Optimization of chemical plant simulation using double collocation

A method has been constructed for the solution of a wide range of chemical plant simulation models including differential equations and optimization. Double orthogonal collocation on finite elements is applied to convert the model into an NLP problem that is solved either by the VF 13AD package based on successive quadratic programming, or by the GRG2 package, based on the generalized reduced gradient method. This approach is termed simultaneous optimization and solution strategy. The objective functional can contain integral terms. The state and control variables can have time delays. Equalities and inequalities containing state and control variables can be included into the model as well as algebraic equations and inequalities. The maximum number of independent variables is 2. Problems containing 3 independent variables can be transformed into problems having 2 independent variables using finite differencing. The maximum number of NLP variables and constraints is 1500. The method is also suitable for solving ordinary and partial differential equations. The state functions are approximated by a linear combination of Lagrange interpolation polynomials. The control function can either be approximated by a linear combination of Lagrange interpolation polynomials or by a piecewise constant function over finite elements. The number of internal collocation points can vary by finite elements. The residual error is evaluated at arbitrarily chosen equidistant grid-points, thus enabling the user to check the accuracy of the solution between collocation points, where the solution is exact. The solution functions can be tabulated. There is an option to use control vector parameterization to solve optimization problems containing initial value ordinary differential equations. When there are many differential equations or the upper integration limit should be selected optimally then this approach should be used. The portability of the package has been addressed converting the package from V AX FORTRAN 77 into IBM PC FORTRAN 77 and into SUN SPARC 2000 FORTRAN 77. Computer runs have shown that the method can reproduce optimization problems published in the literature. The GRG2 and the VF I 3AD packages, integrated into the optimization package, proved to be robust and reliable. The package contains an executive module, a module performing control vector parameterization and 2 nonlinear problem solver modules, GRG2 and VF I 3AD. There is a stand-alone module that converts the differential-algebraic optimization problem into a nonlinear programming problem.

Supporting interoperable interpolation: the INTAMAP approach

In many Environmental Information Systems the actual observations arise from a discrete monitoring network which might be rather heterogeneous in both location and types of measurements made. In this paper we describe the architecture and infrastructure for a system, developed as part of the EU FP6 funded INTAMAP project, to provide a service oriented solution that allows the construction of an interoperable, automatic, interpolation system. This system will be based on the Open Geospatial Consortium’s Web Feature Service (WFS) standard. The essence of our approach is to extend the GML3.1 observation feature to include information about the sensor using SensorML, and to further extend this to incorporate observation error characteristics. Our extended WFS will accept observations, and will store them in a database. The observations will be passed to our R-based interpolation server, which will use a range of methods, including a novel sparse, sequential kriging method (only briefly described here) to produce an internal representation of the interpolated field resulting from the observations currently uploaded to the system. The extended WFS will then accept queries, such as ‘What is the probability distribution of the desired variable at a given point’, ‘What is the mean value over a given region’, or ‘What is the probability of exceeding a certain threshold at a given location’. To support information-rich transfer of complex and uncertain predictions we are developing schema to represent probabilistic results in a GML3.1 (object-property) style. The system will also offer more easily accessible Web Map Service and Web Coverage Service interfaces to allow users to access the system at the level of complexity they require for their specific application. Such a system will offer a very valuable contribution to the next generation of Environmental Information Systems in the context of real time mapping for monitoring and security, particularly for systems that employ a service oriented architecture.

Projected sequential Gaussian processes: flexible interpolation for large data sets

Recently within the machine learning and spatial statistics communities many papers have explored the potential of reduced rank representations of the covariance matrix, often referred to as projected or fixed rank approaches. In such methods the covariance function of the posterior process is represented by a reduced rank approximation which is chosen such that there is minimal information loss. In this paper a sequential framework for inference in such projected processes is presented, where the observations are considered one at a time. We introduce a C++ library for carrying out such projected, sequential estimation which adds several novel features. In particular we have incorporated the ability to use a generic observation operator, or sensor model, to permit data fusion. We can also cope with a range of observation error characteristics, including non-Gaussian observation errors. Inference for the variogram parameters is based on maximum likelihood estimation. We illustrate the projected sequential method in application to synthetic and real data sets. We discuss the software implementation and suggest possible future extensions.

A web processing service for validating interpolation

An interoperable web processing service (WPS) for the automatic interpolation of environmental data has been developed in the frame of the INTAMAP project. In order to assess the performance of the interpolation method implemented, a validation WPS has also been developed. This validation WPS can be used to perform leave one out and K-fold cross validation: a full dataset is submitted and a range of validation statistics and diagnostic plots (e.g. histograms, variogram of residuals, mean errors) is received in return. This paper presents the architecture of the validation WPS and a case study is used to briefly illustrate its use in practice. We conclude with a discussion on the current limitations of the system and make proposals for further developments.

INTAMAP: an interoperable automated interpolation web service

INTAMAP is a web processing service for the automatic interpolation of measured point data. Requirements were (i) using open standards for spatial data such as developed in the context of the open geospatial consortium (OGC), (ii) using a suitable environment for statistical modelling and computation, and (iii) producing an open source solution. The system couples the 52-North web processing service, accepting data in the form of an observations and measurements (O&M) document with a computing back-end realized in the R statistical environment. The probability distribution of interpolation errors is encoded with UncertML, a new markup language to encode uncertain data. Automatic interpolation needs to be useful for a wide range of applications and the algorithms have been designed to cope with anisotropies and extreme values. In the light of the INTAMAP experience, we discuss the lessons learnt.

Lessons learned from INTAMAP, an interoperable web service for the real-time interpolation of environmental variables

Interpolated data are an important part of the environmental information exchange as many variables can only be measured at situate discrete sampling locations. Spatial interpolation is a complex operation that has traditionally required expert treatment, making automation a serious challenge. This paper presents a few lessons learnt from INTAMAP, a project that is developing an interoperable web processing service (WPS) for the automatic interpolation of environmental data using advanced geostatistics, adopting a Service Oriented Architecture (SOA). The “rainbow box” approach we followed provides access to the functionality at a whole range of different levels. We show here how the integration of open standards, open source and powerful statistical processing capabilities allows us to automate a complex process while offering users a level of access and control that best suits their requirements. This facilitates benchmarking exercises as well as the regular reporting of environmental information without requiring remote users to have specialized skills in geostatistics.

INTAMAP: the design and implementation of an interoperable automated interpolation web service

INTAMAP is a Web Processing Service for the automatic spatial interpolation of measured point data. Requirements were (i) using open standards for spatial data such as developed in the context of the Open Geospatial Consortium (OGC), (ii) using a suitable environment for statistical modelling and computation, and (iii) producing an integrated, open source solution. The system couples an open-source Web Processing Service (developed by 52°North), accepting data in the form of standardised XML documents (conforming to the OGC Observations and Measurements standard) with a computing back-end realised in the R statistical environment. The probability distribution of interpolation errors is encoded with UncertML, a markup language designed to encode uncertain data. Automatic interpolation needs to be useful for a wide range of applications and the algorithms have been designed to cope with anisotropy, extreme values, and data with known error distributions. Besides a fully automatic mode, the system can be used with different levels of user control over the interpolation process.

Projection and interpolation based techniques for structured and impulsive noise filtering

In this chapter we present the relevant mathematical background to address two well defined signal and image processing problems. Namely, the problem of structured noise filtering and the problem of interpolation of missing data. The former is addressed by recourse to oblique projection based techniques whilst the latter, which can be considered equivalent to impulsive noise filtering, is tackled by appropriate interpolation methods.

Spline interpolation to evaluate foveation parameters in congenital nystagmus recordings

Infantile Nystagmus Syndrome, or Congenital Nystagmus, is an ocular-motor disorder characterized by involuntary, conjugated and bilateral to and fro ocular oscillations. Good visual acuity in congenital nystagmus can be achieved during the foveation periods in which eye velocity slows down while the target image crosses the fovea. Visual acuity was found to be mainly dependent on the duration of the foveation periods. In this work a new approach is proposed for estimation of foveation parameters: a cubic spline interpolation of the nystagmus recording before localizing the start point of foveation window and to estimate its duration. The performances of the proposed algorithm were assessed in comparison with a previously developed algorithm, used here as gold standard. The obtained results suggest that the spline interpolation could be a useful tool to filter the eye movement recordings before applying an algorithm to estimate the foveation window parameters. © 2013 IEEE.

Measurement of intervertebral cervical motion by means of dynamic X-ray image processing and data interpolation

Accurate measurement of intervertebral kinematics of the cervical spine can support the diagnosis of widespread diseases related to neck pain, such as chronic whiplash dysfunction, arthritis, and segmental degeneration. The natural inaccessibility of the spine, its complex anatomy, and the small range of motion only permit concise measurement in vivo. Low dose X-ray fluoroscopy allows time-continuous screening of cervical spine during patient's spontaneous motion. To obtain accurate motion measurements, each vertebra was tracked by means of image processing along a sequence of radiographic images. To obtain a time-continuous representation of motion and to reduce noise in the experimental data, smoothing spline interpolation was used. Estimation of intervertebral motion for cervical segments was obtained by processing patient's fluoroscopic sequence; intervertebral angle and displacement and the instantaneous centre of rotation were computed. The RMS value of fitting errors resulted in about 0.2 degree for rotation and 0.2 mm for displacements. © 2013 Paolo Bifulco et al.

PSD modifications of FHRV due to interpolation and CTG storage rate

Cardiotocographic data provide physicians information about foetal development and permit to assess conditions such as foetal distress. An incorrect evaluation of the foetal status can be of course very dangerous. To improve interpretation of cardiotocographic recordings, great interest has been dedicated to foetal heart rate variability spectral analysis. It is worth reminding, however, that foetal heart rate is intrinsically an uneven series, so in order to produce an evenly sampled series a zero-order, linear or cubic spline interpolation can be employed. This is not suitable for frequency analyses because interpolation introduces alterations in the foetal heart rate power spectrum. In particular, interpolation process can produce alterations of the power spectral density that, for example, affects the estimation of the sympatho-vagal balance (computed as low-frequency/high-frequency ratio), which represents an important clinical parameter. In order to estimate the frequency spectrum alterations of the foetal heart rate variability signal due to interpolation and cardiotocographic storage rates, in this work, we simulated uneven foetal heart rate series with set characteristics, their evenly spaced versions (with different orders of interpolation and storage rates) and computed the sympatho-vagal balance values by power spectral density. For power spectral density estimation, we chose the Lomb method, as suggested by other authors to study the uneven heart rate series in adults. Summarising, the obtained results show that the evaluation of SVB values on the evenly spaced FHR series provides its overestimation due to the interpolation process and to the storage rate. However, cubic spline interpolation produces more robust and accurate results. © 2010 Elsevier Ltd. All rights reserved.

Projected sequential Gaussian processes:a C++ tool for interpolation of large datasets with heterogeneous noise

Heterogeneous datasets arise naturally in most applications due to the use of a variety of sensors and measuring platforms. Such datasets can be heterogeneous in terms of the error characteristics and sensor models. Treating such data is most naturally accomplished using a Bayesian or model-based geostatistical approach; however, such methods generally scale rather badly with the size of dataset, and require computationally expensive Monte Carlo based inference. Recently within the machine learning and spatial statistics communities many papers have explored the potential of reduced rank representations of the covariance matrix, often referred to as projected or fixed rank approaches. In such methods the covariance function of the posterior process is represented by a reduced rank approximation which is chosen such that there is minimal information loss. In this paper a sequential Bayesian framework for inference in such projected processes is presented. The observations are considered one at a time which avoids the need for high dimensional integrals typically required in a Bayesian approach. A C++ library, gptk, which is part of the INTAMAP web service, is introduced which implements projected, sequential estimation and adds several novel features. In particular the library includes the ability to use a generic observation operator, or sensor model, to permit data fusion. It is also possible to cope with a range of observation error characteristics, including non-Gaussian observation errors. Inference for the covariance parameters is explored, including the impact of the projected process approximation on likelihood profiles. We illustrate the projected sequential method in application to synthetic and real datasets. Limitations and extensions are discussed. © 2010 Elsevier Ltd.