target tracking problem over delayed channels.

Some Statistical Approximation based on Post-Widder operators

In the present paper, we study the convergence of real and modified Post-Widder operators in C, which is known as the extension of approximation of these operators from the real axis in the complex plane. In this direction, we also investigate error estimation in simultaneous approximation and a Voronovskaya-type asymptotic formula.

A Novel Numerical Method for Geothermal Reservoirs Embedded with Fracture Networks and Parameter Optimization for Power Generation

Geothermal recovery involves a coupled thermo-hydro-mechanical (THM) process in fractured rocks. A fluid transient equilibrium equation, considering thermal conduction, convection, and heat exchange, is established. The evolution of the reservoir permeability and the variance in the fracture aperture due to a change in the stress field are derived simultaneously. THM coupling is accomplished through iterative hydromechanical and thermo-hydro processes. To overcome the difficulty of geometric discretization, a three-dimensional THM coupler model embedded with discrete fracture networks, using a zero-thickness surface and line elements to simulate fractures and injection/production wells, is established to evaluate the geothermal production. The reliability of the method is verified by a case study. Then, this method is applied to evaluate the influence of the geometric topological characteristics of fracture networks and the fracture aperture on the reservoir temperature evolution and heat extraction effectiveness. The results show that the power generation efficiency and geothermal depletion rate are significantly affected by the injection–production pressure. Injection wells and production wells with pressures higher than the initial fluid pressure in the fractures can be used to significantly increase power generation, but the consumption of geothermal energy and loss of efficiency are significant and rapid. To achieve better benefits for the geothermal recovery system, an optimization algorithm based on simultaneous perturbation stochastic approximation (SPSA) is proposed; it takes the power generation efficiency as the objective function, and the corresponding program is developed using MATLAB to optimize the position and pressure values for each production well. The results show that the heat transfer for the entire EGS reservoir becomes more uniform after optimization, and the heat transfer efficiency is greatly improved.

Efficient parameter tuning to enhance practicability of a model-free vibration controller based on a virtual controlled object

In active vibration control, controller tuning is necessary to obtain a sufficient damping performance. This study presents an efficient tuning technique for model-free vibration controller based on a virtual controlled object (VCO). A model-free vibration control system is constructed by using an actuator model and the VCO. A reference controlled object (RCO), which is a designer-defined single-degree-of-freedom (SDOF) structure, is used to tune the VCO-controller offline. A novel parameter tuning technique based on the RCO and the simultaneous perturbation stochastic approximation (SPSA) is proposed considering the difference in scale of the tuning parameters. The proposed tuning scheme automatically determines the tuning parameter of the controller without manual trial-and-errors such as experiments with actual plants and is much faster than previous methods. Therefore, the proposed method significantly enhances the practicability of the VCO-based model-free vibration suppression, bridging the gap between the basic study of the VCO-method and its implementation. The effectiveness of the proposed method is verified by applying it to the VCO-based H2 controller.

Neural networks singular evolutive interpolated Kalman filter and its application to data assimilation for 2D water pollution model

Abstract The present study promotes a new algorithm for estimating the water pollution propagation with the primary goal of providing more reliable and high quality estimates to decision makers. To date, the widely used variational method suffers from the large computational burden, which limits its application in practice. Moreover, this method, considering the initial state as a control variable, is very sensitive in specifying initial error, especially for unstable dynamical systems. The Neural Network Filter (NNF), proposed in the present paper, is aimed at overcoming these two drawbacks in the variational method: by its nature, the NNF is sequential (no batch large assimilation window used) and stable even for unstable dynamics, with the gain parameters as control variables. The NNF, developed in the present paper, is a Neural Network Filter (NNF) version of the Singular Evolutive Interpolated Kalman Filter (SEIKF). One of the new versions of this NNF is that it uses structure of the gain of SEIKF0 taken by the SEIKF at the first time moment of correction process. To deal with the uncertainty of the system parameters and of the noise covariance, the proposed Neural Network SEIKF0 named by NNSEIKF0 makes use of the covariance of a reduced rank iterated during assimilation process and of some pertinent gain parameters tuned adaptively to yield the minimum prediction error for the system output. The computational burden in implementation of the NNSEIKF0 is reduced drastically due to applying the optimization tool known as a simultaneous perturbation stochastic approximation (SPSA) algorithm, which requires only two integrations of the numerical model. No iterative loop is required at each assimilation instant as usually happens with the standard gradient descent optimization algorithms. Data assimilation experiment, carried out by the SEIKF0 and NNSEIKF0, is implemented for the Thanh Nhan Lake in Hanoi and the performance comparison between the NNSEIKF0 and SEIKF0 is given to show the high efficiency of the proposed NNSEIKF0.

Measure theoretic laws for limsup sets defined by rectangles

In this paper, we present a general principle for the Lebesgue measure theory of limsup sets defined by rectangles under the hypothesis of ubiquity for rectangles. Applications are given for simultaneous Diophantine approximation, Khintchine-Groshev theorem for rectangles and higher dimensional shrinking target problems.

Idiographic Dynamic Modeling for Behavioral Interventions with Mixed Data Partitioning and Discrete Simultaneous Perturbation Stochastic Approximation.

This paper presents the use of discrete simultaneous perturbation stochastic approximation (DSPSA) as a routine method to efficiently determine features and parameters of idiographic (i.e. single subject) dynamic models for personalized behavioral interventions using various partitions of estimation and validation data. DSPSA is demonstrated as a valuable method to search over model features and regressor orders of AutoRegressive with eXogenous input estimated models using participant data from Just Walk (a behavioral intervention to promote physical activity in sedentary adults); results of DSPSA are compared to those of exhaustive search. In Just Walk, DSPSA efficiently and quickly estimates models of walking behavior, which can then be used to develop control systems to optimize the impacts of behavioral interventions. The use of DSPSA to evaluate models using various partitions of individual data into estimation and validation data sets also highlights data partitioning as an important feature of idiographic modeling that should be carefully considered.

Physics-guided identification of Euler–Bernoulli beam PDE model from full-field displacement response with SimultaNeous basis function Approximation and Parameter Estimation (SNAPE)

Full-field measurements of the continuous spatiotemporal response of the physical processes such as structural vibration or fluid flow generate large datasets. In many scientific fields, such continuous spatiotemporal dynamic models are represented by partial differential equations (PDEs). In the past, attempts have been made to identify the PDE models from the measured response by inferring its parameters by the use of either regression or deep learning-based techniques. But the previously presented regression-based methods fail to estimate the parameters of the higher-order PDE models in the presence of moderate noise. Likewise, the deep learning-based methods lack the much-needed property of repeatability and robustness in the identification of PDE models from the measured response. The authors introduced the method of SimultaNeous Basis Function Approximation and Parameter Estimation (SNAPE) in a recent paper which addresses such drawbacks by fitting basis functions to the measured response and simultaneously infer the parameters of the PDE model. In this paper the theory and formulation of SNAPE is presented to perform physics-guided identification of the Euler–Bernoulli beam PDE model which is widely applied in the modeling of large scale infrastructures to nanoscale structures.The domain knowledge of the physics is used as a constraint in the formulation of the optimization framework. The alternating direction method of multipliers (ADMM) algorithm is used to simultaneously optimize the loss function over the parameter space of the PDE model and coefficient space of the basis functions. The proposed method not only infers the parameters but also estimates a continuous function that approximates the solution to the PDE model. The efficacy of the method is both numerically and experimentally validated on noise corrupted full-field vibration response. The method neither requires the knowledge of the initial or boundary conditions of the beam nor comprises of model discretization error as in the case of finite element model updating. SNAPE demonstrates its applicability on various homogeneous and time-varying nonhomogeneous boundary conditions.

Simultaneous perturbation stochastic approximation: towards one-measurement per iteration

When measuring the value of a function to be minimized is not only expensive but also with noise, the popular simultaneous perturbation stochastic approximation (SPSA) algorithm requires only two function values in each iteration. In this paper, we present a method requiring only one function measurement value per iteration in the average sense. We prove the strong convergence and asymptotic normality of the new algorithm. Limited experimental results demonstrate the effectiveness and potential of our algorithm for solving low-dimensional problems.

Non-conforming Interface Conditions for the Second-Order Wave Equation

Imposition methods of interface conditions for the second-order wave equation with non-conforming grids is considered. The spatial discretization is based on high order finite differences with summation-by-parts properties. Previously presented solution methods for this problem, based on the simultaneous approximation term (SAT) method, have shown to introduce significant stiffness. This can lead to highly inefficient schemes. Here, two new methods of imposing the interface conditions to avoid the stiffness problems are presented: (1) a projection method and (2) a hybrid between the projection method and the SAT method. Numerical experiments are performed using traditional and order-preserving interpolation operators. Both of the novel methods retain the accuracy and convergence behavior of the previously developed SAT method but are significantly less stiff.

On the dimension drop conjecture for diagonal flows on the space of lattices

Let X=G/Γ, where G is a Lie group and Γ is a lattice in G, let U be an open subset of X, and let {gt} be a one-parameter subgroup of G. Consider the set of points in X whose gt-orbit misses U; it has measure zero if the flow is ergodic. It has been conjectured that this set has Hausdorff dimension strictly smaller than the dimension of X. This conjecture is proved when X is compact or when G is a simple Lie group of real rank 1. In this paper we prove this conjecture for the case G=SLm+n(R), Γ=SLm+n(Z) and gt=diag(ent,…,ent,e−mt,…,e−mt), in fact providing an effective estimate for the codimension. The proof uses exponential mixing of the flow together with the method of integral inequalities for height functions on SLm+n(R)/SLm+n(Z). We also discuss an application to the problem of improving Dirichlet's theorem in simultaneous Diophantine approximation.

Threshold Optimized CSWPL Behavioral Model for RF Power Transistors Based on Particle Swarm Algorithm

In this letter, the particle swarm optimization (PSO) is employed to optimize the threshold values of the canonical section-wise piecewise linear (CSWPL) function-based behavioral model of power transistors. The effectiveness of the proposed method is validated through measurements carried out on a 10-W GaN power transistor produced by Wolfspeed. Compared with the existing simultaneous perturbation stochastic approximation (SPSA) method, the developed modeling technique not only provides superior prediction performance across different input power levels but also finds the optimal thresholds much more efficiently.

On the Best Simultaneous Approximation of Functions in the Bergman Space B2

On the Best Simultaneous Approximation of Functions in the Bergman Space B2

Approximation Properties of the Vector Weak Rescaled Pure Greedy Algorithm

We first study the error performances of the Vector Weak Rescaled Pure Greedy Algorithm for simultaneous approximation with respect to a dictionary D in a Hilbert space. We show that the convergence rate of the Vector Weak Rescaled Pure Greedy Algorithm on A1(D) and the closure of the convex hull of the dictionary D is optimal. The Vector Weak Rescaled Pure Greedy Algorithm has some advantages. It has a weaker convergence condition and a better convergence rate than the Vector Weak Pure Greedy Algorithm and is simpler than the Vector Weak Orthogonal Greedy Algorithm. Then, we design a Vector Weak Rescaled Pure Greedy Algorithm in a uniformly smooth Banach space setting. We obtain the convergence properties and error bound of the Vector Weak Rescaled Pure Greedy Algorithm in this case. The results show that the convergence rate of the VWRPGA on A1(D) is sharp. Similarly, the Vector Weak Rescaled Pure Greedy Algorithm is simpler than the Vector Weak Chebyshev Greedy Algorithm and the Vector Weak Relaxed Greedy Algorithm.

Two-Dimensional Mechanical Model of Human Stability in External Force-Caused Fall

The paper proposes a two-dimensional model of human stability. The model allows for gaining data on forces and moments of forces being the effect of musculoskeletal interactions of body parts at human joints. The necessary input data are registered with the use of an optoelectronic motion capture system, two force plates, and a dynamometer. The latter measures the magnitude of the external force that is randomly applied to cause a backward fall. Therefore, the position of the participant’s body during the experiment, the external force used to cause the fall, and the ground reaction forces are known. The model proposed has the structure of a closed kinematic chain with one loop and two open subchains and uses the Euler–Newton approach to analyse the motion. In addition to external loads, linear and angular accelerations are also needed. To calculate those, we used a variant of approximation. This consists in carrying out a simultaneous approximation of a given function and its first and second derivatives in the subintervals that shift along the whole data range. The experiments conducted include two activities while maintaining a state of equilibrium and the fall itself. This gives the basis for examining whether the internal forces and moments at human joints differ significantly in both these states. The results and conclusions are discussed in the final part of the paper.

Diophantische Approximationen

This workshop was focused on a large variety of problems which have seen important progress during the last few years, such as extensions and refinements of Schmidt Subspace Theorem, together with new applications, works on the Zilber-Pink conjecture and on unlikely intersections, geometry of numbers, simultaneous Diophantine approximation, theory of heights, continued fractions, and arithmetic dynamics.

Modeling and inversion in acoustic-elastic coupled media using energy-stable summation-by-parts operators

Seismic data acquired at the seafloor are valuable in characterizing the subsurface and monitoring producing hydrocarbon fields. To fully use such data, a stable, accurate, and efficient numerical scheme is needed that accounts for acoustic and elastic wave propagation, their interaction at the seafloor interface, and for sources and receivers placed on either side of that interface. Existing methods either make incorrect assumptions or have high implementation and computational costs. We have developed a high-order finite-difference summation-by-parts framework for the acoustic-elastic wave equations in second-order form (in terms of displacements in the solid and an extension of velocity potential in the fluid). Our modified discretization of the elastic operator overcomes the dispersion errors known to plague displacement-based schemes in the high VP/ VS limit. We weakly impose boundary and interface conditions using simultaneous approximation terms, leading to an energy-stable numerical scheme that rigorously handles point injection and extraction. The fully discrete system is self-adjoint after a time reversal and a sign flip and is furthermore a high-order accurate discretization of the continuous problem. The self-adjointness ensures that forward and adjoint wavefields are computed with similar accuracy and simplifies gradient computation for inversion purposes. We find that our numerical scheme achieves accuracy comparable to a more computationally expensive spectral-element method and demonstrate its application to full-waveform inversion using the Marmousi2 model.

High Dimensional Origin Destination Calibration Using Metamodel Assisted Simultaneous Perturbation Stochastic Approximation

The huge traffic data generated by intelligent transportation system (ITS) leads to the development of many advanced traffic models. These traffic models consist of many adjustable parameters which need to be calibrated before they are used in practice. This paper focuses on the offline calibration of origin-destination (OD) input parameters of a simulation-based traffic model to match its output with sensor data. An improved Simultaneous Perturbation Stochastic Approximation (SPSA) algorithm called Metamodel Assisted Simultaneous Perturbation Stochastic Approximation (MSPSA) is proposed in this paper to calibrate high-dimensional OD parameters within a tight computational budget. The proposed MSPSA combines the gradient of SPSA with the gradient of a differentiable metamodel function to improve the calibration efficiency. An integer program is also used to fine-tune the OD estimates. The proposed MSPSA algorithm is tested on a simple synthetic toy network and complex road network of Kuala Lumpur (KL), Malaysia. The proposed MSPSA algorithm is compared against SPSA and state-of-the-art Weighted Simultaneous Perturbation Stochastic Approximation (WSPSA) in both transportation networks. For KL network, synthetic and real-world sensor measurements are used as ground truth references to evaluate the performance of each approach. Based on the simulation results, the proposed MSPSA algorithm is able to gain at least 50% of improvement as compared to SPSA and WSPSA in both synthetic and real-world scenarios.

Simultaneous UAV having actively sweep angle morphing wing and flight control system design

PurposeThe purpose of this paper is to improve autonomous flight performance of an unmanned aerial vehicle (UAV) having actively sweep angle morphing wing using simultaneous UAV and flight control system (FCS) design.Design/methodology/approachAn UAV is remanufactured in the ISTE Unmanned Aerial Vehicle Laboratory. Its wing sweep angle can vary actively during flight. FCS parameters and wing sweep angle are simultaneously designed to optimize autonomous flight performance index using a stochastic optimization method called as simultaneous perturbation stochastic approximation (SPSA). Results obtained are applied for flight simulations.FindingsUsing simultaneous design process of an UAV having actively sweep angle morphing wing and FCS design, autonomous flight performance index is maximized.Research limitations/implicationsAuthorization of Directorate General of Civil Aviation in Turkey is crucial for real-time UAV flights.Practical implicationsSimultaneous UAV having actively sweep angle morphing wing and FCS design process is so beneficial for recovering UAV autonomous flight performance index.Social implicationsSimultaneous UAV having actively sweep angle morphing wing and FCS design process achieves confidence, high autonomous performance index and simple service demands of UAV operators.Originality/valueComposing a novel approach to improve autonomous flight performance index (e.g. less settling and rise time, less overshoot meanwhile trajectory tracking) of an UAV and creating an original procedure carrying out simultaneous UAV having actively sweep angle morphing wing and FCS design idea.