Interval Iteration Research Articles

Enclosing solutions of complex linear systems of equations iteratively

AbstractWe consider the interval iteration [x ]k +1 = [A ][x ]k + [b ] in different interval arithmetics with the aim to enclose solutions of x = Ax +b in the case that A and b are only known to be contained in some given intervals. We give necessary and sufficient criteria for the convergence of the interval iteration for every initial interval vector [x ]0 to some [x ]* = [x ]*([x ]0) with respect to the considered interval arithmetic. Such a limit is a solution of the interval system [x ] = [A ][x ] + [b ].If we compare the interval arithmetics with respect to the behavior of [x ]k +1 = [A ][x ]k + [b ] we come to the conclusion, that the special choice of the arithmetic has a sensitive influence on the convergence of the sequence. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Enclosing Solutions of Singular Interval Systems Iteratively

Richardson splitting applied to a consistent system of linear equations Cx = b with a singular matrix C yields to an iterative method x k+1 = Ax k + b where A has the eigenvalue one. It is known that each sequence of iterates is convergent to a vector x* = x* (x0) if and only if A is semi-convergent. In order to enclose such vectors we consider the corresponding interval iteration $$[x]^{k+1} = [A][x]^k+[b]$$ with ρ(|[A]|) = 1 where |[A]| denotes the absolute value of the interval matrix [A]. If |[A]| is irreducible we derive a necessary and sufficient criterion for the existence of a limit $$[x]^* = [x]^*([x]^0)$$ of each sequence of interval iterates. We describe the shape of $$[x]^*$$ and give a connection between the convergence of ( $$[x]^k$$ ) and the convergence of the powers $$[A]^k$$ of [A].

The investigation of mole fraction dependence of mobility for In xGa 1−xN alloy

We have calculated the mobility of In x Ga 1− x N alloy by the iterative method interval x = 0 - 1 in low field. We have investigated x mole fraction dependence of calculated mobility. The Boltzmann transport equation involved in relevant scattering mechanisms has been solved. These scattering mechanisms are polar optic phonon scattering, ionized impurity scattering, acoustic phonon deformation potential scattering, acoustic phonon piezoelectric scattering and alloy scattering. Vegard's law was used for the lattice constants of these compounds. Furthermore, Phillip's electronegative difference theorem and the values of alloy potential in Ref. [11] were used to calculate the alloy scattering potential for alloy scattering and for comparison.

USING MOLECULAR ALLELIC VARIATION TO UNDERSTAND DOMESTICATION PROCESSES AND CONSERVE DIVERSITY IN BRASSICA CROPS

USING MOLECULAR ALLELIC VARIATION TO UNDERSTAND DOMESTICATION PROCESSES AND CONSERVE DIVERSITY IN BRASSICA CROPS

A new simultaneous method of fourth order for finding complex zeros in circular interval arithmetic

Starting from disjoint disks which contain polynomial complex zeros, the new iterative interval method for simultaneous finding of inclusive disks for complex zeros is formulated. The convergence theorem and the conditions for convergence are considered, and the convergence is shown to be fourth. Numerical examples are included.

Interval Finite Element Analysis of Structural Systems with Uncertainties. Static Analyses of Truss Structures with Geometrical Uncertainties.

In the Interval Finite Element Analysis of structure the uncertainties given by the interval are studied within the finite element formulation to determine the sharp bounds of the displacement response. Two approaches namely, the Direct Interval Finite Element Analysis (DIFEA) and the Iterative Interval Finite Element Analysis (IIFEA) are proposed to obtain the sharp solution as much as possible. In DIFEA, each step of the standard FEA, using ordinary real arithmetic, is simply replaced with the corresponding interval arithmetic step. In IIFEA, the interval structural analysis is formulated as a nonlinear optimization problem, while components of the displacement vector as well as parameters of the structural system with uncertainties are dealt as decision variables. A specific component of the displacement vector is considered as the objective function, which is minimized and maximized to obtain the lower and upper bounds. Validation of these methods is tested to the analysis of truss structural systems with uncertainties in nodal coordinates.

The Preliminary Enclosing of the ODE Solutions on the Base of the Cauchy-Duhamel Identity

An iterative interval algorithm of a preliminary enclosing of the integral curve in the next step of an integration process is described. It is based on the well-known Cauchy-Duhamel identity. The conditions of the ending of the process are derived. The criterion of existence and uniqueness of the solution is discussed, too.

An interval method for global nonlinear analysis

In this paper, the problem of finding the set of all real solutions to a system of n nonlinear equations contained in a given n-dimensional box [the global nonlinear analysis (GNA) problem] is considered. A new iterative interval method for solving the GNA problem is suggested. It is based on the following techniques: (1) transformation of the original system into an augmented system of n'=n+m equations of n' variables by introducing m auxiliary variables, the augmented system being of the so-called semiseparable form; (2) enclosure of the nonlinear augmented system at each iteration by a specific linear interval system of size n'/spl times/n'; (3) elimination of the auxiliary variables; and (4) solution of the resulting reduced size n/spl times/n linear system, using the so-called constraint propagation approach. The method suggested shows a significant improvement over previous techniques for the numerical examples solved.

Mapping quantitative trait loci affecting sternopleural bristle number in Drosophila melanogaster using changes of marker allele frequencies in divergently selected lines.

Quantitative trait loci (QTLs) responsible for variation in sternopleural bristle number in crosses between the laboratory lines of Drosophila melanogaster OregonR and CantonS were mapped using information from allele frequency changes of two families of retrotransposon markers in divergently selected populations. QTL effects and positions were inferred by likelihood, using transition matrix iteration and Monte Carlo interval mapping. Individuals from the selected population were genotyped for markers spaced at an average distance 4.4 cM. Four QTLs of moderate effect ranging from 0.6 to 1.32 bristles accounted for most of the selection response. A permutation test of the correspondence between the mapped QTLs and the positions of bristle number candidate genes suggested that alleles at these candidate genes were no more strongly associated with selected changes in marker allele frequency than were randomly chosen positions in the genome.

Explaining curd and spear geometry in broccoli, cauliflower and `romanesco': quantitative variation in activity of primary meristems

Brassica oleracea L. is highly polymorphic and includes varieties which exhibit a headed phenotype (a large preinflorescence): the curd of cauliflower and `romanesco' (var. botrytis), and the spear of broccoli (var. italica). This headed phenotype results from highly iterative patterns of activity at the primary meristems. Differences in the morphology of curds and spears are accounted for by three quantitative variables: the rate of production of branch primordia on the flanks of the apical meristems (RPP); the number of branch primordia produced before the first formed begin producing their own branch primordia (the iteration interval, ITI); and the duration of the preinflorescence stage (before production of flower primordia). Relatively stable iteration parameters (RPP and ITI) during curd development lead to the production of semi-spherical curds with a smooth surface in cauliflower and broccoli, whereas in `romanesco' RPP and ITI increase throughout curd development, inducing a pyramidal curd with an angular surface. A relatively long preinflorescence stage in cauliflower and `romanesco' results in the curd surface being composed largely of branch primordia, whereas in broccoli this stage is short and the spear surface is made up of flower buds. Simplified growth models for these three headed types are presented. The implications for the genetic control of the B. oleracea L. headed phenotype and the relationships between shoot apical meristem size, phyllotaxis and curd/spear morphology are discussed.

Analyzing Asynchronous Pipeline Schedules

Asynchronous pipelining is a form of parallelism that is useful in both distributed and shared memory systems. We show that asynchronous pipeline schedules are a generalization of both noniterative DAG (directed acyclic graph) schedules as well as simpler pipeline schedules, unifying these two types of scheduling. We generalize previous work on determining if a pipeline schedule will deadlock, and generalize Reiter's well-known formula for determining the iteration interval of a deadlock-free schedule, which is the primary measure of the execution time of a schedule. Our generalizations account for nonzero communication times (easy) and the assignment of multiple tasks to processors (nontrivial). A key component of our generalized approach to pipeline schedule analysis is the use of pipeline scheduling edges with potentially negative data dependence distances. We also discuss implementation of an asynchronous pipeline schedule at runtime; show how to efficiently simulate pipeline execution on a sequential processor; define and derive bounds on the startup time of a schedule, which is a secondary schedule performance measure; and describe a new algorithm for evaluating the iteration interval formula.

Existence Test for Asynchronous Interval Iteration

In the search for regions that contain fixed points of a real function of several variables, tests based on interval calculations can be used to establish existence or non-existence of fixed points in regions that are examined in the course of the search. The search can e.g. be performed as a synchronous (sequential) interval iteration: In each iteration step all components of the iterate are calculated based on the previous iterate. In this case it is straight forward to base simple interval existence and non-existence tests on the calculations done in each step of the iteration. The search can also be performed as an asynchronous (parallel) iteration: Only a few components are changed in each step and this calculation is in general based on components from different previous iterates. For the asynchronous iteration it turns out that simple tests of existence and non-existence can be based on the component wise calculations done in the course of the iteration. These component wise tests are useful for parallel implementation of the search, since the tests can then be performed local to each processor and only when a test is successful does a processor communicate this result to other processors.

Acceleration of the back propagation through dynamic adaptation of the learning rate

Standard backpropagation, as with many gradient based optimization methods converges slowly as neural networks training problems become larger and more complex. In this paper, we present a new algorithm, dynamic adaptation of the learning rate to accelerate steepest descent. The underlying idea is to partition the iteration number domain into n intervals and a suitable value for the learning rate is assigned for each respective iteration interval. We present a derivation of the new algorithm and test the algorithm on several classification problems. As compared to standard backpropagation, the convergence rate can be improved immensely with only a minimal increase in the complexity of each iteration.

An interval iteration for multiple roots of transcendental equations

An iteration method for roots of algebraic functions with roots of multiplicity greater than one is established using tools and techniques from interval arithmetic. The method is based on an interval iteration functions for multiple roots and it retains the convergence order of the underlying iteration method while preserving global convergence over an initial interval. A number of simple examples are provided to show that the method is feasible and that it produces reasonable results.

On the application of certain interval iteration methods to solving a system of nonlinear algebraic equations

We consider the problem of finding real solutions of a system of nonlinear algebraic equations using interval analysis. Several versions of Newton and Runge interval iteration methods are presented. The computational aspects of their application are explained.

Applications of Pre-Stack Depth Migration

Complex shallow faulting is a common problem which can lead to poor results when using the CDP assumption. Wave-field distortion makes it impossible to select a stacking velocity that does not deteriorate stack response. The only way to achieve a high-quality stack is to predict and remove the wave-field distortion before stack, that is, by correctly migrating the pre-stack gather. Predicting the distortion is difficult, as it requires the estimation of the actual interval velocity field. Many methods have been proposed for estimating velocities, three of which are iterative pre-stack depth migration, ray-trace modelling, and interactive interval velocity analysis. Each of these has advantages and disadvantages, but iterative pre-stack depth migration is generally the most useful. Reflection tomography can also be useful. Once accurate velocities have been obtained, many algorithms are available to do the actual migration. Two methods are compared; a shot-record finite-difference algorithm and a common-offset, Kirchhoff algorithm. The latter algorithm has practical advantages that lends itself more to iterative pre-stack depth-migration velocity analysis and generally makes it more attractive. Pre-stack depth migration, with both algorithms, is demonstrated on a real data set from offshore north-western Australia. The examples show that iterative pre-stack depth migration can successfully improve seismic images in difficult areas.

Decomposition of arithmetic expressions to improve the behavior of interval iteration for nonlinear systems

Interval iteration can be used, in conjunction with other techniques, for rigorously bounding all solutions to a nonlinear system of equations within a given region, or for verifying approximate solutions. However, because of overestimation which occurs when the interval Jacobian matrix is accumulated and applied, straightforward linearization of the original nonlinear system sometimes leads to nonconvergent iteration. In this paper, we examine interval iterations based on an expanded system obtained from the intermediate quantities in the original system. In this system, there is no overestimation in entries of the interval Jacobi matrix, and nonlinearities can be taken into account to obtain sharp bounds. We present an example in detail, algorithms, and detailed experimental results obtained from applying our algorithms to the example. — Author's Abstract

Decomposition of Dynamic Programming by Nonlinear Programming and Parallel Processing

To solve optimal control problems in which the state variables are of high dimension, the conventional numerical dynamic programming algorithm is in fact inapplicable owing to the well known difficulty in relation to the “ curse of dimensionality”. In this paper, a new technique called generalized time interval iteration algorithm is advanced which eliminates the difficulty of conventional dynamic programming thoroughly and can be applied to a fairly large class of optimal control problems of practical importance. The basic idea is to decompose the dynamic programming model into a series of nonlinear programming models with various scales in such a way to fit the structural feature of the problem in hand so as to reduce the total computation time. Hypotheses with rather mild conditions are proposed to guarantee the convergence of the algorithm. An important advantage of the technique proposed is that it is particularly suitable for parallel processing, and the total computation time can be reduced in a factor proportional to the number of time intervals provided that the quantity of the processors for parallel processing is enough.

Interval Newton method: Hansen-Greenberg approach—some procedural improvements

E.R. Hansen and R.I. Greenberg have presented an interval Newton method to solve a system of n nonlinear equations in n real variables. Following R.E. Moore, they have linearized the system using a mean value expansion, and used preconditioning technique due to Hansen and R.R. Smith to modify the system. The modified system is subjected to a Hansen-Sengupta step to obtain an updated interval containing the solution. Hansen and Greenberg have in fact used the subalgorithms of preconditioning and Hansen-Sengupta step witha real (local, noninterval) iteration and an elimination procedure to provide an algorithm of greater efficiency. In this paper, we indicate procedures which will further improve the efficiency of the Hansen-Greenberg algorithm. Firstly, a better approximating matrix for the identity is obtained. Secondly, a successive overrelaxation (SOR) technique is introduced to replace the Hansen-Sengupta step if necessary. Finally, an interval iteration is suggested to provide an alternative to the real (noninterval) inner iteration introduced by Hansen and Greenberg. Examples are included to show improvement in results and/or the forms the new procedures would take. the additional procedures provide more efficient results besides improving the techniques.

A self-validating numerical method for the matrix exponential

An algorithm is presented, which produces highly accurate and automatically verified bounds for the matrix exponential function. Our computational approach involves iterative defect correction, interval analysis and advanced computer arithmetic. The algorithm presented is based on the “scaling and squaring” scheme, utilizing Pade approximations and safe error monitoring. A PASCAL-SC program is reported and numerical results are discussed.