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21
Autovalidating von Neumann rejection sampling from small
, 2009
"... which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Background: In phylogenetic inference one is interested in obtaining samples from the posterior distribution over the tree space on the basis of some observed DNA sequence data ..."
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which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Background: In phylogenetic inference one is interested in obtaining samples from the posterior distribution over the tree space on the basis of some observed DNA sequence data. One of the simplest sampling methods is the rejection sampler due to von Neumann. Here we introduce an autovalidating version of the rejection sampler, via interval analysis, to rigorously draw samples from posterior distributions over small phylogenetic tree spaces. Results: The posterior samples from the autovalidating sampler are used to rigorously (i) estimate posterior probabilities for different rooted topologies based on mitochondrial DNA from human, chimpanzee and gorilla, (ii) conduct a nonparametric test of rate variation between proteincoding and tRNAcoding sites from three primates and (iii) obtain a posterior estimate of the humanneanderthal divergence time. Conclusion: This solves the open problem of rigorously drawing independent and identically distributed samples from the posterior distribution over rooted and unrooted small tree spaces (3 or 4 taxa) based on any multiplyaligned sequence data.
Optimal centers in branchandprune algorithms for univariate global optimization
 APPLIED MATHEMATICS AND COMPUTATION
, 2005
"... We present an interval branchandprune algorithm for computing verified enclosures
for the global minimum and all global minimizers of univariate functions subject
to bound constraints. The algorithm works within the branchandbound framework
and uses first order information of the objective funct ..."
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We present an interval branchandprune algorithm for computing verified enclosures
for the global minimum and all global minimizers of univariate functions subject
to bound constraints. The algorithm works within the branchandbound framework
and uses first order information of the objective function. In this context, we investigate
valuable properties of the optimal center of a mean value form and prove optimality.
We also establish an inclusion function selection criterion between natural interval
extension and an optimal mean value form for the bounding process. Based on optimal
centers, we introduce linear (inner and outer) pruning steps that are responsible for the
branching process. The proposed algorithm incorporates the above techniques in order
to accelerate the search process. Our algorithm has been implemented and tested on a
test set and compared with three other methods. The method suggested shows a significant
improvement on previous methods for the numerical examples solved.
Mathematica Connectivity to Interval Libraries filib++ and CXSC
 Numerical Validation in Current Hardware Architectures. LNCS
, 2009
"... Abstract. Building interval software interoperability can be a good solution when reusing highquality legacy code or when accessing functionalities unavailable natively in one of the software packages. In this work we present the integration of programs based on the interval libraries filib++ and ..."
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Abstract. Building interval software interoperability can be a good solution when reusing highquality legacy code or when accessing functionalities unavailable natively in one of the software packages. In this work we present the integration of programs based on the interval libraries filib++ and CXSC into Mathematica via MathLink communication protocol. On some small easily readable programs we demonstrate: i) some details of MathLink technology, ii) the transparency of numerical data communication without any conversion, iii) the advantage of symbolic manipulation interfaces — the access to the external compiled language functionality from within Mathematica is often even more convenient than from its own native environment.
S.: Interval subroutine library mission
 Reliable Implementation of Real Number Algorithms: Theory and Practice. Number 06021 in Dagstuhl Seminar Proceedings, Internationales Begegnungs und Forschungszentrum fur Informatik, Schloss Dagstuhl
, 2006
"... Abstract. We propose the collection, standardization, and distribution of a fullfeatured, production quality library for reliable scientific computing with routines using interval techniques for use by the wide community of applications developers. 1 Vision – Why are we doing this? The interval/rel ..."
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Abstract. We propose the collection, standardization, and distribution of a fullfeatured, production quality library for reliable scientific computing with routines using interval techniques for use by the wide community of applications developers. 1 Vision – Why are we doing this? The interval/reliable computing research community has long worked to attract practicing scientists and engineers to use its results. We use any of the terms interval, reliable, verified computation in the sense of producing rigorous bounds on true results [1, 2]. The Interval Subroutine Library (ISL) is a project to place interval tools into the hands of people we believe will benefit from their use by gathering and refining existing tools from many interval authors. We acknowledge that intervals carry a steep learning curve, and that they sometimes have been overpromised. The winning strategy for widespread adoption of interval technologies is the development of “killer applications ” that are so much better (in some sense) than current practice that practicing scientists and engineers
An AutoValidating, TransDimensional, Universal Rejection Sampler for Locally Lipschitz Arithmetical Expressions
, 2013
"... The sample space of a transdimensional random vector is a union of spaces with different dimensions. We introduce a transdimensional extension of the rejection sampler of von Neumann. Our construction of the rejection sampler is based on interval analysis and provides a universal method that is ca ..."
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The sample space of a transdimensional random vector is a union of spaces with different dimensions. We introduce a transdimensional extension of the rejection sampler of von Neumann. Our construction of the rejection sampler is based on interval analysis and provides a universal method that is capable of producing independent and identically distributed (IID) samples from a large class of transdimensional target densities with locally Lipschitz arithmetical expressions. We illustrate the efficiency of the sampler by theory and by examples in up to ten dimensions. Our sampler is immune to the ‘pathologies’ of some infamous densities that were previously considered unsamplable and can rigorously draw IID transdimensional posterior samples from small binomial partition models and phylogenetic tree spaces.
Prototype implementation of certified approximation of the Medial Axis of smooth curves
"... In this report we give an algorithm to construct an approximate medial axis of a connected domain without any holes, in 2D, also known as the simply connected domain. The first part of the report concentrates on the algorithm which gives the approximate medial axis of a domain bounded by simple, ..."
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In this report we give an algorithm to construct an approximate medial axis of a connected domain without any holes, in 2D, also known as the simply connected domain. The first part of the report concentrates on the algorithm which gives the approximate medial axis of a domain bounded by simple, closed C 3 curve. In the second part of the report we look at a modified version of which looks at domain bounded by splines, which are piecewise C 3. We report the results of our algorithm for tangent continuous cubic splines which do not have any extremums of the Euclidean curvature at the point of tangency of two splines. Our algorithm uses interval arithmetic to compute the leaf and the branch points. Using simple combinatorial properties of the Medial Axis we guarantee topological correctness of the output.
Communicating Functional Expressions from Mathematica to CXSC ⋆
"... Abstract. This work focuses on a mechanism (and software) which communicates (via MathLink protocol) and provides compatibility between the representation of nonlinear functions specified as Mathematica expressions and objects of suitable classes supported by the CXSC automatic differentiation modu ..."
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Abstract. This work focuses on a mechanism (and software) which communicates (via MathLink protocol) and provides compatibility between the representation of nonlinear functions specified as Mathematica expressions and objects of suitable classes supported by the CXSC automatic differentiation modules. The application of the developed communication software is demonstrated by MathLink compatible programs embedding in Mathematica the CXSC modules for automatic differentiation as packages. The design methodology, some implementation issues and the use of the developed software are discussed. 1
A SOLVER FOR COMPLEXVALUED PARAMETRIC LINEAR SYSTEMS *
"... Abstract. This work reports on a new software for solving linear systems involving affinelinear dependencies between complexvalued interval parameters. We discuss the implementation of a parametric residual iteration for linear interval systems by advanced communication between the system Mathemat ..."
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Abstract. This work reports on a new software for solving linear systems involving affinelinear dependencies between complexvalued interval parameters. We discuss the implementation of a parametric residual iteration for linear interval systems by advanced communication between the system Mathematica and the library CXSC supporting rigorous complex interval arithmetic. An example of AC electrical circuit illustrates the use of the presented software. 1. Introduction. Scientific and engineering problems described by systems of linear algebraic equations involving uncertain model parameters include problems in engineering analysis or design [3, 5, 8], control engineering [2], etc. Significant research in this field is directed towards the use of intervals to represent the uncertain quantities in such systems.
Solving discrete minimax problems using interval arithmetic
 PACIFIC JOURNAL OF OPTIMIZATION
, 2006
"... Abstract. We present an interval algorithm for solving discrete minimax problems where the constituent minimax functions are continuously differentiable functions of one real variable. Our approach is based on smoothing the maxtype function by exploiting the Jaynes’s maximum entropy [Phys. Rev., 10 ..."
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Abstract. We present an interval algorithm for solving discrete minimax problems where the constituent minimax functions are continuously differentiable functions of one real variable. Our approach is based on smoothing the maxtype function by exploiting the Jaynes’s maximum entropy [Phys. Rev., 106:620–630, 1957]. The algorithm works within the branchandbound framework and uses first order information of the entropic objective function by means of an interval evaluation. First order information aids threefold. Firstly, to check monotonicity. Secondly, to apply mean value form for bounding the range of the function, and finally, to prune the search interval using the current upper bound of the global minimum. Our numerical results show that the proposed algorithm guarantees computationally rigorous bounds for the global minimum and all global minimizers.