Sequential Approximation Method Research Articles

Harnessing 12-lead ECG and MRI data to personalise repolarisation profiles in cardiac digital twin models for enhanced virtual drug testing

Cardiac digital twins are computational tools capturing key functional and anatomical characteristics of patient hearts for investigating disease phenotypes and predicting responses to therapy. When paired with large-scale computational resources and large clinical datasets, digital twin technology can enable virtual clinical trials on virtual cohorts to fast-track therapy development. Here, we present an open-source automated pipeline for personalising ventricular electrophysiological function based on routinely acquired magnetic resonance imaging (MRI) data and the standard 12-lead electrocardiogram (ECG).Using MRI-based anatomical models, a sequential Monte-Carlo approximate Bayesian computational inference method is extended to infer electrical activation and repolarisation characteristics from the ECG. Fast simulations are conducted with a reaction-Eikonal model, including the Purkinje network and biophysically-detailed subcellular ionic current dynamics for repolarisation. For each patient, parameter uncertainty is represented by inferring an envelope of plausible ventricular models rather than a single one, which means that parameter uncertainty can be propagated to therapy evaluation. Furthermore, we have developed techniques for translating from reaction-Eikonal to monodomain simulations, which allows more realistic simulations of cardiac electrophysiology. The pipeline is demonstrated in three healthy subjects, where our inferred pseudo-diffusion reaction-Eikonal models reproduced the patient's ECG with a median Pearson's correlation coefficient of 0.9, and then translated to monodomain simulations with a median correlation coefficient of 0.84 across all subjects. We then demonstrate our digital twins for virtual evaluation of Dofetilide with uncertainty quantification. These evaluations using our cardiac digital twins reproduced dose-dependent QTc and T peak to T end prolongations that are in keeping with large population drug response data.The methodologies for cardiac digital twinning presented here are a step towards personalised virtual therapy testing and can be scaled to generate virtual populations for clinical trials to fast-track therapy evaluation. The tools developed for this paper are open-source, documented, and made publicly available.

CALCULATION OF FLAT SHELLS BY THE BOUNDARY ELEMENT METHOD

This article describes the iterative process of solving the problem of the stress-strain state of a long flexible cylindrical panel. Stress-strain state and the study of the stability of flat shells taking into account the physical, mechanical properties of the material, the process of deformation change. It is based on sequential approximation methods and the boundary element method (MGE). The results of algorithmic weighting are presented in the form of a table showing what value each band represents. The equations are determined by solving twelve unknown values of functions at the ends of the segment. Linear problems of elasticity theory and plate theory fundamental solutions have a simple form, so the method is widely used here. For flat shells, the matrix of fundamental solutions is determined by complex volumetric expressions, and for flat shells - by special functions. Therefore, there is little research on solving problems in the theory of flat shells by the boundary element method. In this regard, an urgent topic of research is the development of methods of boundary integral equations for solving linear and nonlinear problems in the theory of flat shells based on the application of fundamental solutions defined by simple analytical expressions. The scientific novelty of the work consists in the development of a methodology for assessing the reliability of thin-walled spatial structures using the boundary element method.

Exact and Approximate Solution of Multi-Higher Order Fractional Differential Equations Via Sawi Transform and Sequential Approximation Method

Exact and Approximate Solution of Multi-Higher Order Fractional Differential Equations Via Sawi Transform and Sequential Approximation Method

ელექტროგამტარ სითხეში წრიული ფოროვანი ფირფიტის ბრუნვის სტაციონარული ამოცანა გაჟონვის სიჩქარის დიდი მნიშვნელობებისათვის თბოგადაცემით, მაგნიტური ველისა და ჯოულის სითბოს გათვალისწინებით

The stationary problem of rotation of a circular porous plate in an electrically conductive fluid is studied by the sequential approximation method (Green's function and small parameter method) for large values of the suction velocity with heat transfer, taking into account a weak magnetic field. The problem considers the case when the influence of dissipative members on heat transfer is very small, the energy equation takes into account the influence of Joule's heat on heat transfer, and it is assumed that the temperature along with the radius of the plate changes according to the square law. For solving the problem, the Navier-Stokes and energy partial differential equations of fluid motion in a magnetic field are reduced to ordinary nonlinear differential equations using generalized Karman embeddings, the solutions of which are sought in the form of infinite series for small values of the suction parameter. The solution of the problem by the means of Green's function is reduced to the solution of integral-differential equations. Recurrence formulas have been obtained, by the means of which it is possible to calculate the solution with any approximation. The first two approximations are clearly calculated. All kinematic characteristics of fluid flow are calculated. Images for heat transfer and pressure distribution on a circular plate are obtained. The moment of resistance to rotation of the plate and the heat transfer coefficient are also calculated.

Iterative sequential approximate solutions method to Hyers–Ulam stability of time-varying delayed fractional-order neural networks

There is a key issue in the field of stability analysis for fractional-order neural networks (FONNs) concerning whether the exact solution of the FONNs exists under the condition that a series of successive approximate solutions of the models can be obtained. If so, can the state differences between the approximate solutions and the exact solution be estimated? Hyers–Ulam stability, as one of the powerful tools to solve this problem, has attracted the attention of scholars, and various related research achievements have been reported. In view of this, this paper aims to investigate the Hyers–Ulam stability of FONNs with time-varying delays. Using the successive approximation method and the modified retarded Henry–Gronwall integral inequality, several conditions guaranteeing the existence and uniqueness of the exact solution of FONNs with time-varying delays are derived. Moreover, the corresponding results on the Hyers–Ulam–Rassias stability of the addressed models are given. Finally, a numerical example is designed to illustrate the efficiency and rationality of the theoretical results.

"Sprężanie cieczy z odwiertów eksploatacyjnych na powierzchnię z zastosowaniem metody sekwencyjnej aproksymacji"

The article discusses the issue of pushing water to the surface by injecting gas into the dome of a water pressure system. The layer is completely filled with liquid. This requires the creation of underground gas storages in the central upper part of the water pressure system. For this purpose, the water must be pressurized from the drilled and unloaded wells. A sequential approximation method is used to solve the problem, and formation of the reservoir occurs due to the compression of the fluid through the operating wells. Fluids and gases that enrich the water pressure system, field system and parameters are known. The boundaries of the latter are the contours of the pressure and the flow. Over time, the water pressure in the reservoir has been determined by changing the sum of the volume of the cavity at the edge of the reservoir, the capacity of the created gas and the amount of injected gas. Under the conditions considered, the movement of water in the areas bounded by the contour of the discharge and flow can be considered as radial. Since this area is not very large, the elasticity of the water and the porosity of the reservoir can be ignored. This issue can be considered as the filtration of incompressible fluid in a non-deformable bed. With a relatively small change in flow rate due to the constant pressure in the discharge circuit, the water compressed by the gas flows freely through the wellhead of the discharge well.

The use of sequential approximation method for risk determination in problems of geotechnical mechanics

Most processes in technical improvements are deterministic, therefore, the concept of risk as a product of the probability of an accident occurrence on the financial costs of its elimination, which is proposed in most articles and regulatory documents, is not acceptable, since none of the project parameters is a random variable. In this regard, it is proposed to define risk as a technical system exceeding the values permissible by technical specifications, that is going beyond the operational capability. Before determining the degree of influence of parameters on the risk amount, it is necessary to determine sensitivity to their changing. Sensitivity analysis allows identifying parameters with the greatest influence on the risk of criterion going beyond the operational capability. However, in practice, it is not always possible to determine criterion sensitivity to the change of one or another parameter. In practice, typical situation is a problem to determine risk under conditions of simultaneous change of all parameters. Thus, a relevant method for risk calculation would be a method which allows determining risk sensitivity to the change of parameters and, at the same time, calculating the risk with simultaneous changes in all parameters. The sequential approximation method (SAM) makes it possible to calculate the risk with simultaneous changes of other parameters within a certain range using the information obtained during determining the risk sensitivity to the change of parameters. In the SAM, risk is represented in a multiplicative form, where the components of the product are the functions of one parameter. If the risk approximation is carried out in the form of a product of power functions, each of which depends on only one parameter, then the risk sensitivity to the change of the parameters can be approximately determined by the power indicators. The higher is the power, the greater is the influence of parameter on the risk. In this way, it is possible not only to make an approximate assessment of the influence of parameters on the criterion itself, but also to make conclusions about the importance of the influence of the system exceeding the permissible limits on the risk. In this work, the efficiency of the SAM method for determining the risks of parameters exceeding the permissible limits is demonstrated by the results of solving a classic problem of determining the stress-strain state in the neighborhood of the roadway with a circular cross-section by the finite element method. An algorithm for calculating risks based on specific examples is presented. In order to demonstrate the satisfactory accuracy of the criterion calculations, surfaces of the tangent stress intensity function obtained by the SAM method is compared with the interpolation surfaces obtained by numerical results. Conclusion is made about the ability of the method to determine the risks of the criterion exceeding the permissible limits and to provide satisfactory accuracy of the obtained results. Keywords: risk of loss of operational capacity, multiplicative form of representation, sensitivity of the function, change of parameters, neighborhood of a point, tangential stresses intensity.

Rectangular plate calculation algorithm for static loads with geometric nonlinearity

The aim of this study is to develop an algorithm for calculating rectangular plates for statistical loads taking into account geometrical nonlinearity on the basis of difference equations of the method of sequential approximations taking into account full and partial contact with elastic base. For our study, we used methods: finite elements method and method of sequential approximations, and theory of plate calculation considering large deflections. Results: a method, algorithm, and calculation algorithm for rectangular plates in the geometrically nonlinear formulation using difference equations of the method of sequential approximations (SEA) taking into account full and partial contact with the elastic base have been developed. Conclusion: An algorithm has been developed for the computation of rectangular plates in geometrically nonlinear formulation using difference equations of the method of sequential approximations. It is recommended to use generalized difference equations of finite difference method when calculating rectangular plates for the action of piecewise distributed transverse loads without taking into consideration interaction with elastic base. Having more compact form of recording, the solution obtained by using them has comparable accuracy as the version using SEA difference equations.

Trajectory Optimization and Channel Allocation for Delay Sensitive Secure Transmission in UAV-Relayed VANETs

This paper investigates the delay sensitive secure transmission problem in unmanned aerial vehicle (UAV)-relayed vehicular ad hoc networks (VANETs). Specifically, we take the security assurance into account and formulate this problem as a total information delay minimization problem by optimizing the UAV relay trajectory and channel allocation. To tackle this optimization problem, we propose an alternating optimization framework, where the UAV relay trajectory is solved by the Newton method, and the channel allocation is obtained by relax-and-round and sequential convex approximation methods. Finally, simulation results show the performance improvement of our algorithm compared with the current works in terms of the total information delay.

Классическое и слабое решение первой смешанной задачи для телеграфного уравнения с нелинейным потенциалом

We study the first mixed problem for the telegraph equation with a nonlinear potential in the first quadrant. We pose the Cauchy conditions on the lower base of the domain and the Dirichlet condition on the lateral boundary. By the method of characteristics, we obtain an expression for the solution of the problem in an implicit analytical form as a solution of some integral equations. To solve these equations, we use the method of sequential approximations. The existence and uniqueness of the classical solution under specific smoothness and matching conditions for given functions are proved. Under inhomogeneous matching conditions, we consider a problem with conjugation conditions. When the given data is not smooth enough, we construct a mild solution.

Performance Analysis and Waveform Optimization of Integrated FD-MIMO Radar-Communication Systems

Multiple-input multiple-output (MIMO) has been used in wireless communications to increase data rates via multiplexing or improve performance via diversity. Additionally, in radar systems, MIMO promises improved parameter identifiability and target resolution. Frequency diverse (FD)-MIMO radar can effectively distinguish targets that are closely spaced in the same angle cell by exploiting its range-angle-dependent transmit beamform. Therefore, in this paper, we first propose an integrated waveform design by embedding weighted, phase-modulated communication signals in the FD-MIMO radar waveform. Then, we derive the Cramer-Rao lower bound (CRLB) of the location estimation and analyze the communication performance achieved when the communication receiver extracts information from the integrated signals. Next, transmit beamforming is optimized to achieve the best tradeoff between the radar and communication performances of the system. Due to the nonconvexity of the optimization problem, we apply sequential parametric convex approximation (SPCA) and semidefinite relaxation (SDR) methods to solve the problem. The simulation results reveal the effects of some parameters, such as the frequency interval, symbol period, and communication symbol sequence, on radar and communication performance and demonstrate the tradeoff between radar and communication obtained by optimizing the transmit beamforming vector.

Stability of rectangular cantilever plates with high elasticity

The problem of cantilever plate stability has been little studied due to the difficulty of solving the corresponding boundary problem. The known approximate solutions mainly concern only the first critical load. In this paper, stability of an elastic rectangular cantilever plate under the action of uniform pressure applied to its edge opposite to the clamped edge is investigated. Under such conditions, thin canopies of buildings made of new materials can be found at sharp gusts of wind in longitudinal direction. At present, cantilever nanoplates are widely used as key components of sensors to create nanoscale transistors where they are exposed to magnetic fields in the plate plane. The aim of the study is to obtain the critical force spectrum and corresponding forms of supercritical equilibrium. The deflection function is selected as a sum of two hyperbolic trigonometric series with adding special compensating summands to the main symmetric solution for the free terms of the decomposition of the functions in the Fourier series by cosines. The fulfillment of all conditions of the boundary problem leads to an infinite homogeneous system of linear algebraic equations with regard to unknown series coefficients. The task of the study is to create a numerical algorithm that allows finding eigenvalues of the resolving system with high accuracy. The search for critical loads (eigenvalues) giving a nontrivial solution of this system is carried out by brute force search of compressive load value in combination with the method of sequential approximations. For the plates with different side ratios, the spectrum of the first three critical loads is obtained, at which new forms of equilibrium emerge. An antisymmetric solution is obtained and studied. 3D images of the corresponding forms are presented.

Joint Beamwidth and Power Optimization in MmWave Hybrid Beamforming-NOMA Systems

The use of directional transmission in millimeter-Wave (mmWave) frequencies results in limited channel coherence time. In this paper, we take the limited channel coherence time into account for non-orthogonal multiple access (NOMA) in mmWave hybrid beamforming systems. Due to the limited coherence time, the beamwidth of the hybrid beamformer affects the beam-training time, which in turn directly impacts the data transmission rate. To investigate this trade-off, we utilize a combined beam-training algorithm. Then, we formulate a sum-rate expression which considers the channel coherence time and beam-training time as well as users' power and other system parameters. Further, a joint power and beamwidth optimization problem is solved by iterating between the power allocation and the beamwidth optimization. When allocating the power, we use the log-exponential reformulation and the sequential parametric convex approximation (SPCA) methods to solve the non-convex problem. Since beamwidth optimization involves too many variables, we propose an algorithm which iterates between clusters of users. Numerical results show that the optimized mmWave hybrid beamforming-NOMA system can achieve much higher sum-rates compared to NOMA with analog beamforming and traditional multiple access techniques.

TRIPLE PROBABILITY DENSITY DISTRIBUTION MODEL IN THE TASK OF AVIATION RISK ASSESSMENT

The probability of an airplane deviation from pre-planned trajectory is a core of aviation safety analysis. We propose to use a mixture of three probability density distribution functions it the task of aviation risk assessment. Proposed model takes into account the effect of navigation system error, flight technical error, and occurrence of rare events. Univariate Generalized Error Distribution is used as a basic component of distribution functions, that configures the error distribution model from the normal error distribution to double exponential distribution function. Statistical fitting of training sample by proposed Triple Univariate Generalized Error Distribution (TUGED) is supported by Maximum Likelihood Method. Optimal set of parameters is estimated by sequential approximation method with defined level of accuracy. The developed density model has been used in risk assessment of airplane lateral deviation from runway centreline during take-off and landing phases of flight. The efficiency of the developed model is approved by Chi-square, Akaike’s, and Bayes information criteria. The results of TUGED fitting indicate better performance in comparison with double probability density distribution model. The risk of airplane veering off the runway is considered as the probability of a rare event occurrence and is estimated as an area under the TUGED.

Joint resource optimisation in cell‐free massive MIMO with low‐resolution ADCs

In this study, the uplink performance of cell-free massive multi-input multi-output (mMIMO) system with multi-antenna access points (APs) and users is investigated, assuming low-resolution analogue–digital converters (ADCs) are employed at the APs. By exploiting the additive quantisation noise model, a tight closed-form rate expression is derived. This tractable finding characterises the impacts of the multi-antenna APs and users, the imperfect quantisation error and the channel estimation error. In order to maximise the uplink sum-rate, a joint quantisation bit and power control problem is formulated, subjecting to the backhaul capacity and each user power constraints. The original resource optimisation problem is non-convex and it is decomposed into two sub-problems, namely quantisation bit design and power allocation problem, to alleviate the difficulties. In particular, the resultant two sub-problems can be efficiently determined by utilising the Lagrange Multiplier and sequential convex approximation methods, respectively. Finally, numerical simulations are presented to examine the analytical findings and evaluate the effectiveness of the proposed algorithm.

Sequential approximate reliability-based design optimization for structures with multimodal random variables

For practical engineering design problems, random variables tend to follow multimodal probability distributions when working at multiple operating conditions. For example, the structural fatigue stress of a steel bridge carrying both highway and railway traffic obeys the bimodal distribution. The existing popular reliability-based design optimization (RBDO) methods are mainly used to treat random variables with only unimodal distributions, which, therefore, tends to result in relatively large computational errors when multimodal random variables are involved. In this paper, a sequential approximate RBDO method is firstly proposed for engineering design involving multimodal random variables. The probability density function (PDF) of the response function is firstly calculated to assess the reliability of each probabilistic constraint, due to the existence of multimodal random variables. Then, a deterministic optimization problem is established to calculate candidate design points, based on the approximation that the response PDF demonstrates only transitional deformation in the optimization process. Three numerical examples and one engineering application of a thermal laptop design are presented to demonstrate the effectiveness of the proposed method.

Kriging Surrogate Model based Optimisation of Welded Bogie Frame for Fatigue Improvement

The welded side frame of intercity Electric Multiple Unit (EMU) bogie frame is taken as the research objective to optimize the layout of its internal stiffeners and to improve its fatigue performance. The fatigue accumulation damage value is computed in side frame for some key welds using Master S-N curve approach and the maximal value of fatigue accumulation damage is considered as a significant constraint to construct a lightweight optimisation model. To solve the optimisation model effectively, a sequential approximate method on the basis of Kriging surrogate model is presented by combining the minimum response surface sampling with multi-island genetic algorithm. The weight of the side frame is reduced by 16% after optimisation, which realized the fatigue improvement.

THE METHOD OF SUCCESSIVE APPROXIMATIONS IN THE MATHEMATICAL THE-ORY OF SHALLOW SHELLS OF ARBITRARY THICKNESS


 
 
 The method of sequential approximations (MSA) in mathematical theory (MT) of transversal-isotropic shallow shells of arbitrary thickness is developed. MT takes into account all components of stress-strain state (SSS). SSS and boundary conditions are considered to be functions of three varia-bles. Three-dimensional problems are reduced to two- dimensional decompositions of all the compo-nents of the SSS into series in the transverse coordinate using Legendre polynomials and using the Reisner variational principle. The boundary conditions for stresses on the front surfaces of the shell are fulfilled precisely. Previous studies have shown the high efficiency of this MT. The boundary-value problem for a shallow shell is reduced to sequences of two boundary-value problems for the respective plates. One sequence describes symmetric deformation relative to the median plane, and the other sequence is skew symmetric. MSA makes it easier to find a common solution of differential equations (DE) for shallow shells. Highly accurate results for SSS are already in the first approxi-mation. MSA can be used when solving problems for shallow shells by other theories.
 
 

Sequential Approximation of Functions in Sobolev Spaces Using Random Samples

We present an iterative algorithm for approximating an unknown function sequentially using random samples of the function values and gradients. This is an extension of the recently developed sequential approximation (SA) method, which approximates a target function using samples of function values only. The current paper extends the development of the SA methods to the Sobolev space and allows the use of gradient information naturally. The algorithm is easy to implement, as it requires only vector operations and does not involve any matrices. We present tight error bound of the algorithm, and derive an optimal sampling probability measure that results in fastest error convergence. Numerical examples are provided to verify the theoretical error analysis and the effectiveness of the proposed SA algorithm.

Equalization Strategy for Multi-Battery Energy Storage Systems Using Maximum Consistency Tracking Algorithm of the Conditional Depreciation

The neglect of the history depreciation imbalance in the conventional equalization strategies may aggravate the lifetime depreciation of the multi-battery energy storage systems (MBESSs) and overuse the high-history-depreciation batteries (HHDBs). This paper proposes an equalization strategy using maximum consistency tracking algorithm of the conditional depreciation to solve the stated problem. Conditional depreciation is innovatively defined to evaluate the operation equitability of the MBESSs. The maximum consistency tracking algorithm consists of two searching levels: the global searching level and the range searching level. A sequential approximation method is utilized at the global searching level to find the variation range consisting of the most equitable conditional depreciation. The newton downhill method is adopted at the range searching level to find the exact value of the most equitable conditional depreciation. The proposed equalization strategy is validated by a hardware-in-the-loop simulation system. Test results show that the proposed equalization strategy minimizes the imbalance of batteries’ conditional depreciations and decreases the HHDBs’ sharing power. Compared to the conventional equalization strategies, the proposed equalization strategy will protect the HHDBs from intermittent overuse and extend the lifetime of the MBESSs.