23 research outputs found
Minimal average degree aberration and the state polytope for experimental designs
For a particular experimental design, there is interest in finding which
polynomial models can be identified in the usual regression set up. The
algebraic methods based on Groebner bases provide a systematic way of doing
this. The algebraic method does not in general produce all estimable models but
it can be shown that it yields models which have minimal average degree in a
well-defined sense and in both a weighted and unweighted version. This provides
an alternative measure to that based on "aberration" and moreover is applicable
to any experimental design. A simple algorithm is given and bounds are derived
for the criteria, which may be used to give asymptotic Nyquist-like
estimability rates as model and sample sizes increase
An improved test set approach to nonlinear integer problems with applications to engineering design
Many problems in engineering design involve the use of nonlinearities
and some integer variables. Methods based on test sets have been
proposed to solve some particular problems with integer variables, but they
have not been frequently applied because of computation costs. The walk-back
procedure based on a test set gives an exact method to obtain an optimal point
of an integer programming problem with linear and nonlinear constraints, but
the calculation of this test set and the identification of an optimal solution
using the test set directions are usually computationally intensive.
In problems for which obtaining the test set is reasonably fast, we show
how the effectiveness can still be substantially improved. This methodology
is presented in its full generality and illustrated on two specific problems: (1)
minimizing cost in the problem of scheduling jobs on parallel machines given
restrictions on demands and capacity, and (2) minimizing cost in the series
parallel redundancy allocation problem, given a target reliability. Our computational
results are promising and suggest the applicability of this approach
to deal with other problems with similar characteristics or to combine it with
mainstream solvers to certify optimalityJunta de Andalucía FQM- 5849Ministerio de Ciencia e Innovación MTM2010-19336Ministerio de Ciencia e Innovación MTM2010-19576Ministerio de Ciencia e Innovación MTM2013-46962- C2-1-PFEDE
Algebraic geometry in experimental design and related fields
The thesis is essentially concerned with two subjects corresponding to the two grants under which the author was research assistant in the last three years. The one presented first, which cronologically comes second, addresses the issues of iden- tifiability for polynomial models via algebraic geometry and leads to a deeper understanding of the classical theory. For example the very recent introduction of the idea of the fan of an experimental design gives a maximal class of models identifiable with a given design. The second area develops a theory of optimum orthogonal fractions for Fourier regression models based on integer lattice designs. These provide alternatives to product designs. For particular classes of Fourier models with a given number of interactions the focus is on the study of orthogonal designs with attention given to complexity issues as the dimension of the model increases. Thus multivariate identifiability is the field of concern of the thesis. A major link between these two parts is given by Part III where the algebraic approach to identifiability is extended to Fourier models and lattice designs. The approach is algorithmic and algorithms to deal with the various issues are to be found throughout the thesis.
Both the application of algebraic geometry and computer algebra in statistics and the analysis of orthogonal fractions for Fourier models are new and rapidly growing fields. See for example the work by Koval and Schwabe (1997) [42] on qualitative Fourier models, Shi and Fang (1995) [67] on ¿/-designs for Fourier regression and Dette and Haller (1997) [25] on one-dimensional incomplete Fourier models. For algebraic geometry in experimental design see Fontana, Pistone and Rogantin (1997) [31] on two-level orthogonal fractions, Caboara and Robbiano (1997) [15] on the inversion problem and Robbiano and Rogantin (1997) [61] on distracted fractions. The only previous extensive application of algebraic geometry in statistics is the work of Diaconis and Sturmfels (1993) [27] on sampling from conditional distributions
Multi-parametric linear programming under global uncertainty
Multi-parametric programming has proven to be an invaluable tool for optimisation under uncertainty. Despite the theoretical developments in this area, the ability to handle uncertain parameters on the left-hand side remains limited and as a result, hybrid, or approximate solution strategies have been proposed in the literature. In this work, a new algorithm is introduced for the exact solution of multi-parametric linear programming problems with simultaneous variations in the objective function's coefficients, the right-hand side and the left-hand side of the constraints. The proposed methodology is based on the analytical solution of the system of equations derived from the first order Karush–Kuhn–Tucker conditions for general linear programming problems using symbolic manipulation. Emphasis is given on the ability of the proposed methodology to handle efficiently the LHS uncertainty by computing exactly the corresponding nonconvex critical regions while numerical studies underline further the advantages of the proposed methodology, when compared to existing algorithms
Berechnung und Anwendungen Approximativer Randbasen
This thesis addresses some of the algorithmic and numerical challenges associated with the computation of approximate border bases, a generalisation of border bases, in the context of the oil and gas industry. The concept of approximate border bases was introduced by D. Heldt, M. Kreuzer, S. Pokutta and H. Poulisse in "Approximate computation of zero-dimensional polynomial ideals" as an effective mean to derive physically relevant polynomial models from measured data. The main advantages of this approach compared to alternative techniques currently in use in the (hydrocarbon) industry are its power to derive polynomial models without additional a priori knowledge about the underlying physical system and its robustness with respect to noise in the measured input data. The so-called Approximate Vanishing Ideal (AVI) algorithm which can be used to compute approximate border bases and which was also introduced by D. Heldt et al. in the paper mentioned above served as a starting point for the research which is conducted in this thesis. A central aim of this work is to broaden the applicability of the AVI algorithm to additional areas in the oil and gas industry, like seismic imaging and the compact representation of unconventional geological structures. For this purpose several new algorithms are developed, among others the so-called Approximate Buchberger Möller (ABM) algorithm and the Extended-ABM algorithm. The numerical aspects and the runtime of the methods are analysed in detail - based on a solid foundation of the underlying mathematical and algorithmic concepts that are also provided in this thesis. It is shown that the worst case runtime of the ABM algorithm is cubic in the number of input points, which is a significant improvement over the biquadratic worst case runtime of the AVI algorithm. Furthermore, we show that the ABM algorithm allows us to exercise more direct control over the essential properties of the computed approximate border basis than the AVI algorithm. The improved runtime and the additional control turn out to be the key enablers for the new industrial applications that are proposed here. As a conclusion to the work on the computation of approximate border bases, a detailed comparison between the approach in this thesis and some other state of the art algorithms is given. Furthermore, this work also addresses one important shortcoming of approximate border bases, namely that central concepts from exact algebra such as syzygies could so far not be translated to the setting of approximate border bases. One way to mitigate this problem is to construct a "close by" exact border bases for a given approximate one. Here we present and discuss two new algorithmic approaches that allow us to compute such close by exact border bases. In the first one, we establish a link between this task, referred to as the rational recovery problem, and the problem of simultaneously quasi-diagonalising a set of complex matrices. As simultaneous quasi-diagonalisation is not a standard topic in numerical linear algebra there are hardly any off-the-shelf algorithms and implementations available that are both fast and numerically adequate for our purposes. To bridge this gap we introduce and study a new algorithm that is based on a variant of the classical Jacobi eigenvalue algorithm, which also works for non-symmetric matrices. As a second solution of the rational recovery problem, we motivate and discuss how to compute a close by exact border basis via the minimisation of a sum of squares expression, that is formed from the polynomials in the given approximate border basis. Finally, several applications of the newly developed algorithms are presented. Those include production modelling of oil and gas fields, reconstruction of the subsurface velocities for simple subsurface geometries, the compact representation of unconventional oil and gas bodies via algebraic surfaces and the stable numerical approximation of the roots of zero-dimensional polynomial ideals
Computer algebra and transputers applied to the finite element method
Recent developments in computing technology have opened new prospects for computationally intensive numerical methods such as the finite element method. More complex and refined problems can be solved, for example increased number and order of the elements improving accuracy. The power of Computer Algebra systems and parallel processing techniques is expected to bring significant improvement in such methods. The main objective of this work has been to assess the use of these techniques in the finite element method. The generation of interpolation functions and element matrices has been investigated using Computer Algebra. Symbolic expressions were obtained automatically and efficiently converted into FORTRAN routines. Shape functions based on Lagrange polynomials and mapping functions for infinite elements were considered. One and two dimensional element matrices for bending problems based on Hermite polynomials were also derived. Parallel solvers for systems of linear equations have been developed since such systems often arise in numerical methods. Both symmetric and asymmetric solvers have been considered. The implementation was on Transputer-based machines. The speed-ups obtained are good. An analysis by finite element method of a free surface flow over a spillway has been carried out. Computer Algebra was used to derive the integrand of the element matrices and their numerical evaluation was done in parallel on a Transputer-based machine. A graphical interface was developed to enable the visualisation of the free surface and the influence of the parameters. The speed- ups obtained were good. Convergence of the iterative solution method used was good for gated spillways. Some problems experienced with the non-gated spillways have lead to a discussion and tests of the potential factors of instability
Displacement Analysis of Under-Constrained Cable-Driven Parallel Robots
This dissertation studies the geometric static problem of under-constrained cable-driven
parallel robots (CDPRs) supported by n cables, with n ≤ 6. The task consists of determining the overall robot configuration when a set of n variables is assigned. When variables
relating to the platform posture are assigned, an inverse geometric static problem (IGP)
must be solved; whereas, when cable lengths are given, a direct geometric static problem (DGP) must be considered. Both problems are challenging, as the robot continues to
preserve some degrees of freedom even after n variables are assigned, with the final configuration determined by the applied forces. Hence, kinematics and statics are coupled and
must be resolved simultaneously.
In this dissertation, a general methodology is presented for modelling the aforementioned
scenario with a set of algebraic equations. An elimination procedure is provided, aimed at
solving the governing equations analytically and obtaining a least-degree univariate polynomial in the corresponding ideal for any value of n. Although an analytical procedure
based on elimination is important from a mathematical point of view, providing an upper
bound on the number of solutions in the complex field, it is not practical to compute these
solutions as it would be very time-consuming. Thus, for the efficient computation of the
solution set, a numerical procedure based on homotopy continuation is implemented. A
continuation algorithm is also applied to find a set of robot parameters with the maximum
number of real assembly modes for a given DGP. Finally, the end-effector pose depends
on the applied load and may change due to external disturbances. An investigation into
equilibrium stability is therefore performed
Combined decision procedures for nonlinear arithmetics, real and complex
We describe contributions to algorithmic proof techniques for deciding the satisfiability
of boolean combinations of many-variable nonlinear polynomial equations and
inequalities over the real and complex numbers.
In the first half, we present an abstract theory of Grobner basis construction algorithms
for algebraically closed fields of characteristic zero and use it to introduce
and prove the correctness of Grobner basis methods tailored to the needs of modern
satisfiability modulo theories (SMT) solvers. In the process, we use the technique of
proof orders to derive a generalisation of S-polynomial superfluousness in terms of
transfinite induction along an ordinal parameterised by a monomial order. We use this
generalisation to prove the abstract (“strategy-independent”) admissibility of a number
of superfluous S-polynomial criteria important for efficient basis construction. Finally,
we consider local notions of proof minimality for weak Nullstellensatz proofs and give
ideal-theoretic methods for computing complex “unsatisfiable cores” which contribute
to efficient SMT solving in the context of nonlinear complex arithmetic.
In the second half, we consider the problem of effectively combining a heterogeneous
collection of decision techniques for fragments of the existential theory of real
closed fields. We propose and investigate a number of novel combined decision methods
and implement them in our proof tool RAHD (Real Algebra in High Dimensions).
We build a hierarchy of increasingly powerful combined decision methods, culminating
in a generalisation of partial cylindrical algebraic decomposition (CAD) which we
call Abstract Partial CAD. This generalisation incorporates the use of arbitrary sound
but possibly incomplete proof procedures for the existential theory of real closed fields
as first-class functional parameters for “short-circuiting” expensive computations during
the lifting phase of CAD. Identifying these proof procedure parameters formally
with RAHD proof strategies, we implement the method in RAHD for the case of
full-dimensional cell decompositions and investigate its efficacy with respect to the
Brown-McCallum projection operator.
We end with some wishes for the future