23,897 research outputs found

    Certification of Bounds of Non-linear Functions: the Templates Method

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    The aim of this work is to certify lower bounds for real-valued multivariate functions, defined by semialgebraic or transcendental expressions. The certificate must be, eventually, formally provable in a proof system such as Coq. The application range for such a tool is widespread; for instance Hales' proof of Kepler's conjecture yields thousands of inequalities. We introduce an approximation algorithm, which combines ideas of the max-plus basis method (in optimal control) and of the linear templates method developed by Manna et al. (in static analysis). This algorithm consists in bounding some of the constituents of the function by suprema of quadratic forms with a well chosen curvature. This leads to semialgebraic optimization problems, solved by sum-of-squares relaxations. Templates limit the blow up of these relaxations at the price of coarsening the approximation. We illustrate the efficiency of our framework with various examples from the literature and discuss the interfacing with Coq.Comment: 16 pages, 3 figures, 2 table

    A Lifting Relation from Macroscopic Variables to Mesoscopic Variables in Lattice Boltzmann Method: Derivation, Numerical Assessments and Coupling Computations Validation

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    In this paper, analytic relations between the macroscopic variables and the mesoscopic variables are derived for lattice Boltzmann methods (LBM). The analytic relations are achieved by two different methods for the exchange from velocity fields of finite-type methods to the single particle distribution functions of LBM. The numerical errors of reconstructing the single particle distribution functions and the non-equilibrium distribution function by macroscopic fields are investigated. Results show that their accuracy is better than the existing ones. The proposed reconstruction operator has been used to implement the coupling computations of LBM and macro-numerical methods of FVM. The lid-driven cavity flow is chosen to carry out the coupling computations based on the numerical strategies of domain decomposition methods (DDM). The numerical results show that the proposed lifting relations are accurate and robust

    Towards an exact adaptive algorithm for the determinant of a rational matrix

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    In this paper we propose several strategies for the exact computation of the determinant of a rational matrix. First, we use the Chinese Remaindering Theorem and the rational reconstruction to recover the rational determinant from its modular images. Then we show a preconditioning for the determinant which allows us to skip the rational reconstruction process and reconstruct an integer result. We compare those approaches with matrix preconditioning which allow us to treat integer instead of rational matrices. This allows us to introduce integer determinant algorithms to the rational determinant problem. In particular, we discuss the applicability of the adaptive determinant algorithm of [9] and compare it with the integer Chinese Remaindering scheme. We present an analysis of the complexity of the strategies and evaluate their experimental performance on numerous examples. This experience allows us to develop an adaptive strategy which would choose the best solution at the run time, depending on matrix properties. All strategies have been implemented in LinBox linear algebra library

    A nested Krylov subspace method to compute the sign function of large complex matrices

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    We present an acceleration of the well-established Krylov-Ritz methods to compute the sign function of large complex matrices, as needed in lattice QCD simulations involving the overlap Dirac operator at both zero and nonzero baryon density. Krylov-Ritz methods approximate the sign function using a projection on a Krylov subspace. To achieve a high accuracy this subspace must be taken quite large, which makes the method too costly. The new idea is to make a further projection on an even smaller, nested Krylov subspace. If additionally an intermediate preconditioning step is applied, this projection can be performed without affecting the accuracy of the approximation, and a substantial gain in efficiency is achieved for both Hermitian and non-Hermitian matrices. The numerical efficiency of the method is demonstrated on lattice configurations of sizes ranging from 4^4 to 10^4, and the new results are compared with those obtained with rational approximation methods.Comment: 17 pages, 12 figures, minor corrections, extended analysis of the preconditioning ste

    Certification of Real Inequalities -- Templates and Sums of Squares

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    We consider the problem of certifying lower bounds for real-valued multivariate transcendental functions. The functions we are dealing with are nonlinear and involve semialgebraic operations as well as some transcendental functions like cos\cos, arctan\arctan, exp\exp, etc. Our general framework is to use different approximation methods to relax the original problem into polynomial optimization problems, which we solve by sparse sums of squares relaxations. In particular, we combine the ideas of the maxplus estimators (originally introduced in optimal control) and of the linear templates (originally introduced in static analysis by abstract interpretation). The nonlinear templates control the complexity of the semialgebraic relaxations at the price of coarsening the maxplus approximations. In that way, we arrive at a new - template based - certified global optimization method, which exploits both the precision of sums of squares relaxations and the scalability of abstraction methods. We analyze the performance of the method on problems from the global optimization literature, as well as medium-size inequalities issued from the Flyspeck project.Comment: 27 pages, 3 figures, 4 table

    On the Complexity and Approximation of Binary Evidence in Lifted Inference

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    Lifted inference algorithms exploit symmetries in probabilistic models to speed up inference. They show impressive performance when calculating unconditional probabilities in relational models, but often resort to non-lifted inference when computing conditional probabilities. The reason is that conditioning on evidence breaks many of the model's symmetries, which can preempt standard lifting techniques. Recent theoretical results show, for example, that conditioning on evidence which corresponds to binary relations is #P-hard, suggesting that no lifting is to be expected in the worst case. In this paper, we balance this negative result by identifying the Boolean rank of the evidence as a key parameter for characterizing the complexity of conditioning in lifted inference. In particular, we show that conditioning on binary evidence with bounded Boolean rank is efficient. This opens up the possibility of approximating evidence by a low-rank Boolean matrix factorization, which we investigate both theoretically and empirically.Comment: To appear in Advances in Neural Information Processing Systems 26 (NIPS), Lake Tahoe, USA, December 201

    An Output-sensitive Algorithm for Computing Projections of Resultant Polytopes

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    We develop an incremental algorithm to compute the Newton polytope of the resultant, aka resultant polytope, or its projection along a given direction. The resultant is fundamental in algebraic elimination and in implicitization of parametric hypersurfaces. Our algorithm exactly computes vertex- and halfspace-representations of the desired polytope using an oracle producing resultant vertices in a given direction. It is output-sensitive as it uses one oracle call per vertex. We overcome the bottleneck of determinantal predicates by hashing, thus accelerating execution from 1818 to 100100 times. We implement our algorithm using the experimental CGAL package {\tt triangulation}. A variant of the algorithm computes successively tighter inner and outer approximations: when these polytopes have, respectively, 90\% and 105\% of the true volume, runtime is reduced up to 2525 times. Our method computes instances of 55-, 66- or 77-dimensional polytopes with 3535K, 2323K or 500500 vertices, resp., within 22hr. Compared to tropical geometry software, ours is faster up to dimension 55 or 66, and competitive in higher dimensions
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