PolyU IR Community:
http://hdl.handle.net/10397/10
2015-07-02T18:18:32ZSpherical tε-design and approximation on the sphere : theory and algorithms
http://hdl.handle.net/10397/7370
Title: Spherical tε-design and approximation on the sphere : theory and algorithms
Authors: Zhou, Yang
Abstract: This thesis concentrates on the spherical tε-designs on the two-sphere, numerical algorithms for finding spherical tε-designs and numerical approximation on the sphere using spherical tε-designs. A set of points on the unit sphere is called a spherical t-design if the average value of any polynomial of degree at most t over the set is equal to the average value of the polynomial over the sphere. Spherical t-designs have many important applications in geophysics and bioengineering, and provide many challenging problems in computational mathematics. As a generalization of spherical t-design, we define a spherical tε-design with 0 ≤ t < 1 which provides an integration rule with a set of points on the unit sphere min weight and positive weights satisfying (1-ε)² ≤ min weight/max weight ≤ 1. The integration rule also gives the exact integral for any polynomial of degree at most t. Due to the flexibility of choice for the weights, the number of points in the integration rule can be less for making the exact integral for any polynomial of degree at most t. To our knowledge, so far there is no theoretical result which proves the existence of a spherical t-design with (t+1)² points for arbitrary t. In 2010 Chen, Frommer and Lang developed a computation-assist proof for the existence of spherical t-designs for t = 1,..., 100 with (t+1)² points. Based on the algorithm proposed in that paper, a series of interval enclosures for spherical t-design was computed. In this thesis we prove that all the point sets arbitrarily chosen in these interval enclosures are spherical tε-designs and give an upper bound of ε. We then study the variational characterization and the worst-case error of spherical tε-design. Based on the reproducing kernel theory and its relationship with the geodesic distance, we propose a way to compute the worst-case error for numerical integration using spherical tε-design in Sobolev space. Moreover, we propose an approach for finding spherical tε-designs. We show that finding a spherical tε-design can be reformulated as a system of polynomial equations with box constraints. Using the projection operator, the system can be written as a nonsmooth nonconvex least squares problem with zero residual. We propose a smoothing trust region filter algorithm for solving such problems. We present convergence theorems of the proposed algorithm to a Clarke stationary point or a global minimizer of the problem under certain conditions. Preliminary numerical experiments show the efficiency of the proposed algorithm for finding spherical tε-designs. Another contribution in this thesis is the numerical approximation on the sphere using regularized least squares approaches. We consider two regularized least squares problems using spherical tε-designs: regularized polynomial approximation on the sphere, and regularized hybrid approximation on the sphere using both radial basis functions and spherical polynomials. For the first approach we apply the ℓ₂ regularized form and give an approximation quality estimation. For the second approach we study its ℓ₁ regularized form and solve the problem using alternating direction method with multipliers. Numerical experiments are given to demonstrate the effectiveness of these two models.
Description: xviii, 105 pages : illustrations ; 30 cm; PolyU Library Call No.: [THS] LG51 .H577P AMA 2014 Zhou2014-01-01T00:00:00ZTesting serial correlation in partially linear additive models
http://hdl.handle.net/10397/7369
Title: Testing serial correlation in partially linear additive models
Authors: Yang, Jin
Abstract: This thesis proposes procedures for testing serial correlation in the partially linear additive models without and with errors in variables, which include the partially linear models and additive models as their special cases. For the partially linear additive models without errors, an empirical-likelihood-based procedure is developed based on the profile least-squares method. It is shown that the proposed test statistic is asymptotically chi-square distributed under the null hypothesis of no serial correlation. Then the rejection region can be constructed using this result. It is noted that the procedures are not only for testing zero first-order serial correlation, but also for testing higher-order serial correlation. For the partially linear additive models with errors, the methods based on the profile least-squares is invalid because of the existence of the errors in variables. By a corrected profile least-squares approach, another empirical-likelihood-based procedure is developed. The asymptotic properties are investigated, based on which the rejection region can be easily constructed. Extensive simulation studies were conducted to assess the finite sample properties of the proposed procedures' sizes and powers.
Description: vii, 66 leaves : illustrations ; 30 cm; PolyU Library Call No.: [THS] LG51 .H577M AMA 2014 Yang2014-01-01T00:00:00ZLower-order penalty methods for nonlinear optimization and complementarity problems
http://hdl.handle.net/10397/7368
Title: Lower-order penalty methods for nonlinear optimization and complementarity problems
Authors: Tian, Boshi
Abstract: The main purpose of this thesis is to propose efficient numerical methods to solve inequality constrained nonlinear programming problems and complementarity problems by virtue of the {190} 1/p (p> 1)-penalty function. We propose an interior-point l1/p -penalty method for inequality constrained optimization problems by introducing a technique of the p-order relaxation to the nonconvex and non-Lipschitzian l1/p -penalty function and combining with an interior-point method. We introduce different kinds of constraint qualifications to establish first-order necessary conditions for the relaxed problem. We employ the modified Newton method to solve a sequence of logarithmic barrier subproblems and detail three reliable algorithms by using the Armijo line search. We prove that the iteration sequence generated by the proposed method converges to some KKT (or FJ) point of the original problem under mild conditions. Preliminary numerical experiments on small, medium and large test problems in the literature show that, comparing with some existing interior-point l1 -penalty methods, the proposed method is competitive in terms of the iteration numbers, better when comparing the number of updating the penalty parameters and more reliable when comparing the relative error. We introduce a box-constrained differentiable penalty method for nonlinear complementarity problems, which not only inherits the same convergence rate as the existing ℓ 1/p -penalty method but also overcomes its disadvantage of the non-Lipschitzianness. We introduce a concept of a uniform ξ-P-function with ξε[1, 2), under which we prove that the solution of box-constrained penalized equations converges to a solution of the original problem at an exponential order. Instead of solving the box-constrained penalized equations directly, we solve a corresponding differentiable least squares problem by using a trust-region Gauss-Newton method to design the globally convergent method that allows arbitrary starting points for solving the complementarity problems. Furthermore, we establish the connection between the local solution of the least squares problem and the solution of the original problem under mild conditions. We carry out the numerical experiments on the test problems from MCPLIB, which show that the proposed method is efficient and robust. We investigate an unconstrained differentiable penalty method for general complementarity problems without introducing artificial variables, which shares the exponential convergence rate under the assumption of a uniform ξ-P-function. Instead of solving the unconstrained penalized equations directly, we solve a corresponding differentiable least squares problem by using a trust-region Gauss-Newton method. Preliminary numerical experiments show that the proposed method is more robust than the box-constrained differentiable penalty method.
Description: ix, 44 leaves ; 30 cm xii, 129 pages ; 30 cm xiii, 126 pages : illustrations ; 30 cm; PolyU Library Call No.: [THS] LG51 .H577P AMA 2014 Tian2014-01-01T00:00:00ZMathematical studies on some models arising in chemotaxis and magnetohydrodynamic turbulence
http://hdl.handle.net/10397/7367
Title: Mathematical studies on some models arising in chemotaxis and magnetohydrodynamic turbulence
Authors: Jin, Haiyang
Abstract: This thesis is mainly focused on the theoretical studies on some models arising in chemotaxis and magnetohydrodynamic turbulence. The main results of this thesis consist of the following three parts. 1. A quasilinear parabolic volume-filling chemotaxis model with critical sensitivity in two dimensions is considered. In this study, a threshold number is explicitly found such that the solution exists globally with uniform-in-time bound or blows up if the initial cell mass is less than or greater than this number. Furthermore we determine the blowup time is infinite under certain conditions on the decay rate of the chemotactic sensitivity. 2. We consider the initial-boundary value problem of the attraction-repulsion Keller-Segel (ARKS) chemotaxis model describing the quorum effect in chemotaxis and the aggregation of Microglia in the central nervous system in Alzhemer' disease. First, we study the asymptotic behavior of solutions to the ARKS chemotaxis model in one dimension, where we obtain the uniform-in-time boundedness of solutions and prove that the model possesses a global attractor. For a special case where the attractive and repulsive chemical signals have the same degradation rate, we show that the solution converges to a stationary solution algebraically as time tends to infinity if the attraction dominates. In two dimensional spaces, we show that if the repulsion dominates over attraction, then the global classical solutions exist with uniform-in-time bound for large initial data. Moreover we present a Lyapunov function at the first time for the irreducible three-component attraction-repulsion chemotaxis model which plays a central role to obtain our results. 3. We establish the asymptotic nonlinear stability of solutions to the Cauchy problem of a strongly coupled Burgers system arising in magnetohydrodynamic (MHD) turbulence. We show that, as time tends to infinity, the solutions of the Cauchy problem converge to constant states or rarefaction waves with large initial data, or viscous shock waves with arbitrarily large amplitude, where the precise asymptotic behavior depends on the relationship between the left and right end states of the initial value. Our results confirm the existence of shock waves (or turbulence) numerically found in the literature.
Description: ix, 44 leaves ; 30 cm xii, 129 pages ; 30 cm; PolyU Library Call No.: [THS] LG51 .H577P AMA 2014 Jin2014-01-01T00:00:00Z