Posteriori Method Research Articles

Adaptive meshing strategies for nanophotonics using a posteriori error estimation

As nanophotonic devices become increasingly complex, computer simulations of such devices are becoming ever more important. Unfortunately, computer simulations of nanophotonic devices are computationally expensive, especially if many simulations are necessary, e.g., when optimizing or inverse designing a device. Here we study adaptive mesh refinement for finite-element method simulations using an a posteriori error estimation method. We demonstrate that the use of adaptive meshing leads to faster convergence with lower memory footprint for complex three-dimensional nanophotonic structures. Nevertheless, one needs to be careful to avoid a mesh propagation effect for adaptive mesh refinement to be a successful strategy.

CLSI-based verification and de novo establishment of reference intervals for common biochemical assays in Croatian newborns.

This study aimed to examine whether the Canadian Laboratory Initiative on Pediatric Reference Intervals (CALIPER) reference intervals for 19 commonly used biochemical assays (potassium, sodium, chloride, calcium, magnesium, inorganic phosphorous, glucose, urea, creatinine, direct and total bilirubin, C-reactive protein (CRP), total protein, albumin, aspartate aminotransferase (AST), alanine aminotransferase (ALT), gamma-glutamyl transferase (GGT), alkaline phosphatase (ALP) and lactate dehydrogenase (LD)) could be applied to the newborn population of one Croatian clinical hospital. Reference interval verification was performed according to the CLSI EP28-A3c guidelines. Samples of healthy newborns were selected using the direct a posteriori sampling method and analyzed on the Beckman Coulter AU680 biochemical analyzer. If verification wasn't satisfactory, further procedure included de novo determination of own reference intervals by analyzing 120 samples of healthy newborns. After the first set of measurements, 14/19 tested reference intervals were adopted for use: calcium, inorganic phosphorous, glucose, urea, creatinine, total bilirubin, CRP, total protein, albumin, AST, ALT, GGT, ALP and LD. A second set of samples was tested for 5 analytes: potassium, sodium, chloride, magnesium and direct bilirubin. The verification results of the additional samples for sodium and chloride were satisfactory, while the results for potassium, magnesium and direct bilirubin remained unsatisfactory and new reference intervals were determined. The CALIPER reference intervals can be implemented into routine laboratory and clinical practice for the tested newborn population for most of the analyzed assays, while own reference intervals for potassium, magnesium and direct bilirubin have been determined.

A Data-Driven Probabilistic Network Approach to Assess Model Similarity in CMIP Ensembles

Abstract The different phases of the Coupled Model Intercomparison Project (CMIP) provide ensembles of past, present, and future climate simulations crucial for climate change impact and adaptation activities. These ensembles are produced using multiple global climate models (GCMs) from different modeling centers with some shared building blocks and interdependencies. Applications typically follow the “model democracy” approach which might have significant implications in the resulting products (e.g., large bias and low spread). Thus, quantifying model similarity within ensembles is crucial for interpreting model agreement and multimodel uncertainty in climate change studies. The classical methods used for assessing GCM similarity can be classified into two groups. The a priori approach relies on expert knowledge about the components of these models, while the a posteriori approach seeks similarity in the GCMs’ output variables and is thus data-driven. In this study, we apply probabilistic network models (PNMs), a well-established machine learning technique, as a new a posteriori method to measure intermodel similarities. The proposed methodology is applied to surface temperature fields of the historical experiments from the CMIP5 multimodel ensemble and different reanalysis gridded datasets. PNMs are able to learn the complex spatial dependency structures present in climate data, including teleconnections operating on multiple spatial scales, characteristic of the underlying GCM. A distance metric building on the resulting PNMs is applied to characterize GCM model dependencies. The results of this approach are in line with those obtained with more traditional methods but have further explanatory potential building on probabilistic model querying. Significance Statement The present study proposes the use of probabilistic network models (PNMs) to quantify model similarity within ensembles of global climate models (GCMs). This is crucial for interpreting model agreement and multimodel uncertainty in climate change studies. When applied to climate data (gridded global surface temperature in this study), PNMs encode the relevant spatial dependencies (local and remote connections). Similarities among the PNMs resulting from different GCMs can be quantified and are shown to capture similar GCM formulations reported in previous studies. Differently to other machine learning methods previously applied to this problem, PNMs are fully explainable (allowing probabilistic querying) and are applicable to high-dimensional gridded raw data.

Model-Driven Bayesian Network Learning for Factory-Level Fault Diagnostics and Resilience

We propose to use engineering models for Bayesian Network (BN) learning for fault diagnostics at the factory-level using key performance indicators (KPIs) such as overall equipment effectiveness (OEE). OEE is widely used in industry and it measures sustainability by capturing product quality (e.g., less scrap) and measures resilience by capturing availability. A major advantage of the proposed approach is that the engineering models are likely to be available long before the corresponding digitalized smart factory becomes fully operational. Specifically, for BN structure learning, we propose to use analytical queueing theory models of the factory to elicit the structure, and to carry out intervention we propose to use designed experiments based on discrete-event simulation models of the factory. For parameter learning, we apply a qualitative maximum a posteriori (QMAP) method and propose additional expert constraints based on the law of propagation of uncertainty from queueing theory. Furthermore, the proposed approach overcomes the challenge of obtaining balanced-class data in BN learning for fault diagnostics. We apply the proposed BN learning approach to (i) a 4-robot cell in our laboratory and (ii) a robotic machining cell in a commercial vehicle factory. In both cases, the proposed method is found to be efficacious in accurately learning the BN structure and parameter, as measured using structural-hamming distance and Kullback–Leibler divergence score, respectively. The proposed approach can pave the way for a new class of resilient and sustainable smart manufacturing systems.

BAYESIAN CALIBRATION WITH ADAPTIVE MODEL DISCREPANCY

We investigate a computer model calibration technique inspired by the well-known Bayesian framework of Kennedy and O'Hagan (KOH). We tackle the full Bayesian formulation where model parameter and model discrepancy hyperparameters are estimated jointly and reduce the problem dimensionality by introducing a functional relationship that we call the full maximum a posteriori (FMP) method. This method also eliminates the need for a true value of model parameters that caused identifiability issues in the KOH formulation. When the joint posterior is approximated as a mixture of Gaussians, the FMP calibration is proven to avoid some pitfalls of the KOH calibration, namely missing some probability regions and underestimating the posterior variance. We then illustrate two numerical examples where both model error and measurement uncertainty are estimated together. Using the solution to the full Bayesian problem as a reference, we show that the FMP results are accurate and robust, and avoid the need for high-dimensional Markov chains for sampling.

A robust optimization approach for inventory management with limited-time discounts and service-level requirement under demand uncertainty

A multi-period inventory management problem is considered for a retailer offering limited-time discounts and having a joint service-level requirement over the discount periods under demand uncertainty. In each period, the retailer, only knowing a reference value and the bounds of the uncertain demand, has to determine the order quantity of the product before demand realizations to maximize the total profit over the planning horizon while achieving the required service level over the discount periods. A budgeted uncertainty set is adopted to accommodate demand uncertainty and a joint chance constraint is used to formulate the service-level requirement. The retailer problem is formulated as an affinely adjustable robust chance-constrained model and is transformed into a linear program. A double-layer iterative approach is proposed to solve the problem. The outer layer uses a posteriori method to update the budget coefficient. The inner layer solves the affinely adjustable robust chance-constrained model with the updated budget coefficient, and obtains the updated order quantities. A case study based on real data demonstrates that the proposed approach outperforms other approaches by better balancing the average realized profit and the realized service level. Moreover, the results show that the retailer can obtain a larger average realized profit by gradually than abruptly changing the prices.

An Assessment of the Laminar Hypersonic Double-Cone Experiments in the LENS-XX Tunnel

This is an investigation on two experimental datasets of laminar hypersonic flows, over a double-cone geometry, acquired in Calspan—University at Buffalo Research Center’s Large Energy National Shock (LENS)-XX expansion tunnel. These datasets have yet to be modeled accurately. A previous paper suggested that this could partly be due to mis-specified inlet conditions. The authors of this paper solved a Bayesian inverse problem to infer the inlet conditions of the LENS-XX test section and found that in one case they lay outside the uncertainty bounds specified in the experimental dataset. However, the inference was performed using approximate surrogate models. In this paper, the experimental datasets are revisited and inversions for the tunnel test-section inlet conditions are performed with a Navier–Stokes simulator. The inversion is deterministic and can provide uncertainty bounds on the inlet conditions under a Gaussian assumption. It was found that deterministic inversion yields inlet conditions that do not agree with what was stated in the experiments. An a posteriori method is also presented to check the validity of the Gaussian assumption for the posterior distribution. This paper contributes to ongoing work on the assessment of datasets from challenging experiments conducted in extreme environments, where the experimental apparatus is pushed to the margins of its design and performance envelopes.

Novel method to remotely measure air temperature distribution for indoor environments with heat sources using thermal infrared spectroradiometer

Air temperature is an important physical indicator of a built space. A novel method for remotely measuring the air temperature distribution using a thermal infrared spectroradiometer is evaluated, and an experiment using the maximum a posteriori method, which uses a priori information, is conducted with a heat source. The a priori information is the probabilistic information that the observer knows about the air temperature prior to the observation. Assuming an indoor scale (10 m), the air temperature distribution is estimated (i.e., retrieved) by discretizing the space into four layers. Accordingly, our method reduces the error in the air temperature estimation compared to the a priori information. When the layer of the heat source is known, the air temperature estimation is sensitive to the air temperature of the heated layer, even when the heat source is away from the sensor (2nd to 4th layers). The root-mean-square error (RMSE) of the heated layer decreased from 10 K (a priori) to 4.8 K. Although the RMSE of the temperature distribution was approximately the same regardless of the number of spectral channels used in the retrieval, the RMSE of the heated layer was smaller when ten spectral channels were used. In the case of a target space and spectrometer, as in this study, ten spectral channels were sufficient. These findings indicate the possibility of remotely measuring the air temperature distribution in a space with a local heat source (e.g., data centers), which is a relatively difficult case for remote sensing methods.

Bayesian network parameter learning using fuzzy constraints

Prior knowledge effectively mitigates low modeling accuracy when insufficient data exist. This idea has been confirmed in many fields, especially the transformation of prior knowledge into constraints, widely employed in Bayesian network (BN) parameter learning. The role of constraints in parameter learning is the focus of this study. If parameter learning results are blindly biased toward constraints, underfitting occurs, whereas the effectiveness of the constraint intervention is weak and can not mitigate overfitting by insufficient data. Therefore, fuzzy theory is introduced in parameter learning. The fuzzy membership function is applied to measure the interference effects of constraints and improve the interpretability and accuracy of the constraint usage. For the maximum a posteriori method, hyperparameters are utilized as virtual samples to realize the intervention of parameter learning. This study proposes a fuzzy maximum a posteriori (FMAP) method, in which the hyperparameter is adjusted to a suitable value using a fuzzy membership function. For the data extension method, a proper extension function is critical to the quality of the extended data. Therefore, this paper proposes a fuzzy bootstrap (FB) method that uses a fuzzy membership function to determine the distribution of the expansion parameters. The two algorithms presented in this paper are verified on 12 standard networks. The experimental results show that the proposed methods effectively improve the accuracy of parameter learning.

A Note on Latent Traits Estimates under IRT Models with Missingness

AbstractMissingness due to not‐reached items and omitted items has received much attention in the recent psychometric literature. Such missingness, if not handled properly, would lead to biased parameter estimation, as well as inaccurate inference of examinees, and further erode the validity of the test. This paper reviews some commonly used IRT based models allowing missingness, followed by three popular examinee scoring methods, including maximum likelihood estimation, maximum a posteriori, and expected a posteriori. Simulation studies were conducted to compare these examinee scoring methods across these commonly used models in the presence of missingness. Results showed that all the methods could infer examinees' ability accurately when the missingness is ignorable. If the missingness is nonignorable, incorporating those missing responses would improve the precision in estimating abilities for examinees with missingness, especially when the test length is short. In terms of examinee scoring methods, expected a posteriori method performed better for evaluating latent traits under models allowing missingness. An empirical study based on the PISA 2015 Science Test was further performed.

Identification of Crack Shapes by Digital Image Correlation Using Joint Estimation Maximum a Posteriori Method

AbstractA method that estimates invisible cracks from the surface based on the surface deformation measured by digital image correlation (DIC) is being developed. An inverse problem is setup to estimate such invisible cracks from the surface deformation. Surface deformation, measured by the DIC method, contains noise. Inverse problems have ill-conditions. The regularization method applied in this study is an extension of the joint estimation maximum a posteriori (JE-MAP) method. The JE-MAP algorithm alternates between MAP method estimation and the grab-cut (GC) method to avoid ill-conditions. The physical constraints on displacement and the forces at the cracks and the crack perimeters (ligaments) are added to the MAP method. The displacement and load at the cracks and the ligaments have a cross-sparse relationship. The MAP method estimates the displacement or the load at the cracks and the ligaments. The estimated result varies greatly at the boundary between the cracks and the ligaments. This boundary is determined by the GC method based on the estimated result. This study amplified the changes at the boundary between the cracks and the ligaments in the estimated results. The amplified results were input into the GC method to improve the boundary-determination accuracy. The regularization method developed from the JE-MAP method was combined with DIC method to estimate the cracks in invisible locations. The method proposed in this study estimated cracks more accurately than L1-norm regularization in inverse problems where the observed data were strain distributions measured by the DIC method.

Reference Intervals for Bone Impact Microindentation in Healthy Adults: A Multi-Centre International Study

Impact microindentation (IMI) is a novel technique for assessing bone material strength index (BMSi) in vivo, by measuring the depth of a micron-sized, spherical tip into cortical bone that is then indexed to the depth of the tip into a reference material. The aim of this study was to define the reference intervals for men and women by evaluating healthy adults from the United States of America, Europe and Australia. Participants included community-based volunteers and participants drawn from clinical and population-based studies. BMSi was measured on the tibial diaphysis using an OsteoProbe in 479 healthy adults (197 male and 282 female, ages 25 to 98 years) across seven research centres, between 2011 and 2018. Associations between BMSi, age, sex and areal bone mineral density (BMD) were examined following an a posteriori method. Unitless BMSi values ranged from 48 to 101. The mean (± standard deviation) BMSi for men was 84.4 ± 6.9 and for women, 79.0 ± 9.1. Healthy reference intervals for BMSi were identified as 71.0 to 97.9 for men and 59.8 to 95.2 for women. This study provides healthy reference data that can be used to calculate T- and Z-scores for BMSi and assist in determining the utility of BMSi in fracture prediction. These data will be useful for positioning individuals within the population and for identifying those with BMSi at the extremes of the population.

Robust Mixed-Effect Estimation of Tumor Growth and Age Based on Gompertz Model

This letter considers the problem of estimating tumor growth and tumor age using the Gompertz model within the nonlinear mixed-effect framework. In particular, the choice for the prior for the population model parameters is the Normal-Laplace distribution. The Maximum A Posteriori (MAP) method provides a robust estimation scheme that leverages the fat-tail properties of such a distribution. Due to the non-differentiability of the distribution, the resulting estimator shows an inaction region in which, for small residues, the best estimate remains the population prior. In addition, a change of variable in the usual Gompertz model sets the tumor age as an extra parameter, which is straightforwardly estimated. This information might provide helpful insights into clinical cancer treatment. Numerical experiments suggest that the proposed estimation scheme renders significantly more accurate estimates in such a vital problem.

Iterative Dichotomiser Posteriori Method Based Service Attack Detection in Cloud Computing

Cloud computing (CC) is an advanced technology that provides access to predictive resources and data sharing. The cloud environment represents the right type regarding cloud usage model ownership, size, and rights to access. It introduces the scope and nature of cloud computing. In recent times, all processes are fed into the system for which consumer data and cache size are required. One of the most security issues in the cloud environment is Distributed Denial of Service (DDoS) attacks, responsible for cloud server overloading. This proposed system ID3 (Iterative Dichotomiser 3) Maximum Multifactor Dimensionality Posteriori Method (ID3-MMDP) is used to overcome the drawback and a relatively simple way to execute and for the detection of (DDoS) attack. First, the proposed ID3-MMDP method calls for the resources of the cloud platform and then implements the attack detection technology based on information entropy to detect DDoS attacks. Since because the entropy value can show the discrete or aggregated characteristics of the current data set, it can be used for the detection of abnormal data flow, User-uploaded data, ID3-MMDP system checks and read risk measurement and processing, bug rating file size changes, or file name changes and changes in the format design of the data size entropy value. Unique properties can be used whenever the program approaches any data error to detect abnormal data services. Finally, the experiment also verifies the DDoS attack detection capability algorithm.

Własność i cywilizacja łacińska. Dystrybucjonizm w myśli polityczno-prawnej Feliksa Konecznego

Przedmiot badań: Dorobek intelektualny Feliksa Konecznego cieszy się współcześnie stosunkowo dużą popularnością, inspirując twórczy dyskurs naukowy w wielu dziedzinach. Najważniejszym elementem owej spuścizny jest oryginalne opracowanie zagadnienia cywilizacji, a w szczególności cywilizacji łacińskiej, którą opisał w sposób wyczerpujący i z którą osobiście się identyfikował. Aby lepiej zrozumieć istotę jego myśli, należy spojrzeć na nią z różnych perspektyw. Przedmiotem niniejszej pracy jest próba zestawienia założeń cywilizacji łacińskiej u Konecznego z ustrojem o potencjalnie podobnej naturze, aby ukazać oba zagadnienia w nowym świetle – lepiej wytłumaczyć źródła obu systemów, jak i wytyczyć potencjalne ścieżki rozwoju.
 Cel badawczy: Celem artykułu jest umiejscowienie cywilizacji łacińskiej obok dystrybucjonizmu i zbadanie, w jakim stopniu dystrybucjonizm spełnia wymagania ustrojowe postawione cywilizacji łacińskiej przez Konecznego. Dystrybucjonizm, koncepcja ustroju społeczno-ekonomicznego opracowana przez Hilaire Belloca i Gilberta Keitha Chestertona zakłada w swojej istocie potrzebę jak najszerszej dystrybucji dóbr materialnych. Autor stawia sobie za cel opisanie, na ile ten cel jest kompatybilny z cywilizacją łacińską.
 Metoda badawcza: Autor za pomocą metody historyczno-porównawczej i krytycznej zestawia ze sobą cywilizację łacińską i dystrybucjonizm. Drogą gruntownej analizy materiałów źródłowych dokonuje określenia genez i podstawowych czynników składowych obu koncepcji, a następnie wskazuje na analogie i pokrewieństwa.
 Wyniki: Wnioskiem końcowym jest stwierdzenie wysokiej kompatybilności cywilizacji łacińskiej z doktryną dystrybucjonizmu. Identyfikacji podlegają liczne tożsame elementy: uznanie znaczenia własności i wolności ekonomicznej; postulat jak najszerszej dystrybucji własności, zwłaszcza własności nieruchomości; aposterioryzm w kształtowaniu ustroju; system prawny sprzyjający podziałowi i utrzymaniu własności; afirmacja roli Kościoła Katolickiego i chrześcijańskiej moralności w kształtowaniu społeczeństw. Uzyskane wyniki badań pozwalają postawić tezę o dystrybucjonizmie jako modelowym ustroju ekonomicznym w cywilizacji łacińskiej. Powodzenie przeprowadzonej analizy porównawczej rokuje pozytywnie na dalszą komparatystykę, mającą za przedmiot typy cywilizacji u Konecznego oraz systemy ekonomiczne opisane przez Belloca i Chestertona.

A posteriori localization of many-body excited states through simultaneous diagonalization.

In this paper we propose a numerical method to localize many-electron excited states. To characterize the electronic structure of the electronic excited states of a system, quantum chemistry methods typically yield a delocalized description of the excitations. Some a priori localization methods have been developed to provide an intuitive local picture of the excited states. They typically require a good strategy to separate the system of interest from its environment, or a set of a priori localized orbitals, that may reduce their computational accuracy. Here, we introduce an a posteriori method to localize delocalized many-body excited states directly obtained from quantum chemistry calculations. A localization metric for the excited states is defined from their representation as electron-hole pairs, which is encoded in the transition density matrix. This novel a posteriori strategy thus allows to localize excitons within a volume around selected fragments of a complex molecular system without tempering with its quantum chemical treatment. The method is tested on π-stacked oligomers of phenanthrenes and pyrenes. It is found to efficiently localize and separate the excitons according to their character while preserving the information about delocalized many-body states at a low computational cost.

A Decomposition based Multi-Objective Heat Transfer Search algorithm for structure optimization

Multi-objective (MO) problems are complex and challenging to solve due to multiple conflicting objectives and incommensurate constraints. Heat Transfer Search is a recently introduced thermodynamic laws-based algorithm for solving many single objective problems. But it faces criticism due to its shortcomings including local optima trap, computational complexity, and reliability issues while solving MO problems. Decompositions is a simple yet efficient framework that aids in resolving fitness assignments and diversity maintenance issues of MO problems. It is also popular for reducing computational complexity and improving solution quality. Therefore, this work proposes and investigates a novel, simple, and robust Decomposition-based Multi-Objective Heat Transfer Search (MOHTS/D) for solving real-world structural problems. To achieve the Pareto optimal solutions and to confirm their coverage behaviour, the evenly generated weight vectors sorting and Euclidean distance strategies were employed in the proposed posteriori method. The performance of MOHTS/D is investigated through eight constrained widely accepted benchmarks. The results contrast with the MOHTS, MO evolutionary algorithm based on decomposition, MO passing vehicle search, MO slime mould, MO symbiotic organisms search, and MO multi-verse optimization. The efficacy of MOHTS/D is evaluated based on hyper-volume, coverage, inverted generational distance, pure diversity, spacing, spread, coverage Pareto front, diversity maintenance, generational distance, and runtime metrics. The results reveal that MOHTS/D is a robust optimization approach compared to others for optimizing real-life structural problems. MOHTS/D was able to find the optimal solution with minor computational complexity, and the obtained solutions demonstrate a better convergence, coverage, diversity, and spread behaviour over Pareto fronts

SISTEM INFORMASI AKUNTANSI ANGGARAN LABA DENGAN METODE A POSTERIORI

Information technology can be used in various human lives, one of which is in the business field. In today's era, it is undeniable that computers with accounting information system equipment are components that must be owned by companies to help ease the task of company performance. One of them is BKM which is specifically aimed at poverty alleviation and community development. UPK itself is a non-profit organization or an organization whose main objective is to support an issue or matter in attracting public attention for a non-commercial purpose, without any attention to things that are for-profit. The allocation has also been divided according to their respective portions.
 This study aims to simplify the presentation of profit budgeting at BKM which is still not structured and does not provide clarity about the results of the funds that should be used as the allocation of community activities. Profit budgeting is also not carried out periodically and is still manual in recording so that profits cannot be known in detail. In overcoming the problems above, by using a programmed system with the a posteriori method, the institution can easily record every operational budget to the profit budget so that it can allocate funds for community activities in a structured manner

Intermolecular Interactions and Spectroscopic Signatures of the Hydrogen-Bonded System-n-Octanol in Experimental and Theoretical Studies.

n-Octanol is the object of experimental and theoretical study of spectroscopic signatures and intermolecular interactions. The FTIR measurements were carried out at 293 K for n-octanol and its deuterated form. Special attention was paid to the vibrational features associated with the O-H stretching and the isotope effect. Density Functional Theory (DFT) in its classical formulations was applied to develop static models describing intermolecular hydrogen bond (HB) and isotope effect in the gas phase and using solvent reaction field reproduced by Polarizable Continuum Model (PCM). The Atoms in Molecules (AIM) theory enabled electronic structure and molecular topology study. The Symmetry-Adapted Perturbation Theory (SAPT) was used for energy decomposition in the dimers of n-octanol. Finally, time-evolution methods, namely classical molecular dynamics (MD) and Car-Parrinello Molecular Dynamics (CPMD) were employed to shed light onto dynamical nature of liquid n-octanol with emphasis put on metric and vibrational features. As a reference, CPMD gas phase results were applied. Nuclear quantum effects were included using Path Integral Molecular Dynamics (PIMD) and a posteriori method by solving vibrational Schrödinger equation. The latter applied procedure allowed to study the deuterium isotope effect.

A posteriori verification of the positivity of solutions to elliptic boundary value problems

The purpose of this paper is to develop a unified a posteriori method for verifying the positivity of solutions of elliptic boundary value problems by assuming neither H^2-regularity nor L^{infty } -error estimation, but only H^1_0 -error estimation. In (J Comput Appl Math 370:112647, 2020), we proposed two approaches to verify the positivity of solutions of several semilinear elliptic boundary value problems. However, some cases require L^{infty } -error estimation and, therefore, narrow applicability. In this paper, we extend one of the approaches and combine it with a priori error bounds for Laplacian eigenvalues to obtain a unified method that has wide application. We describe how to evaluate some constants required to verify the positivity of desired solutions. We apply our method to several problems, including those to which the previous method is not applicable.