Symbolic preconditioning techniques for linear systems of partial differential equations

Some algorithmic aspects of systems of PDEs based simulations can be better clarified by means of symbolic computation techniques. This is very important since numerical simulations heavily rely on solving systems of PDEs. For the large-scale problems we deal with in today's standard applications, it is necessary to rely on iterative Krylov methods that are scalable (i.e., weakly dependent on the number of degrees on freedom and number of subdomains) and have limited memory requirements. They are preconditioned by domain decomposition methods, incomplete factorizations and multigrid preconditioners. These techniques are well understood and efficient for scalar symmetric equations (e.g., Laplacian, biLaplacian) and to some extent for non-symmetric equations (e.g., convection-diffusion). But they have poor performances and lack robustness when used for symmetric systems of PDEs, and even more so for non-symmetric complex systems (fluid mechanics, porous media ...). As a general rule, the study of iterative solvers for systems of PDEs as opposed to scalar PDEs is an underdeveloped subject. We aim at building new robust and efficient solvers, such as domain decomposition methods and preconditioners for some linear and well-known systems of PDEs

Developing a labelled object-relational constraint database architecture for the projection operator

Current relational databases have been developed in order to improve the handling of stored data, however, there are some types of information that have to be analysed for which no suitable tools are available. These new types of data can be represented and treated as constraints, allowing a set of data to be represented through equations, inequations and Boolean combinations of both. To this end, constraint databases were defined and some prototypes were developed. Since there are aspects that can be improved, we propose a new architecture called labelled object-relational constraint database (LORCDB). This provides more expressiveness, since the database is adapted in order to support more types of data, instead of the data having to be adapted to the database. In this paper, the projection operator of SQL is extended so that it works with linear and polynomial constraints and variables of constraints. In order to optimize query evaluation efficiency, some strategies and algorithms have been used to obtain an efficient query plan. Most work on constraint databases uses spatiotemporal data as case studies. However, this paper proposes model-based diagnosis since it is a highly potential research area, and model-based diagnosis permits more complicated queries than spatiotemporal examples. Our architecture permits the queries over constraints to be defined over different sets of variables by using symbolic substitution and elimination of variables.Ministerio de Ciencia y Tecnología DPI2006-15476-C02-0

Enhanced LFR-toolbox for MATLAB and LFT-based gain scheduling

We describe recent developments and enhancements of the LFR-Toolbox for MATLAB for building LFT-based uncertainty models and for LFT-based gain scheduling. A major development is the new LFT-object definition supporting a large class of uncertainty descriptions: continuous- and discrete-time uncertain models, regular and singular parametric expressions, more general uncertainty blocks (nonlinear, time-varying, etc.). By associating names to uncertainty blocks the reusability of generated LFT-models and the user friendliness of manipulation of LFR-descriptions have been highly increased. Significant enhancements of the computational efficiency and of numerical accuracy have been achieved by employing efficient and numerically robust Fortran implementations of order reduction tools via mex-function interfaces. The new enhancements in conjunction with improved symbolical preprocessing lead generally to a faster generation of LFT-models with significantly lower orders. Scheduled gains can be viewed as LFT-objects. Two techniques for designing such gains are presented. Analysis tools are also considered

AutoBayes: A System for Generating Data Analysis Programs from Statistical Models

Data analysis is an important scientific task which is required whenever information needs to be extracted from raw data. Statistical approaches to data analysis, which use methods from probability theory and numerical analysis, are well-founded but difficult to implement: the development of a statistical data analysis program for any given application is time-consuming and requires substantial knowledge and experience in several areas. In this paper, we describe AutoBayes, a program synthesis system for the generation of data analysis programs from statistical models. A statistical model specifies the properties for each problem variable (i.e., observation or parameter) and its dependencies in the form of a probability distribution. It is a fully declarative problem description, similar in spirit to a set of differential equations. From such a model, AutoBayes generates optimized and fully commented C/C++ code which can be linked dynamically into the Matlab and Octave environments. Code is produced by a schema-guided deductive synthesis process. A schema consists of a code template and applicability constraints which are checked against the model during synthesis using theorem proving technology. AutoBayes augments schema-guided synthesis by symbolic-algebraic computation and can thus derive closed-form solutions for many problems. It is well-suited for tasks like estimating best-fitting model parameters for the given data. Here, we describe AutoBayes's system architecture, in particular the schema-guided synthesis kernel. Its capabilities are illustrated by a number of advanced textbook examples and benchmarks

SqFreeEVAL: An (almost) optimal real-root isolation algorithm

Let f be a univariate polynomial with real coefficients, f in R[X]. Subdivision algorithms based on algebraic techniques (e.g., Sturm or Descartes methods) are widely used for isolating the real roots of f in a given interval. In this paper, we consider a simple subdivision algorithm whose primitives are purely numerical (e.g., function evaluation). The complexity of this algorithm is adaptive because the algorithm makes decisions based on local data. The complexity analysis of adaptive algorithms (and this algorithm in particular) is a new challenge for computer science. In this paper, we compute the size of the subdivision tree for the SqFreeEVAL algorithm. The SqFreeEVAL algorithm is an evaluation-based numerical algorithm which is well-known in several communities. The algorithm itself is simple, but prior attempts to compute its complexity have proven to be quite technical and have yielded sub-optimal results. Our main result is a simple O(d(L+ln d)) bound on the size of the subdivision tree for the SqFreeEVAL algorithm on the benchmark problem of isolating all real roots of an integer polynomial f of degree d and whose coefficients can be written with at most L bits. Our proof uses two amortization-based techniques: First, we use the algebraic amortization technique of the standard Mahler-Davenport root bounds to interpret the integral in terms of d and L. Second, we use a continuous amortization technique based on an integral to bound the size of the subdivision tree. This paper is the first to use the novel analysis technique of continuous amortization to derive state of the art complexity bounds

Semiclassical quantization of the diamagnetic hydrogen atom with near action-degenerate periodic-orbit bunches

The existence of periodic orbit bunches is proven for the diamagnetic Kepler problem. Members of each bunch are reconnected differently at self-encounters in phase space but have nearly equal classical action and stability parameters. Orbits can be grouped already on the level of the symbolic dynamics by application of appropriate reconnection rules to the symbolic code in the ternary alphabet. The periodic orbit bunches can significantly improve the efficiency of semiclassical quantization methods for classically chaotic systems, which suffer from the exponential proliferation of orbits. For the diamagnetic hydrogen atom the use of one or few representatives of a periodic orbit bunch in Gutzwiller's trace formula allows for the computation of semiclassical spectra with a classical data set reduced by up to a factor of 20.Comment: 10 pages, 9 figure