Approximation Method Research Articles

Efficient, multimodal, and derivative-free bayesian inference with Fisher–Rao gradient flows

Abstract In this paper, we study efficient approximate sampling for probability distributions known up to normalization constants. We specifically focus on a problem class arising in Bayesian inference for large-scale inverse problems in science and engineering applications. The computational challenges we address with the proposed methodology are: (i) the need for repeated evaluations of expensive forward models; (ii) the potential existence of multiple modes; and (iii) the fact that gradient of, or adjoint solver for, the forward model might not be feasible.

While existing Bayesian inference methods meet some of these challenges individually, we propose a framework that tackles all three systematically. Our approach builds upon the Fisher-Rao gradient flow in probability space, yielding a dynamical system for probability densities that converges towards the target distribution at a uniform exponential rate. This rapid convergence is advantageous for the computational burden outlined in (i). We apply Gaussian mixture approximations with operator splitting techniques to simulate the flow numerically; the resulting approximation can capture multiple modes thus addressing (ii). Furthermore, we employ the Kalman methodology to facilitate a derivative-free update of these Gaussian components and their respective weights, addressing the issue in (iii).

The proposed methodology results in an efficient derivative-free posterior approximation method, flexible enough to handle multi-modal distributions: Gaussian Mixture Kalman Inversion (GMKI). The effectiveness of GMKI is demonstrated both theoretically and numerically in several experiments with multimodal target distributions, including proof-of-concept and two-dimensional examples, as well as a large-scale application: recovering the Navier-Stokes initial condition from solution data at positive times.



Application of three-dimensional MHD equilibrium calculation coupled with plasma response to island divertor experiments on J-TEXT

Three-dimensional (3D) equilibrium calculations, including the plasma rotation shielding effect to resonant magnetic perturbations (RMPs) produced by the island divertor (ID) coils, were carried out using the HINT and MARS-F codes on J-TEXT. Validation of 3D equilibrium calculations with experimental observations demonstrates that the shielding effect will prevent the penetration of the edge m/n = 3/1 mode component when the ID coil current is 4 kA, while change the size of magnetic islands once the current exceeds the penetration threshold. This indicates that equilibrium calculations including the plasma rotation shielding effect to RMPs can lead to better agreements with experimental observations compared to the vacuum approximation method. Additionally, the magnetic topology at the boundary undergoes changes, impacting the interaction between the plasma and the target plate. These results may be important in understanding RMP effects on edge transport and magnetohydrodynamic (MHD) instability control, as well as divertor heat and particle flux distribution control.

Towards non-virtual inertia control of renewable energy for frequency regulation: Modeling, analysis and new control scheme

Currently, when renewable generation participates in frequency regulation, the traditional control method is to emulate synchronous generators through virtual inertia control. However, virtual inertia has a time delay, so essentially, it is a fast power response. Meanwhile, virtual inertia control is likely to be affected by frequency fluctuation since it responds to the derivative of frequency. Hence, it’s worth exploring non-virtual inertia control for renewable energy when participating in frequency regulation. For this reason, a novel two-segment droop control scheme for renewable energy frequency regulation is proposed in this research. Firstly, the extended system frequency regulation (SFR) model, which contains virtual inertia with time delay, is built and analytically solved by order decrement based on the Routh approximation method. Afterwards, according to the analytical solution, time delay affects the frequency response of renewable energy. It can also be analytically proved that the non-virtual inertia control, e.g., sole droop control, could replace virtual inertia under the same frequency deviation. Still, more energy may be needed for frequency regulation. Furthermore, a novel two-segment droop control is presented, and to analytically prove its ability to replace virtual inertia, the impulse function balancing principle and the integration by parts algorithm were adopted to address the initial conditions of the differential equation. Based on the analytical expression, it can be analytically proved that a lower frequency deviation can be obtained under the same frequency regulation energy. Accordingly, a parameter-setting method for two-segment droop control was proposed. Finally, the effectiveness of the proposed method is verified by using a two-area system frequency response model, and the results reveal that it can be used to replace virtual inertia and has better performance.

Evaluation system of vocational education construction in China based on linked TOPSIS analysis

In response to the task of vocational education reform in the new era, in order to cultivate innovative skilled talents, it is necessary to improve traditional evaluation methods to enhance their efficiency. This study proposes an evaluation model for vocational education construction that combines grey relational analysis and approximate ideal point method (TOPSIS). By selecting indicators and combining subjective analytic hierarchy process and objective entropy weight method for weight allocation, the weights of the indicators are determined using subjective analytic hierarchy process and objective entropy weight method. Through experimental analysis, the results showed that among the 11 analysis indicators, the weight of policy environment indicators was the highest at 0.124, while the weight of employment rate and contract signing rate indicators was the lowest at 0.050. The above results indicate that the TOPSIS comprehensive evaluation system based on grey correlation analysis has better application effects in practical scenarios and is more suitable for the development requirements of vocational education.

Large-scale annotation of biochemically relevant pockets and tunnels in cognate enzyme–ligand complexes

Tunnels in enzymes with buried active sites are key structural features allowing the entry of substrates and the release of products, thus contributing to the catalytic efficiency. Targeting the bottlenecks of protein tunnels is also a powerful protein engineering strategy. However, the identification of functional tunnels in multiple protein structures is a non-trivial task that can only be addressed computationally. We present a pipeline integrating automated structural analysis with an in-house machine-learning predictor for the annotation of protein pockets, followed by the calculation of the energetics of ligand transport via biochemically relevant tunnels. A thorough validation using eight distinct molecular systems revealed that CaverDock analysis of ligand un/binding is on par with time-consuming molecular dynamics simulations, but much faster. The optimized and validated pipeline was applied to annotate more than 17,000 cognate enzyme–ligand complexes. Analysis of ligand un/binding energetics indicates that the top priority tunnel has the most favourable energies in 75% of cases. Moreover, energy profiles of cognate ligands revealed that a simple geometry analysis can correctly identify tunnel bottlenecks only in 50% of cases. Our study provides essential information for the interpretation of results from tunnel calculation and energy profiling in mechanistic enzymology and protein engineering. We formulated several simple rules allowing identification of biochemically relevant tunnels based on the binding pockets, tunnel geometry, and ligand transport energy profiles.Scientific contributionsThe pipeline introduced in this work allows for the detailed analysis of a large set of protein–ligand complexes, focusing on transport pathways. We are introducing a novel predictor for determining the relevance of binding pockets for tunnel calculation. For the first time in the field, we present a high-throughput energetic analysis of ligand binding and unbinding, showing that approximate methods for these simulations can identify additional mutagenesis hotspots in enzymes compared to purely geometrical methods. The predictor is included in the supplementary material and can also be accessed at https://github.com/Faranehhad/Large-Scale-Pocket-Tunnel-Annotation.git. The tunnel data calculated in this study has been made publicly available as part of the ChannelsDB 2.0 database, accessible at https://channelsdb2.biodata.ceitec.cz/.

Mathematical Evaluation and Dynamic Transmissions of a Tumor Growth Model Using a Generalized Singular and Non-Local Kernel

Currently, fractional calculus plays a critical role in improving control techniques, analyzing disease transmission dynamics, and solving several other real-world problems. This research investigates the time-fractional tumor growth model using an innovative approach. The new modified fractional derivative operator employs a singular and non-local kernel, based on Atangana and Baleanu's concepts with the Caputo derivative. The tumor growth model used the newly modified fractional operator, which provided numerical simulation. With the introduction of this new operator, we provide significant analysis for the tumor growth epidemic model. We have proven the uniqueness and stability conditions of the model by utilizing Banach's fixed point theory and the Picard successive approximation method. Using the Laplace-Adomian decomposition method (LADM), we found the numerical solution to the Modified Atangana-Baleanu-Caputo derivative model. We have verified the convergence analysis of the suggested scheme. We ultimately utilize the suggested method to obtain numeric outcomes and simulations for the tumor growth model. The study investigates the effect of multiple biological variables on the transmission of tumor growth dynamics.

Application of experimental planning methods for an analytical description of the areas of operability of marine electromechanical systems

The topic of the study is to consider the possibility of using experimental planning methods to mathematically describe areas of performance in relation to marine electromechanical systems. A definition is given and a graphic illustration of the field of operability is provided. It is shown that information about the boundary of the field of operability allows solving the most important tasks of controlling the state of ship electromechanical systems of various functional purposes. Possible ways of solving the problem of constructing a field of efficiency and the difficulties of their practical implementation are given. The method of analytical description of the operability domain is considered, which assumes an independent approximation of each hypersurface that makes up the operability domain. It is shown that for such an approximation it is convenient to use polynomial dependencies, which can be obtained as a result of using experimental planning methods. The main advantages of this approach are considered, including the possibility of solving the problem in the case of a high dimension of the space of primary parameters, in the space of which the area of operability is built. It is noted that in order to obtain a unified analytical description of the field of operability, the properties of logical R-functions can be used, which make it possible to make the transition from logical functions to analytical dependencies. It is proved that polynomial dependencies have the properties of R-functions and can be used in such a transformation. The resulting analytical description of the health area is universal and does not depend on the configuration of the area itself. Unlike the known methods of analytical approximation of the field of operability, it is characterized by an exceptionally low methodological error and allows us to solve with high reliability the problem of parametric synthesis of marine electromechanical systems according to the criterion of the reserve of operability, as well as the tasks of evaluating and predicting their condition during operation. The considered example clearly demonstrated the sequence of solving the problem of analytical description of the field of operability and allowed us to conclude about the complex configuration of this area even for special cases and the simplest electromechanical systems.

Requirements for Standard and I/O Cell Library Domestic Characterization System Development

The article is devoted to define requirements and discover specific featuresfor domestic standard cell and I/O libraries characterization EDA tool. A general structure of the library characterization system is presented. The basic requirements indicted for the system are considered and analyzed in the article. A comparative analysis of correspondence to these requirements for domestic and foreign characterization tools is done. The specific features the domestic library characterization system should satisfy are formulated. Among the main features the following are outlined in the article: compatibility and support of domestic OS, support of well-known foreign and domestic Spice-simulators, Russian language support for documentation, scalability, ensuring confidence of the output data. A brief review of typical tasks, intrinsic for characterization tool development and corresponding algorithms, required for its resolving is given. The following issues along with appropriate proposals for its solution are mentioned: automated definition of optimal input waveforms for sequential cells initialization, searching for maximum capacitance load for the cell, setup and hold parameters calculation for sequential cells, method of optimal output current PWL approximation for CCS parameters calculation. As a conclusion of the investigation the current status of domestic characterization systems development is analyzed. In the outcome a problem of domestic library characterization tool that aligned with all the requirements and support the specific features defined above has been raised.

More Efficient Smaller Multi-Cancer Screening Trials.

The NHS-Galleri trial has demonstrated feasibility for multi-cancer screening trial design where all participants provide a 'sample' for screening, but only samples from the intervention arm are tested and acted upon during the trial. We assess efficiency of analysis methods when the control arm may be retrospectively tested at time of analysis. Analyses considered are: (1, traditional) by randomised allocation with all events included; (2, 'intended-effect') nested in those who tested positive in both arms and all events therein; and (3, targeted) by randomised allocation but with endpoint 'test-positive event'. They are compared using approximate statistical methods and scenario analysis. Provided the number who die from cancer after a test-positive sample is a small fraction of the total number who die from cancer, intended-effect and targeted analyses require a much smaller sample size to evaluate cancer-specific mortality than the traditional approach. Intended-effect analysis has a smaller sample size requirement than targeted analysis. This gain is only substantial when the risk of cancer death in test positives is high. Intended-effect or targeted analysis will substantially reduce the sample size needed to evaluate cancer-specific mortality in blood-based screening trials. Targeted analysis requires many fewer retrospective tests and avoids potential effects of needing to inform those whose stored samples test positive. Trialists should consider the trade-off of costs between sample size and retrospective testing requirements when choosing the analysis.

Algorithm XXXX: The Delaunay Density Diagnostic

Accurate approximation of a real-valued function depends on two aspects of the available data: the density of inputs within the domain of interest and the variation of the outputs over that domain. There are few methods for assessing whether the density of inputs is sufficient to identify the relevant variations in outputs—i.e., the “geometric scale” of the function—despite the fact that sampling density is closely tied to the success or failure of an approximation method. In this paper, we introduce a general purpose, computational approach to detecting the geometric scale of real-valued functions over a fixed domain using a deterministic interpolation technique from computational geometry. The algorithm is intended to work on scalar data in moderate dimensions (2-10). Our algorithm is based on the observation that a sequence of piecewise linear interpolants will converge to a continuous function at a quadratic rate (in \(L^{2}\) norm) if and only if the data are sampled densely enough to distinguish the feature from noise (assuming sufficiently regular sampling). We present numerical experiments demonstrating how our method can identify feature scale, estimate uncertainty in feature scale, and assess the sampling density for fixed (i.e. static) datasets of input–output pairs. We include analytical results in support of our numerical findings and have released lightweight code that can be adapted for use in a variety of data science settings.

Distributionally Robust Portfolio Optimization under Marginal and Copula Ambiguity

AbstractWe investigate a new family of distributionally robust optimization problem under marginal and copula ambiguity with applications to portfolio optimization problems. The proposed model considers the ambiguity set of portfolio returns in which the marginal distributions and their copula are close—in terms of the Wasserstein distance—to their nominal counterparts. We develop a cutting-surface method to solve the proposed problem, in which the distribution separation subproblem is nonconvex and includes bilinear terms. We propose three approaches to solve the bilinear formulation, namely (1) linear relaxation via McCormick inequalities, (2) exact mixed-integer linear program reformulation via disjunctive inequalities, and (3) inner approximation method via a novel iterative procedure that exploits the structural properties of the bilinear optimization problem. We further carry out a comprehensive set of computational experiments with distributionally robust portfolios featuring Conditional Value-at-Risk (CVaR) measures. These tests aim to compare the accuracy of the proposed algorithms, analyze the impact of the radius of the Wasserstein ambiguity ball on the portfolio, and assess portfolio performance. We use a rolling-horizon approach to conduct the out-of-sample tests, which show the superior performance of the portfolios under marginal and copula ambiguity over the equally weighted and ambiguity-free Mean-CVaR benchmark portfolios.

A multi-objective optimization scheduling approach of integrated energy system considering the exergy efficiency using the variable step-size approximation method

As an important part of the new power system, it is crucial to consider the efficiency, economy and environment of the integrated energy system simultaneously. In order to improve the efficiency of solving multi-objective optimization scheduling problems for integrated energy systems, a variable step-size approximation method is proposed in this paper. Firstly, a model of integrated energy system is constructed, and the source-side and load-side exergy equations of different energy networks are given, as well as the mathematical expressions of equipment by combining the idea of unification. Secondly, in comparison with the ergodic weights approach, the variable step-size approximation method theoretical minimum reduction percentage of solution time is deduced by mathematical methodology under the m objective functions. Finally, the results from an integrated energy system including a modified New England 30-bus system, a 6-node heat network and a 7-node gas network demonstrate that the multi-objective function proposed compromises the economy, environment and efficiency of integrated energy system compared to the traditional optimization problem. In comparison with the ergodic weights approach, the step size approximation method can save over 97 % of the solution time, and the percentage of average error can be as low as 0 % under the multi-objective functions.

Classical and Bayesian inference of Cpc$\mathcal {C}_{pc}$ for Wilson–Hilferty distribution under progressively first‐failure type‐II censoring samples

AbstractThis study uses two frequentist approaches and the Bayesian method of estimation using progressively first‐failure type‐II censored data to estimate process capability index (PCI), , for the Wilson–Hilferty (WH) distribution. A competitive maximum product of spacing (MPS) method for estimation of is proposed in the frequentist method as an alternative to conventional likelihood (LK)‐based estimation. We have also taken into account the PS function in the Bayesian setup as an alternative to the conventional LK function, and both LK and PS‐based Bayesian estimates are given for . Further, the Bayes estimates are evaluated by applying Tierney–Kadane (TK) approximation method and Markov chain Monte Carlo (MCMC) method with the help of both LK and PS functions and Jeffrey's prior. Additionally, the approximate confidence intervals based on the suggested frequentist techniques as well as the Bayes highest posterior density (HPD) credible intervals are built for the index . In addition, two bootstrap confidence intervals are obtained. In the simulation exercise, the performance of the Bayes and traditional estimates of is assessed in terms of their mean squared errors, and the average width and coverage probabilities of the CIs and HPD intervals are compared. An actual data set from the electronic industries is reanalyzed in order to show the efficacy of the proposed index and estimation methodology.

Comparative view of the modified Vogel’s approximation methods in transportation problem

Recently, the whole world is experiencing economic hardship and one fast road to ply is to cut out some expenses. Xlife group is equally affected and so the study proposes a means to minimize its transportation cost if effected. A comparative surgery on Vogel’s approximation with its modified versions were conducted on minimizing the cost of transporting manufactured goods from (Xlife group’s) points of supply to certain destinations was undertaken. The Vogel's Approximation Method (VAM) and the Modified Vogel's Approximation Method based on standard deviation, distribution indicators and average cost were applied to determine an initial basic feasible solution and optimal solution for certain transportation problems. All methods yield identical minimum transportation cost except at an instance.

Selection Procedure of the Approximation Methods for Deriving Priorities: A Case of Inconsistent Pairwise Comparisons

Selection Procedure of the Approximation Methods for Deriving Priorities: A Case of Inconsistent Pairwise Comparisons

A Semi-Analytical numerical non-stationary creep analysis of a rotating disk made of Polyamid66 under mechanical and thermal loads

A semi-analytical numerical non-stationary creep analysis of constant and variable thickness rotating disks made of polyamide 66 under mechanical and thermal loads is presented using Mendelson’s method of successive approximation. The material properties are time, temperature, and stress-dependent, which vary along the disk’s radius due to effective stress variation and temperature distribution and are represented by Burger’s model. A non-homogeneous differential equation in terms of radial displacement with variable and time-dependent coefficients was established using equilibrium, strain-displacement, and stress-strain relations. First, the initial thermo-elastic stresses at zero time were calculated using the Galerkin method and Hermitian elements. The history of stresses and strains was then calculated using Burger’s viscoelastic model and Prandtl-Reuss relations. To validate the results, an analysis of a disk with constant thickness made of PVDF material was also carried out using the same method employed in this study. Very good agreement of the results was observed in both elastic and creep conditions. Creep analysis was performed for PA66 disks of constant and variable thickness conditions, and the results were compared together. It has been found that the radial displacements and effective stresses are increasing with time for both disks. However, effective stresses and radial displacements of variable-thickness disks are much lower than constant-thickness disks.

A first order in time wave equation modeling nonlinear acoustics

In this paper we focus on a small amplitude approximation of a Navier-Stokes-Fourier system modeling nonlinear acoustics. Omitting all third and higher order terms with respect to certain small parameters, we obtain a first order in time system containing linear and quadratic pressure and velocity terms. Subsequently, the well-posedness of the derived system is shown using the classical method of Galerkin approximation in combination with a fixed point argument. We first prove the well-posedness of a linearized equation using energy estimates and then the well-posedness of the nonlinear system using a Newton-Kantorovich type argument. Based on this, we also obtain global in time well-posedness for small enough data and exponential decay. This is in line with semigroup results for a linear part of the system that we provide as well.

An almost sure limit theorem for the maximum interpoint distance of random vectors in spaces of growing dimension

Let { p n ; n ≥ 1 } be a sequence of positive integers. Let X 1 , … , X p n be i.i.d. random vectors in ℝ n with i.i.d. components that are centered and have variance 1. Consider the maximum interpoint distance M n = max 1 ≤ i < j ≤ p n ‖ X i − X j ‖ 2 . This article presents the almost sure limit theorem for M n under the polynomial and the exponential growth regimes for the dimension. The proofs rely on the moderate deviation principle of the partial sum of i.i.d. random variables, the Chen–Stein Poisson approximation method, and Gaussian approximation techniques.

Iterative Methods for Vecchia-Laplace Approximations for Latent Gaussian Process Models

Latent Gaussian process (GP) models are flexible probabilistic non-parametric function models. Vecchia approximations are accurate approximations for GPs to overcome computational bottlenecks for large data, and the Laplace approximation is a fast method with asymptotic convergence guarantees to approximate marginal likelihoods and posterior predictive distributions for non-Gaussian likelihoods. Unfortunately, the computational complexity of combined Vecchia-Laplace approximations grows faster than linearly in the sample size when used in combination with direct solver methods such as the Cholesky decomposition. Computations with Vecchia-Laplace approximations can thus become prohibitively slow precisely when the approximations are usually the most accurate, i.e., on large data sets. In this article, we present iterative methods to overcome this drawback. Among other things, we introduce and analyze several preconditioners, derive new convergence results, and propose novel methods for accurately approximating predictive variances. We analyze our proposed methods theoretically and in experiments with simulated and real-world data. In particular, we obtain a speed-up of an order of magnitude compared to Cholesky-based calculations and a threefold increase in prediction accuracy in terms of the continuous ranked probability score compared to a state-of-the-art method on a large satellite data set. All methods are implemented in a free C++ software library with high-level Python and R packages. 4

Robust Multi-UAV Cooperative Trajectory Planning and Power Control for Reliable Communication in the Presence of Uncertain Jammers

Unmanned aerial vehicles (UAVs) have become a promising application for future communication and spectrum awareness due to their favorable features such as low cost, high mobility, and ease of deployment. Nevertheless, the jamming resistance appears to be a new challenge in multi-UAV cooperative communication scenarios. This paper focuses on designing trajectory planning and power allocation for efficient control and reliable communication in a ground control unit (GCU)-controlled UAV network, where the GCU coordinates multi-UAV systems to execute tasks amidst multiple jammers with imperfect location and power information. Specifically, this paper formulates a nonconvex semi-infinite optimization problem to maximize the average worst-case signal-to-interference-plus-noise ratio (SINR) among multiple UAVs by designing robust flight paths and power control strategy under stringent energy and mobility constraints. To efficiently address this issue, this paper proposes a powerful iterative algorithm utilizing the S-procedure and the successive convex approximation (SCA) method. Extensive simulations validate the effectiveness of the proposed strategy.