In this paper we will show how guaranteed bounds for eigenvalues (together with eigenvectors) are obtained and how non-existence of eigenvalues in a concrete region could be assured. Some examples for several types of operators in bounded and unbounded domains will be presented. We will furthermore discuss possible future applications to eigenvalue enclosing/excluding of Schrodinger operator, hopefully in its spectral gaps.
{"title":"Validated computation for infinite dimensional eigenvalue problems","authors":"K. Nagatou","doi":"10.1109/SCAN.2006.48","DOIUrl":"https://doi.org/10.1109/SCAN.2006.48","url":null,"abstract":"In this paper we will show how guaranteed bounds for eigenvalues (together with eigenvectors) are obtained and how non-existence of eigenvalues in a concrete region could be assured. Some examples for several types of operators in bounded and unbounded domains will be presented. We will furthermore discuss possible future applications to eigenvalue enclosing/excluding of Schrodinger operator, hopefully in its spectral gaps.","PeriodicalId":388600,"journal":{"name":"12th GAMM - IMACS International Symposium on Scientific Computing, Computer Arithmetic and Validated Numerics (SCAN 2006)","volume":"447 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2006-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"131607565","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper, we describe the software toolkit Inter-VerdiKom which implements a variety of reliable workload analysis techniques for queues with general or semi-Markovian arrival processes used in stochastic traffic modeling. We compute the workload at a network element in an open queuing network using transient and steady state analysis methods in interval arithmetic. We also present a modeling approach that employs a genetic programming optimization technique, providing an accurate model for real-life data in a compact state space, which is required for a successful verification. The resulting overflow probabilities of both the model and the empirical data are compared.
{"title":"A workload analysis tool for discrete-time semi-Markovian servers","authors":"Sebastian Kempken","doi":"10.1109/SCAN.2006.6","DOIUrl":"https://doi.org/10.1109/SCAN.2006.6","url":null,"abstract":"In this paper, we describe the software toolkit Inter-VerdiKom which implements a variety of reliable workload analysis techniques for queues with general or semi-Markovian arrival processes used in stochastic traffic modeling. We compute the workload at a network element in an open queuing network using transient and steady state analysis methods in interval arithmetic. We also present a modeling approach that employs a genetic programming optimization technique, providing an accurate model for real-life data in a compact state space, which is required for a successful verification. The resulting overflow probabilities of both the model and the empirical data are compared.","PeriodicalId":388600,"journal":{"name":"12th GAMM - IMACS International Symposium on Scientific Computing, Computer Arithmetic and Validated Numerics (SCAN 2006)","volume":"52 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2006-09-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"122757651","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We give a survey on criteria for the feasibility and non- feasibility of the interval Gaussian algorithm. In particular, we consider generalized diagonally dominant matrices, appropriate sparse matrices, and Hessenberg matrices. Moreover, we recall alternative representations and pivoting.
{"title":"On the interval Gaussian algorithm","authors":"G. Mayer","doi":"10.1109/SCAN.2006.34","DOIUrl":"https://doi.org/10.1109/SCAN.2006.34","url":null,"abstract":"We give a survey on criteria for the feasibility and non- feasibility of the interval Gaussian algorithm. In particular, we consider generalized diagonally dominant matrices, appropriate sparse matrices, and Hessenberg matrices. Moreover, we recall alternative representations and pivoting.","PeriodicalId":388600,"journal":{"name":"12th GAMM - IMACS International Symposium on Scientific Computing, Computer Arithmetic and Validated Numerics (SCAN 2006)","volume":"54 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2006-09-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"124222005","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
When considering systems of equations, it often happens that parameters are known with some uncertainties. This leads to continua of solutions that are usually approximated using the interval theory. A wider set of useful situations can be modeled if one allows furthermore different quantifications of the parameters in their domains. In particular, quantified solution sets where universal quantifiers are constrained to precede existential quantifiers are called AE-solution sets. A state of the art on the approximation of linear AE- solution sets in the framework of generalized intervals (intervals whose bounds are not constrained to be ordered increasingly) is presented in a new and unifying way. Then two new generalized interval operators dedicated to the approximation of quantified linear interval systems are proposed and investigated.
{"title":"On the Approximation of Linear AE-Solution Sets","authors":"A. Goldsztejn, G. Chabert","doi":"10.1109/SCAN.2006.33","DOIUrl":"https://doi.org/10.1109/SCAN.2006.33","url":null,"abstract":"When considering systems of equations, it often happens that parameters are known with some uncertainties. This leads to continua of solutions that are usually approximated using the interval theory. A wider set of useful situations can be modeled if one allows furthermore different quantifications of the parameters in their domains. In particular, quantified solution sets where universal quantifiers are constrained to precede existential quantifiers are called AE-solution sets. A state of the art on the approximation of linear AE- solution sets in the framework of generalized intervals (intervals whose bounds are not constrained to be ordered increasingly) is presented in a new and unifying way. Then two new generalized interval operators dedicated to the approximation of quantified linear interval systems are proposed and investigated.","PeriodicalId":388600,"journal":{"name":"12th GAMM - IMACS International Symposium on Scientific Computing, Computer Arithmetic and Validated Numerics (SCAN 2006)","volume":"24 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2006-09-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"132646573","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This paper gives details on architecture of an arithmetic unit built on principles of continued logarithms and provides its sample characterization using FPGA technology. Continued logarithms can be used for exact arithmetic, but similarly to other exact/reliable methods they face the instant problem of poor performance. Their naturally binary character, however, gives them a good potential to be realized directly in hardware. We prove feasibility of this approach by constructing a continued logarithm unit and quantifying its possible performance.
{"title":"Hardware Implementation of Continued Logarithm Arithmetic","authors":"T. Brabec","doi":"10.1109/SCAN.2006.23","DOIUrl":"https://doi.org/10.1109/SCAN.2006.23","url":null,"abstract":"This paper gives details on architecture of an arithmetic unit built on principles of continued logarithms and provides its sample characterization using FPGA technology. Continued logarithms can be used for exact arithmetic, but similarly to other exact/reliable methods they face the instant problem of poor performance. Their naturally binary character, however, gives them a good potential to be realized directly in hardware. We prove feasibility of this approach by constructing a continued logarithm unit and quantifying its possible performance.","PeriodicalId":388600,"journal":{"name":"12th GAMM - IMACS International Symposium on Scientific Computing, Computer Arithmetic and Validated Numerics (SCAN 2006)","volume":"30 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2006-09-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"131410633","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Validated integration of ordinary differential equations with uncertain initial conditions and uncertain parameters is important for many practical applications. If guaranteed bounds for the uncertainties are known, interval methods can be applied to obtain validated enclosures of all states. However, validated computations are often affected by overestimation, which, in naive implementations, might even lead to meaningless results. Parallelepiped and QR preconditioning of the state equations, Taylor model arithmetic, as well as simulation techniques employing splitting and merging routines are a few existing approaches for reduction of overestimation. In this paper, the recently developed validated solver ValEncIA-IVP and several methods implemented there for reduction of overestimation are described. Furthermore, a detailed comparison of this solver with COSY VI and VNODE, two of the most well- known validated ODE solvers, is presented. Simulation results for simplified system models in mechanical and bio- process engineering show specific properties, advantages, and limitations of each tool.
{"title":"VALENCIA-IVP: A Comparison with Other Initial Value Problem Solvers","authors":"A. Rauh, E. Hofer, E. Auer","doi":"10.1109/SCAN.2006.47","DOIUrl":"https://doi.org/10.1109/SCAN.2006.47","url":null,"abstract":"Validated integration of ordinary differential equations with uncertain initial conditions and uncertain parameters is important for many practical applications. If guaranteed bounds for the uncertainties are known, interval methods can be applied to obtain validated enclosures of all states. However, validated computations are often affected by overestimation, which, in naive implementations, might even lead to meaningless results. Parallelepiped and QR preconditioning of the state equations, Taylor model arithmetic, as well as simulation techniques employing splitting and merging routines are a few existing approaches for reduction of overestimation. In this paper, the recently developed validated solver ValEncIA-IVP and several methods implemented there for reduction of overestimation are described. Furthermore, a detailed comparison of this solver with COSY VI and VNODE, two of the most well- known validated ODE solvers, is presented. Simulation results for simplified system models in mechanical and bio- process engineering show specific properties, advantages, and limitations of each tool.","PeriodicalId":388600,"journal":{"name":"12th GAMM - IMACS International Symposium on Scientific Computing, Computer Arithmetic and Validated Numerics (SCAN 2006)","volume":"10 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2006-09-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"114550071","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This paper presents a new approach for constructing the convex polyhedral enclosure of an interval-based hierarchical structure of any dimension. To reduce the number of points in the hull construction considered, only relevant vertices on the boundary-called presumable extreme points- are involved. Additionally, a suitable update of the presumable extreme points enhances the performance whenever the maximum level of the hierarchical structure is changed. This method utilizes interval arithmetic and combines adaptation of the concept of presumable extreme points to higher dimensions with a convex-hull algorithm based on an interval linear solver.
{"title":"A Reliable Convex-Hull Algorithm for Interval-Based Hierarchical Structures","authors":"E. Dyllong","doi":"10.1109/SCAN.2006.5","DOIUrl":"https://doi.org/10.1109/SCAN.2006.5","url":null,"abstract":"This paper presents a new approach for constructing the convex polyhedral enclosure of an interval-based hierarchical structure of any dimension. To reduce the number of points in the hull construction considered, only relevant vertices on the boundary-called presumable extreme points- are involved. Additionally, a suitable update of the presumable extreme points enhances the performance whenever the maximum level of the hierarchical structure is changed. This method utilizes interval arithmetic and combines adaptation of the concept of presumable extreme points to higher dimensions with a convex-hull algorithm based on an interval linear solver.","PeriodicalId":388600,"journal":{"name":"12th GAMM - IMACS International Symposium on Scientific Computing, Computer Arithmetic and Validated Numerics (SCAN 2006)","volume":"7 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2006-09-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"129971815","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
V. Kreinovich, S. Starks, R. Araiza, G. Xiang, A. Velasco, M. Averill, G. Keller
In many real-life situations, we have several types of uncertainty: measurement uncertainty can lead to probabilistic and/or interval uncertainty, expert estimates come with interval and/or fuzzy uncertainty, etc. In many situations, in addition to measurement uncertainty, we have prior knowledge coming from prior data processing and/or prior knowledge coming from prior interval constraints. In this paper, on the example of the seismic inverse problem, we show how to combine these different types of uncertainty.
{"title":"Towards Combining Probabilistic, Interval, Fuzzy Uncertainty, and Constraints: An Example Using the Inverse Problem in Geophysics","authors":"V. Kreinovich, S. Starks, R. Araiza, G. Xiang, A. Velasco, M. Averill, G. Keller","doi":"10.1109/SCAN.2006.45","DOIUrl":"https://doi.org/10.1109/SCAN.2006.45","url":null,"abstract":"In many real-life situations, we have several types of uncertainty: measurement uncertainty can lead to probabilistic and/or interval uncertainty, expert estimates come with interval and/or fuzzy uncertainty, etc. In many situations, in addition to measurement uncertainty, we have prior knowledge coming from prior data processing and/or prior knowledge coming from prior interval constraints. In this paper, on the example of the seismic inverse problem, we show how to combine these different types of uncertainty.","PeriodicalId":388600,"journal":{"name":"12th GAMM - IMACS International Symposium on Scientific Computing, Computer Arithmetic and Validated Numerics (SCAN 2006)","volume":"116 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2006-09-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"115267416","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In most applications in control engineering not all state variables can be measured. Consequently, state estimation is performed to reconstruct the non-measurable states taking into account both system dynamics and the measurement model. If the system is subject to interval bounded uncertainties, an interval observer provides a guaranteed estimation of all states. The estimation consists of a recursive application of prediction and correction steps. The prediction step corresponds to a verified integration of the system model describing the system dynamics between two points of time at which measured data is available. In this paper, a Taylor model based integrator is used. Considering the state enclosures obtained in the prediction step, the correction step reconstructs states and parameters from the uncertain measurements with the help of a measurement model. The enclosures of states and parameters determined by the interval observer are consistent with both system and measurement models as well as all uncertainties.
{"title":"Interval Observer Design Based on Taylor Models for Nonlinear Uncertain Continuous-Time Systems","authors":"M. Kletting, A. Rauh, E. Hofer, H. Aschemann","doi":"10.1109/SCAN.2006.26","DOIUrl":"https://doi.org/10.1109/SCAN.2006.26","url":null,"abstract":"In most applications in control engineering not all state variables can be measured. Consequently, state estimation is performed to reconstruct the non-measurable states taking into account both system dynamics and the measurement model. If the system is subject to interval bounded uncertainties, an interval observer provides a guaranteed estimation of all states. The estimation consists of a recursive application of prediction and correction steps. The prediction step corresponds to a verified integration of the system model describing the system dynamics between two points of time at which measured data is available. In this paper, a Taylor model based integrator is used. Considering the state enclosures obtained in the prediction step, the correction step reconstructs states and parameters from the uncertain measurements with the help of a measurement model. The enclosures of states and parameters determined by the interval observer are consistent with both system and measurement models as well as all uncertainties.","PeriodicalId":388600,"journal":{"name":"12th GAMM - IMACS International Symposium on Scientific Computing, Computer Arithmetic and Validated Numerics (SCAN 2006)","volume":"12 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2006-09-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"131264416","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Reliable methods for the analysis of tolerance-affected analog circuits are of great importance in nowadays microelectronics. It is impossible to produce circuits with exactly those parameter specifications proposed in the design process. Interval arithmetic can be used to obtain a worst-case analysis of the influence of component tolerances. This paper focuses on a new approach for interval-valued frequency-response analysis of linear analog circuits, which consist of current and voltage sources as well as resistors, capacitances, inductances, and all variants of controlled sources. Part and parcel of this strategy is the handling of fill-in patterns for those parameters related to uncertain components. Such systems can efficiently be solved by successive application of the Sherman-Morrison formula. The approach is also extended to complex-valued systems from frequency- domain analysis of linear circuits. Crude bounds can be obtained by treating real and imaginary part as different variables. The latter is improved by considering the correlations in order to obtain tighter enclosures of the solution.
{"title":"Interval Analysis of Linear Analog Circuits","authors":"Alexander Dreyer","doi":"10.1109/SCAN.2006.24","DOIUrl":"https://doi.org/10.1109/SCAN.2006.24","url":null,"abstract":"Reliable methods for the analysis of tolerance-affected analog circuits are of great importance in nowadays microelectronics. It is impossible to produce circuits with exactly those parameter specifications proposed in the design process. Interval arithmetic can be used to obtain a worst-case analysis of the influence of component tolerances. This paper focuses on a new approach for interval-valued frequency-response analysis of linear analog circuits, which consist of current and voltage sources as well as resistors, capacitances, inductances, and all variants of controlled sources. Part and parcel of this strategy is the handling of fill-in patterns for those parameters related to uncertain components. Such systems can efficiently be solved by successive application of the Sherman-Morrison formula. The approach is also extended to complex-valued systems from frequency- domain analysis of linear circuits. Crude bounds can be obtained by treating real and imaginary part as different variables. The latter is improved by considering the correlations in order to obtain tighter enclosures of the solution.","PeriodicalId":388600,"journal":{"name":"12th GAMM - IMACS International Symposium on Scientific Computing, Computer Arithmetic and Validated Numerics (SCAN 2006)","volume":"94 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2006-09-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"134291864","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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