A nonlinear model of an anaerobic digester wastewater treatment process is considered. Assuming that the model parameters are unknown but bounded, the asymptotic stabilizability of the control system is studied and a new adaptive stabilizing feedback control law is proposed. Computer simulations are also presented to illustrate the theoretical results.
{"title":"Nonlinear Adaptive Control of an UncertainWastewater Treatment Model","authors":"N. Dimitrova, M. Krastanov","doi":"10.1109/SCAN.2006.31","DOIUrl":"https://doi.org/10.1109/SCAN.2006.31","url":null,"abstract":"A nonlinear model of an anaerobic digester wastewater treatment process is considered. Assuming that the model parameters are unknown but bounded, the asymptotic stabilizability of the control system is studied and a new adaptive stabilizing feedback control law is proposed. Computer simulations are also presented to illustrate the theoretical results.","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":"134589134","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}
S. Piskorski, L. Lacassagne, M. Kieffer, D. Etiemble
This paper presents the implementation of a 16-bit interval floating-point unit on a soft-core processor to allow interval computations for embedded systems. The distributed localization of a source using a network of sensors is presented to compare the performance of the proposed processor to that obtained with a general- purpose processor.
{"title":"Efficient 16-bit Floating-Point Interval Processor for Embedded Systems and Applications","authors":"S. Piskorski, L. Lacassagne, M. Kieffer, D. Etiemble","doi":"10.1109/SCAN.2006.15","DOIUrl":"https://doi.org/10.1109/SCAN.2006.15","url":null,"abstract":"This paper presents the implementation of a 16-bit interval floating-point unit on a soft-core processor to allow interval computations for embedded systems. The distributed localization of a source using a network of sensors is presented to compare the performance of the proposed processor to that obtained with a general- purpose processor.","PeriodicalId":388600,"journal":{"name":"12th GAMM - IMACS International Symposium on Scientific Computing, Computer Arithmetic and Validated Numerics (SCAN 2006)","volume":"121 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":"124148700","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 article, we describe a new library for computing guaranteed bounds of the solutions of Initial Value Problems (IVP). Given an initial value problem and an end point, our library computes a sequence of approximation points together with a sequence of approximation errors such that the distance to the true solution of the IVP is below these error terms at each approximation point. These sequences are computed using a classical Runge-Kutta method for which truncation and roundoff errors may be over-approximated. We also compute the propagation of local errors to obtain an enclosure of the global error at each computation step. These techniques are implemented in a C++ library which provides an easy-to-use framework for the rigorous approximation of IVP. This library implements an error control technique based on step size reduction in order to reach a certain tolerance on local errors.
{"title":"GRKLib: a Guaranteed Runge Kutta Library","authors":"O. Bouissou, M. Martel","doi":"10.1109/SCAN.2006.20","DOIUrl":"https://doi.org/10.1109/SCAN.2006.20","url":null,"abstract":"In this article, we describe a new library for computing guaranteed bounds of the solutions of Initial Value Problems (IVP). Given an initial value problem and an end point, our library computes a sequence of approximation points together with a sequence of approximation errors such that the distance to the true solution of the IVP is below these error terms at each approximation point. These sequences are computed using a classical Runge-Kutta method for which truncation and roundoff errors may be over-approximated. We also compute the propagation of local errors to obtain an enclosure of the global error at each computation step. These techniques are implemented in a C++ library which provides an easy-to-use framework for the rigorous approximation of IVP. This library implements an error control technique based on step size reduction in order to reach a certain tolerance on local errors.","PeriodicalId":388600,"journal":{"name":"12th GAMM - IMACS International Symposium on Scientific Computing, Computer Arithmetic and Validated Numerics (SCAN 2006)","volume":"11 3","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2006-09-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"120918876","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}
Expressions are not functions. Confusing the two concepts or failing to define the function that is computed by an expression weakens the rigour of interval arithmetic. We give such a definition and continue with the required re-statements and proofs of the fundamental theorems of interval arithmetic and interval analysis.
{"title":"The Fundamental Theorems of Interval Analysis","authors":"M. H. Emden, B. Moa","doi":"10.1109/scan.2006.43","DOIUrl":"https://doi.org/10.1109/scan.2006.43","url":null,"abstract":"Expressions are not functions. Confusing the two concepts or failing to define the function that is computed by an expression weakens the rigour of interval arithmetic. We give such a definition and continue with the required re-statements and proofs of the fundamental theorems of interval arithmetic and interval analysis.","PeriodicalId":388600,"journal":{"name":"12th GAMM - IMACS International Symposium on Scientific Computing, Computer Arithmetic and Validated Numerics (SCAN 2006)","volume":"80 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":"121180741","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 numerical computations the accuracy of the result quite often depends on a few expressions. In numerical linear algebra, e.g., summations or dot products should very often be computed with additional precision or accuracy. Corresponding algorithms have been developed for a long time and only recently revisited. The usage of these algorithms would be facilitated, if we had a means in a programming language to specify the accuracy requirements of an expression evaluation. In this paper we present a precision aware C+ + template library (PTL)for matrix / vector operations that provides several algorithms with different accuracy or precision characteristics for matrix multiplication and related operations. A matrix is a template parameterized with the number of rows and columns, the element type, a type representing the shape, and an evaluation strategy. Currently only two shapes are implemented, fixed or dynamically adaptable dense arrays. We distinguish between row and column vectors. The access to submatrices and -vectors is accomplished by an overloaded function template. It is possible to adapt the expression system to types declared in other libraries or declared by the user. The concept of expression templates is extended in a way that allows the user to specify rules for the evaluation strategy. The expression tree is constructed by overloading the operators for the expression type. In a second but still compile-time step the evaluation strategy is chosen and the trees are transformed and prepared for run-time execution. The strategy is determined by the tag type of the result, but it can be explicitly set using the index operator. The evaluation strategies can be combined with loop unrolling or loop fusion. Note that the latter not only increases the precision but also the accuracy of the result, since this strategy directly implements the dotprecision expression evaluation in the XSC languages. The library provides evaluation strategies for matrix and vector expressions with k-fold precision and with least bit accuracy. Efficiency and accuracy of the algorithms are tested vs. the Gnu multiple precision library GMP.
{"title":"Expression Defined Accuracy","authors":"A. Pokorny, J. W. von Gudenberg","doi":"10.1109/SCAN.2006.17","DOIUrl":"https://doi.org/10.1109/SCAN.2006.17","url":null,"abstract":"In numerical computations the accuracy of the result quite often depends on a few expressions. In numerical linear algebra, e.g., summations or dot products should very often be computed with additional precision or accuracy. Corresponding algorithms have been developed for a long time and only recently revisited. The usage of these algorithms would be facilitated, if we had a means in a programming language to specify the accuracy requirements of an expression evaluation. In this paper we present a precision aware C+ + template library (PTL)for matrix / vector operations that provides several algorithms with different accuracy or precision characteristics for matrix multiplication and related operations. A matrix is a template parameterized with the number of rows and columns, the element type, a type representing the shape, and an evaluation strategy. Currently only two shapes are implemented, fixed or dynamically adaptable dense arrays. We distinguish between row and column vectors. The access to submatrices and -vectors is accomplished by an overloaded function template. It is possible to adapt the expression system to types declared in other libraries or declared by the user. The concept of expression templates is extended in a way that allows the user to specify rules for the evaluation strategy. The expression tree is constructed by overloading the operators for the expression type. In a second but still compile-time step the evaluation strategy is chosen and the trees are transformed and prepared for run-time execution. The strategy is determined by the tag type of the result, but it can be explicitly set using the index operator. The evaluation strategies can be combined with loop unrolling or loop fusion. Note that the latter not only increases the precision but also the accuracy of the result, since this strategy directly implements the dotprecision expression evaluation in the XSC languages. The library provides evaluation strategies for matrix and vector expressions with k-fold precision and with least bit accuracy. Efficiency and accuracy of the algorithms are tested vs. the Gnu multiple precision library GMP.","PeriodicalId":388600,"journal":{"name":"12th GAMM - IMACS International Symposium on Scientific Computing, Computer Arithmetic and Validated Numerics (SCAN 2006)","volume":"121 3 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":"132788222","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}
Consider a linear system A(p)x = b(p) whose input data depend on a number of uncertain parameters p = (p1,...,pk) varying within given intervals [p]. The objective is to verify by numerical computations monotonic (and convexity/concavity) dependence of a solution component xi(p) with respect to a parameter pj over the interval box [p], or more general, to prove if some boundary inf / sup xi(p) for all p isin [p] is attained at the end-points of [p]. Such knowledge is useful in many applications in order to facilitate the solution of some underlying linear parametric problem involving uncertainties. In this paper we present a technique, for proving the desired properties of the parametric solution, which is alternative to the approaches based on extreme point computations. The proposed computer-aided proof is based on guaranteed interval enclosures for the partial derivatives of the parametric solution for all p isin [p]. The availability of self-validated methods providing guaranteed enclosure of a parametric solution set by floating-point computations is a key for the efficiency and the expanded scope of applicability of the proposed approach. Linear systems involving nonlinear parameter dependencies, and dependencies between A(p) and b(p), as well as non-square linear parametric systems can be handled successfully. Presented are details of the algorithm design and mathematica tools implementing the proposed approach. Numerical examples from structural mechanics illustrate its application.
{"title":"Computer-Assisted Proofs in Solving Linear Parametric Problems","authors":"E. Popova","doi":"10.1109/SCAN.2006.12","DOIUrl":"https://doi.org/10.1109/SCAN.2006.12","url":null,"abstract":"Consider a linear system A(p)x = b(p) whose input data depend on a number of uncertain parameters p = (p1,...,pk) varying within given intervals [p]. The objective is to verify by numerical computations monotonic (and convexity/concavity) dependence of a solution component xi(p) with respect to a parameter pj over the interval box [p], or more general, to prove if some boundary inf / sup xi(p) for all p isin [p] is attained at the end-points of [p]. Such knowledge is useful in many applications in order to facilitate the solution of some underlying linear parametric problem involving uncertainties. In this paper we present a technique, for proving the desired properties of the parametric solution, which is alternative to the approaches based on extreme point computations. The proposed computer-aided proof is based on guaranteed interval enclosures for the partial derivatives of the parametric solution for all p isin [p]. The availability of self-validated methods providing guaranteed enclosure of a parametric solution set by floating-point computations is a key for the efficiency and the expanded scope of applicability of the proposed approach. Linear systems involving nonlinear parameter dependencies, and dependencies between A(p) and b(p), as well as non-square linear parametric systems can be handled successfully. Presented are details of the algorithm design and mathematica tools implementing the proposed approach. Numerical examples from structural mechanics illustrate its application.","PeriodicalId":388600,"journal":{"name":"12th GAMM - IMACS International Symposium on Scientific Computing, Computer Arithmetic and Validated Numerics (SCAN 2006)","volume":"33 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":"116724748","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 general, models of biological or technical applications are represented by nonlinear systems. Moreover, these systems contain multiple uncertain or unknown parameters. These uncertainties are the reason for some numerical and analytical problems in finding guaranteed bounds for the solution of the state space representation. Unfortunately, several industrial applications are demanding exactly these guaranteed bounds in order to fulfil regulations set by the state authorities. To get an idea of the solution of systems with uncertainties the numerical integration of the system's differential equations has to be done with randomly selected values for the unknown parameters. This computation is done several times, in some circumstances more than a thousand times. This approach is well known as the Monte-Carlo method, but this stochastic approach cannot deliver guaranteed bounds for the domain of the system's solution. Thus, we developed a method to find guaranteed bounds which uses linear Lyapunov-like functions to solve this problem. In this work we combine this method with a theory first introduced by Midler. Differential inequalities are used by Mutter to obtain guaranteed bounds. Intersecting the results of both methods provides improved and tight bounds for the original uncertain system. Another approach is shown using a midpoint method providing guaranteed bounds. We achieve guaranteed and finite simulation bounds as a result of our approaches. The results can be used as an initial interval for further methods based on interval arithmetic. An example of a bioreactor with two state variables is shown in this paper to illustrate the methods.
{"title":"Guaranteed Bounds for Uncertain Systems: Methods Using Linear Lyapunov-like Functions, Differential Inequalities and a Midpoint Method","authors":"M. Gennat, B. Tibken","doi":"10.1109/SCAN.2006.21","DOIUrl":"https://doi.org/10.1109/SCAN.2006.21","url":null,"abstract":"In general, models of biological or technical applications are represented by nonlinear systems. Moreover, these systems contain multiple uncertain or unknown parameters. These uncertainties are the reason for some numerical and analytical problems in finding guaranteed bounds for the solution of the state space representation. Unfortunately, several industrial applications are demanding exactly these guaranteed bounds in order to fulfil regulations set by the state authorities. To get an idea of the solution of systems with uncertainties the numerical integration of the system's differential equations has to be done with randomly selected values for the unknown parameters. This computation is done several times, in some circumstances more than a thousand times. This approach is well known as the Monte-Carlo method, but this stochastic approach cannot deliver guaranteed bounds for the domain of the system's solution. Thus, we developed a method to find guaranteed bounds which uses linear Lyapunov-like functions to solve this problem. In this work we combine this method with a theory first introduced by Midler. Differential inequalities are used by Mutter to obtain guaranteed bounds. Intersecting the results of both methods provides improved and tight bounds for the original uncertain system. Another approach is shown using a midpoint method providing guaranteed bounds. We achieve guaranteed and finite simulation bounds as a result of our approaches. The results can be used as an initial interval for further methods based on interval arithmetic. An example of a bioreactor with two state variables is shown in this paper to illustrate the methods.","PeriodicalId":388600,"journal":{"name":"12th GAMM - IMACS International Symposium on Scientific Computing, Computer Arithmetic and Validated Numerics (SCAN 2006)","volume":"157 16 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":"128335216","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}
A linear programming problem whose coefficients are prescribed by intervals is called strongly unbounded if each linear programming problem obtained by fixing coefficients in these intervals is unbounded. In the main result of this paper a necessary and sufficient condition for strong unboundedness of an interval linear programming problem is described. In order to have a full picture we also show conditions for strong feasibility and strong solvability of this problem. The necessary and sufficient conditions for strong feasibility, strong solvability and strong unboundedness can be verified by checking the appropriate properties by the finite algorithms. Checking strong feasibility and checking strong solvability are NP-hard. We show that checking strong unboundedness is NP-hard as well.
{"title":"Strong Unboundedness of Interval Linear Programming Problems","authors":"J. Koničková","doi":"10.1109/SCAN.2006.42","DOIUrl":"https://doi.org/10.1109/SCAN.2006.42","url":null,"abstract":"A linear programming problem whose coefficients are prescribed by intervals is called strongly unbounded if each linear programming problem obtained by fixing coefficients in these intervals is unbounded. In the main result of this paper a necessary and sufficient condition for strong unboundedness of an interval linear programming problem is described. In order to have a full picture we also show conditions for strong feasibility and strong solvability of this problem. The necessary and sufficient conditions for strong feasibility, strong solvability and strong unboundedness can be verified by checking the appropriate properties by the finite algorithms. Checking strong feasibility and checking strong solvability are NP-hard. We show that checking strong unboundedness is NP-hard as well.","PeriodicalId":388600,"journal":{"name":"12th GAMM - IMACS International Symposium on Scientific Computing, Computer Arithmetic and Validated Numerics (SCAN 2006)","volume":"45 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":"133650658","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 present an interval version of the backward differentiation method for solving the initial value problem (IVP) for ordinary differential equations (ODEs). The method considered belongs to a group of the interval multistep methods. A number of the explicit and implicit interval multistep methods are tested on selected problems and the comparison is given.
{"title":"An Interval Version of the Backward Differentiation (BDF) Method","authors":"M. Jankowska, A. Marcinak","doi":"10.1109/SCAN.2006.8","DOIUrl":"https://doi.org/10.1109/SCAN.2006.8","url":null,"abstract":"We present an interval version of the backward differentiation method for solving the initial value problem (IVP) for ordinary differential equations (ODEs). The method considered belongs to a group of the interval multistep methods. A number of the explicit and implicit interval multistep methods are tested on selected problems and the comparison is given.","PeriodicalId":388600,"journal":{"name":"12th GAMM - IMACS International Symposium on Scientific Computing, Computer Arithmetic and Validated Numerics (SCAN 2006)","volume":"29 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":"121254422","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}
A new approach is described for the deterministic global optimization of dynamic systems, including problems in parameter estimation and optimal control. The method is based on interval analysis and Taylor models, and employs a sequential approach using a type of branch-and-reduce strategy. A key feature of the method is the use of a new validated solver for parametric ODEs, which is used to produce guaranteed bounds on the solutions of dynamic systems with interval-valued parameters. This is combined with a new technique for domain reduction based on using Taylor models in an efficient constraint propagation scheme. The result is that problems can be solved to global optimality with both mathematical and computational certainty. Examples are presented to demonstrate the computational efficiency of the method.
{"title":"Deterministic Global Optimization for Dynamic Systems Using Interval Analysis","authors":"Youdong Lin, M. Stadtherr","doi":"10.1109/SCAN.2006.14","DOIUrl":"https://doi.org/10.1109/SCAN.2006.14","url":null,"abstract":"A new approach is described for the deterministic global optimization of dynamic systems, including problems in parameter estimation and optimal control. The method is based on interval analysis and Taylor models, and employs a sequential approach using a type of branch-and-reduce strategy. A key feature of the method is the use of a new validated solver for parametric ODEs, which is used to produce guaranteed bounds on the solutions of dynamic systems with interval-valued parameters. This is combined with a new technique for domain reduction based on using Taylor models in an efficient constraint propagation scheme. The result is that problems can be solved to global optimality with both mathematical and computational certainty. Examples are presented to demonstrate the computational efficiency of the method.","PeriodicalId":388600,"journal":{"name":"12th GAMM - IMACS International Symposium on Scientific Computing, Computer Arithmetic and Validated Numerics (SCAN 2006)","volume":"2 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":"128527590","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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