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I.d.l. Bogle - One of the best experts on this subject based on the ideXlab platform.
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Balanced Random and Adaptive Interval Arithmetic for Systems of Linear Interval Equations
Adaptive Computing in Design and Manufacture VI, 2020Co-Authors: Julius Zilinskas, I.d.l. BogleAbstract:The paper concerns Interval methods — valuable tools for solving engineering problems — for finding outer approximations for the solution set of systems of linear Interval equations. The paper shows how Interval methods are used to analyse steady-state concentrations of systems of coupled reactors. The results of the experiments evaluating the outer approximations for the solution set of systems of linear Interval equations using Gaussian elimination with standard and balanced random Interval Arithmetic are given. Adaptive Interval Arithmetic is proposed to overcome disadvantages of balanced random Interval Arithmetic.
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Balanced random Interval Arithmetic in market model estimation
European Journal of Operational Research, 2006Co-Authors: Julius Zilinskas, I.d.l. BogleAbstract:The possibility of estimating bounds for the econometric likelihood function using balanced random Interval Arithmetic is experimentally investigated. The experiments on the likelihood function with data from housing starts have proved the assumption that distributions of centres and radii of evaluated balanced random Intervals are normal. Balanced random Interval Arithmetic can therefore be used to estimate bounds for this function and global optimization algorithms based on this Arithmetic are applicable to optimize it. The Interval branch and bound algorithms with bounds calculated using standard and balanced random Interval Arithmetic were used to optimize the likelihood function. Results of the experiments show that when reliability is essential the algorithm with standard Interval Arithmetic should be used, but when speed of optimization is more important, the algorithm with balanced random Interval Arithmetic should be used which in this case finishes faster and provides good, although not always optimal, values.
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Balanced random Interval Arithmetic in market model estimation
EUROPEAN JOURNAL OF OPERATIONAL RESEARCH, 2006Co-Authors: I.d.l. BogleAbstract:The possibility of estimating bounds for the econometric likelihood function using balanced random Interval Arithmetic is experimentally investigated. The experiments on the likelihood function with data from housing starts have proved the assumption that distributions of centres and radii of evaluated balanced random Intervals are normal. Balanced random Interval Arithmetic can therefore be used to estimate bounds for this function and global optimization algorithms based on this Arithmetic are applicable to optimize it. The Interval branch and bound algorithms with bounds calculated using standard and balanced random Interval Arithmetic were used to optimize the likelihood function. Results of the experiments show that when reliability is essential the algorithm with standard Interval Arithmetic should be used, but when speed of optimization is more important, the algorithm with balanced random Interval Arithmetic should be used which in this case finishes faster and provides good, although not always optimal, values. (c) 2005 Elsevier B.V. All rights reserved.
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balanced random Interval Arithmetic
Computers & Chemical Engineering, 2004Co-Authors: Julius Zilinskas, I.d.l. BogleAbstract:A version of random Interval Arithmetic, balanced random Interval Arithmetic, is proposed. Random Interval Arithmetic, useful in Interval based optimisation algorithms, utilizes a mix of standard and inner Interval operations to obtain Interval bounding of a function. The guarantee of enclosure which standard Interval operations give is sacrificed but there is a very high probability of success and the computational cost is significantly reduced. Here, random Interval Arithmetic is generalized in the sense that different probabilities of standard Interval operations are allowed in the mix of standard and inner Interval operations instead of 0.5 in the original proposal. The influence of the probabilities of standard and inner Interval operations on the estimates of bounds for the ranges of function values are investigated experimenting with a system of linear Interval equations and with two typical test problems from global optimisation. The test results seem promising for exploitation with hybrid global optimisation algorithms based on the ideas of statistical inference and Interval methods. As in stochastic global optimisation methods there will not be an absolute guarantee of finding the global minimum but there is a very strong likelihood.
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evaluation ranges of functions using balanced random Interval Arithmetic
Informatica (lithuanian Academy of Sciences), 2003Co-Authors: Julius žilinskas Thauth, I.d.l. BogleAbstract:The results of experimental testing of balanced random Interval Arithmetic with typical mathematical test functions and practical problem are presented and discussed. The possibility of evaluation ranges of functions using balanced random Interval Arithmetic is investigated. The influence of the predefined probabilities of standard and inner Interval operations to the ranges of functions is experimentally investigated. The author wishes to acknowledge the financial support through the Royal Society/NATO Postdoctoral Fellowship for the research “Stochastic Interval Methods for Process Optimisation”.
E.e. Swartzlander - One of the best experts on this subject based on the ideXlab platform.
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A variable-precision, Interval Arithmetic processor
Journal of Applied Mathematics and Mechanics, 1996Co-Authors: M.j. Schulte, E.e. SwartzlanderAbstract:This paper presents the software interface and hardware design of a processor which supports variable-precision, Interval Arithmetic. The software interface gives the programmer the ability to specify the precision of the computation, determine the accuracy of the result, and recompute inaccurate results with higher precision. Special instructions and hardware features are provided for operations on matrices, vectors and complex numbers. Providing direct hardware support for variable-precision, Interval Arithmetic greatly improves the accuracy and reliability of the computation, and is much faster than existing software techniques for controlling numerical error. Area and delay estimates indicate that the processor can be implemented on a single chip with a cycle time that is comparable to existing IEEE double-precision floating point processors.
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Variable-precision, Interval Arithmetic coprocessors
Reliable Computing, 1996Co-Authors: M.j. Schulte, E.e. SwartzlanderAbstract:This paper presents hardware designs, Arithmetic algorithms, and numerical applications for variable-precision, Interval Arithmetic coprocessors. These coprocessors give the programmer the ability to set the initial precision of the computation, determine the accuracy of the results, and recompute inaccurate results with higher precision. Variable-precision, Interval Arithmetic algorithms are used to reduce the execution times of numerical applications. Three hardware designs with data paths of 16, 32, and 64 bits are examined. These designs are compared based on their estimated chip area, cycle time, and execution times for various numerical applications. Each coprocessor can be implemented on a single chip with a cycle time that is comparable to IEEE double-precision floating point coprocessors. For certain numerical applications, the coprocessors are two to four orders of magnitude faster than a conventional software package for variable-precision, Interval Arithmetic. Представлены конструкция аппаратуры, испояьзуемые арифметические алгоритмы и приложения к решению численных задач для интервальных арифметических сопроцессоров переменной разрядности. Эти сопроцессоры позволяют программисту устанавливать начальную разрядность вычислений, определять точность результатов и заново вычислять неточные результаты с большей разрядностью. Для уменьшения времени выполнения в численных приложениях используются пнтервально-арифмети ческие алгоритмы переменной разрядности. Рассмотрены три аппаратные схемы с щиной данных щириной 16, 32 и 64 бита. Эти схемы сравниваются по требуемой площади крисгалла, продолжительности рабочего пикяа и быстродействию в различных численных приложениях. Каджый из этих сопроцессоров может быть реализован на одном кристалле с рабочей частотой, сравнимой с сопроцессорами плавающей точки двойной точности стандарта IEEE. В некоторых численных приложениях наши сопроцессоры на два-четыре порядка быстрее, чем распространенные программные пакеты, реализующие интервальную арифметику переменной разрядности.
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A processor for staggered Interval Arithmetic
Proceedings The International Conference on Application Specific Array Processors, 1995Co-Authors: M.j. Schulte, E.e. SwartzlanderAbstract:The paper presents the design of a high-speed processor which performs staggered Interval Arithmetic. Each staggered Interval is represented as the sum of a set of floating point numbers plus an Interval, which consists of two floating point endpoints. Staggered Interval Arithmetic allows the precision of the computation to be specified and the accuracy of the result to be determined. Efficient Arithmetic algorithms, which reduce the number of floating point operations needed to perform staggered Interval Arithmetic, are introduced. To achieve high performance, the processor employs an array of pipelined floating point Arithmetic units and two long accumulators. The processor provides direct hardware support for accurate and numerically reliable vector and matrix computations.
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ASAP - A processor for staggered Interval Arithmetic
Proceedings The International Conference on Application Specific Array Processors, 1995Co-Authors: M.j. Schulte, E.e. SwartzlanderAbstract:The paper presents the design of a high-speed processor which performs staggered Interval Arithmetic. Each staggered Interval is represented as the sum of a set of floating point numbers plus an Interval, which consists of two floating point endpoints. Staggered Interval Arithmetic allows the precision of the computation to be specified and the accuracy of the result to be determined. Efficient Arithmetic algorithms, which reduce the number of floating point operations needed to perform staggered Interval Arithmetic, are introduced. To achieve high performance, the processor employs an array of pipelined floating point Arithmetic units and two long accumulators. The processor provides direct hardware support for accurate and numerically reliable vector and matrix computations.
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ASAP - A variable-precision Interval Arithmetic processor
Proceedings of IEEE International Conference on Application Specific Array Processors (ASSAP'94), 1994Co-Authors: M.j. Schulte, E.e. SwartzlanderAbstract:This paper presents a special-purpose processor which implements variable-precision, Interval Arithmetic. Variable-precision Arithmetic allows the precision of the computation to be specified, based on the problem to be solved and the required accuracy of the computation. Interval Arithmetic produces two values for each result, such that the true result is guaranteed to be between the two values. The distance between the two values gives an upper bound on the error. Direct hardware support for variable-precision, Interval Arithmetic greatly improves the accuracy of the computation, and is much faster than existing software methods for controlling numerical error. Area and delay estimates indicate that the processor can be implemented on a single chip with a cycle time which is comparable to existing IEEE double-precision floating point processors. For computationally intensive problems, an application-specific array of variable-precision, Interval Arithmetic processors can execute in parallel to provide high-performance and numerically reliable results. >
Julius Zilinskas - One of the best experts on this subject based on the ideXlab platform.
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Comparison of Packages for Interval Arithmetic
Informatica (lithuanian Academy of Sciences), 2020Co-Authors: Julius ZilinskasAbstract:In this paper public available C and C++ packages for Interval Arithmetic are investigated and experimentally compared. The results of comparison give suggestions which packages and when are preferable.
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Balanced Random and Adaptive Interval Arithmetic for Systems of Linear Interval Equations
Adaptive Computing in Design and Manufacture VI, 2020Co-Authors: Julius Zilinskas, I.d.l. BogleAbstract:The paper concerns Interval methods — valuable tools for solving engineering problems — for finding outer approximations for the solution set of systems of linear Interval equations. The paper shows how Interval methods are used to analyse steady-state concentrations of systems of coupled reactors. The results of the experiments evaluating the outer approximations for the solution set of systems of linear Interval equations using Gaussian elimination with standard and balanced random Interval Arithmetic are given. Adaptive Interval Arithmetic is proposed to overcome disadvantages of balanced random Interval Arithmetic.
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Interval Arithmetic based optimization in nonlinear regression
Informatica (lithuanian Academy of Sciences), 2010Co-Authors: Antanas žilinskas, Julius ZilinskasAbstract:The optimization problems occurring in nonlinear regression normally cannot be proven unimodal. In the present paper applicability of global optimization algorithms to this problem is investigated with the focus on Interval Arithmetic based algorithms.
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Balanced random Interval Arithmetic in market model estimation
European Journal of Operational Research, 2006Co-Authors: Julius Zilinskas, I.d.l. BogleAbstract:The possibility of estimating bounds for the econometric likelihood function using balanced random Interval Arithmetic is experimentally investigated. The experiments on the likelihood function with data from housing starts have proved the assumption that distributions of centres and radii of evaluated balanced random Intervals are normal. Balanced random Interval Arithmetic can therefore be used to estimate bounds for this function and global optimization algorithms based on this Arithmetic are applicable to optimize it. The Interval branch and bound algorithms with bounds calculated using standard and balanced random Interval Arithmetic were used to optimize the likelihood function. Results of the experiments show that when reliability is essential the algorithm with standard Interval Arithmetic should be used, but when speed of optimization is more important, the algorithm with balanced random Interval Arithmetic should be used which in this case finishes faster and provides good, although not always optimal, values.
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Estimation of Functional Ranges Using Standard and Inner Interval Arithmetic
Informatica (lithuanian Academy of Sciences), 2006Co-Authors: Julius ZilinskasAbstract:New ways to estimate ranges of values of functions from standard and inner Interval Arithmetic have been proposed. Using the proposed ways ranges of values of mathematical test functions for global optimization and of objective functions for practical global optimization problems have been estimated and compared. Results of the experiments show that it is promising to use proposed balanced Interval Arithmetic in Interval global optimization.
M.j. Schulte - One of the best experts on this subject based on the ideXlab platform.
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Intrinsic Compiler Support for Interval Arithmetic
Numerical Algorithms, 2004Co-Authors: A. Akkas, M.j. Schulte, James E. StineAbstract:Interval Arithmetic provides an efficient method for monitoring errors in numerical computations and for solving problems that cannot be efficiently solved with floating-point Arithmetic. To support Interval Arithmetic, several software tools have been developed including Interval Arithmetic libraries, extended scientific programming languages, and Interval-enhanced compilers. The main disadvantage of these software tools is their speed, since Interval operations are implemented using function calls. In this paper, compiler support for Interval Arithmetic is investigated. In particular, the performance benefits of having the compiler inline Interval operations to eliminate function call overhead is researched. Interval operations are inlined with the GNU gcc compiler and the performance of Interval Arithmetic is evaluated on a superscalar architecture. To implement Interval operations with compiler support, the compiler produces sequences of instructions that use existing floating point hardware. Simulation results show that the compiler implementation of Interval Arithmetic is approximately 4 to 5 times faster than a functionally equivalent Interval Arithmetic software implementation with function call overhead and approximately 1.2 to 1.5 times slower than a dedicated Interval Arithmetic hardware implementation.
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Compiler support for Interval Arithmetic
IMTC 99. Proceedings of the 16th IEEE Instrumentation and Measurement Technology Conference (Cat. No.99CH36309), 1999Co-Authors: M.j. Schulte, A. Akkas, V.a. Zelov, J.c. BurleyAbstract:Interval Arithmetic provides an efficient method for representing uncertainty in data that results from physical measurements. It is also useful in evaluating the accuracy of measurements that have been processed by software. To facilitate the development of Interval Arithmetic software, the GNU Fortran compiler has been modified to support Interval Arithmetic. This paper gives an overview of the Interval-enhanced GNU Fortran compiler and its use in developing Interval Arithmetic software. Support for Interval Arithmetic is also being added to GNU C and C++ compilers.
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A variable-precision, Interval Arithmetic processor
Journal of Applied Mathematics and Mechanics, 1996Co-Authors: M.j. Schulte, E.e. SwartzlanderAbstract:This paper presents the software interface and hardware design of a processor which supports variable-precision, Interval Arithmetic. The software interface gives the programmer the ability to specify the precision of the computation, determine the accuracy of the result, and recompute inaccurate results with higher precision. Special instructions and hardware features are provided for operations on matrices, vectors and complex numbers. Providing direct hardware support for variable-precision, Interval Arithmetic greatly improves the accuracy and reliability of the computation, and is much faster than existing software techniques for controlling numerical error. Area and delay estimates indicate that the processor can be implemented on a single chip with a cycle time that is comparable to existing IEEE double-precision floating point processors.
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Variable-precision, Interval Arithmetic coprocessors
Reliable Computing, 1996Co-Authors: M.j. Schulte, E.e. SwartzlanderAbstract:This paper presents hardware designs, Arithmetic algorithms, and numerical applications for variable-precision, Interval Arithmetic coprocessors. These coprocessors give the programmer the ability to set the initial precision of the computation, determine the accuracy of the results, and recompute inaccurate results with higher precision. Variable-precision, Interval Arithmetic algorithms are used to reduce the execution times of numerical applications. Three hardware designs with data paths of 16, 32, and 64 bits are examined. These designs are compared based on their estimated chip area, cycle time, and execution times for various numerical applications. Each coprocessor can be implemented on a single chip with a cycle time that is comparable to IEEE double-precision floating point coprocessors. For certain numerical applications, the coprocessors are two to four orders of magnitude faster than a conventional software package for variable-precision, Interval Arithmetic. Представлены конструкция аппаратуры, испояьзуемые арифметические алгоритмы и приложения к решению численных задач для интервальных арифметических сопроцессоров переменной разрядности. Эти сопроцессоры позволяют программисту устанавливать начальную разрядность вычислений, определять точность результатов и заново вычислять неточные результаты с большей разрядностью. Для уменьшения времени выполнения в численных приложениях используются пнтервально-арифмети ческие алгоритмы переменной разрядности. Рассмотрены три аппаратные схемы с щиной данных щириной 16, 32 и 64 бита. Эти схемы сравниваются по требуемой площади крисгалла, продолжительности рабочего пикяа и быстродействию в различных численных приложениях. Каджый из этих сопроцессоров может быть реализован на одном кристалле с рабочей частотой, сравнимой с сопроцессорами плавающей точки двойной точности стандарта IEEE. В некоторых численных приложениях наши сопроцессоры на два-четыре порядка быстрее, чем распространенные программные пакеты, реализующие интервальную арифметику переменной разрядности.
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A processor for staggered Interval Arithmetic
Proceedings The International Conference on Application Specific Array Processors, 1995Co-Authors: M.j. Schulte, E.e. SwartzlanderAbstract:The paper presents the design of a high-speed processor which performs staggered Interval Arithmetic. Each staggered Interval is represented as the sum of a set of floating point numbers plus an Interval, which consists of two floating point endpoints. Staggered Interval Arithmetic allows the precision of the computation to be specified and the accuracy of the result to be determined. Efficient Arithmetic algorithms, which reduce the number of floating point operations needed to perform staggered Interval Arithmetic, are introduced. To achieve high performance, the processor employs an array of pipelined floating point Arithmetic units and two long accumulators. The processor provides direct hardware support for accurate and numerically reliable vector and matrix computations.
F.l. Alvarado - One of the best experts on this subject based on the ideXlab platform.
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Interval Arithmetic in power flow analysis
IEEE Transactions on Power Systems, 1992Co-Authors: Z. Wang, F.l. AlvaradoAbstract:The authors describe a method for taking uncertainty into account during power flow solutions with uncertain input data. The method is based on Interval Arithmetic which takes into consideration the uncertainty of the nodal information, and is able to provide strict bounds for the solutions to the problem: all possible solutions are included within the bounds given by Interval Arithmetic. Results are compared with those obtainable by Monte Carlo simulations and by the use of stochastic power flows. Object-oriented programming techniques make it possible to use Interval Arithmetic with minimal modifications to existing software. However, to reduce the conservatism inherent in all Interval Arithmetic computations, an iterative method used to obtain the hull of the solution set is described.
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Interval Arithmetic in power flow analysis
[Proceedings] Conference Papers 1991 Power Industry Computer Application Conference, 1991Co-Authors: Z. Wang, F.l. AlvaradoAbstract:Power flow analysis is the fundamental tool for the study of power systems. The data for this problem are subject to uncertainty. Interval Arithmetic is used to solve the power flow problem. Interval Arithmetic takes into consideration the uncertainty of the nodal information, and is able to provide strict bounds for the solutions to the problem: all possible solutions are included within the bounds given by Interval Arithmetic. Results are compared with those obtainable by Monte Carlo simulations and by the use of stochastic power flows. Object-oriented programming techniques makes it possible to use Interval Arithmetic with minimal modifications to existing software. However, to reduce the conservatism inherent in all Interval Arithmetic computations, an iterative method is used to obtain the hull of the solution set.
