Langer U., Paule P. (eds.) Numerical and Symbolic Scientific Computing. Progress and Prospects

Springer, 2012. — 361 p.For more than 10 years, the numerical analysis and symbolic computation groups at the Johannes Kepler University Linz (JKU) have made serious efforts to combine two different worlds of scientific computing, numerics and symbolics. This work has been carried out in the frame of two excellence programs of the Austrian Science Funds (FWF), a special research program (SFB, 1998–2008) and a doctoral program (DK, 2008–). In addition to the JKU institutes for Applied Geometry, Computational Mathematics, Industrial Mathematics, and the Research Institute for Symbolic Computation (RISC), the Radon Institute for Computational and Applied Mathematics (RICAM), a branch of the Austrian Academy of Sciences, have been partners in this enterprise.
This book presents an offspring of this initiative. It contains surveys of the state of the art and of results achieved after more than 10 years of SFB/DK work. In addition, we included chapters that go beyond, this means, which set pointers for future developments. All of the chapters have been carefully refereed.Most of them center around the theme of partial differential equations. Major aspects are: fast solvers in elastoplasticity, symbolic analysis for boundary problems (from rewriting to parametrized Groebner bases), symbolic treatment of operators, use of computer algebra in the finite element method for the construction of recurrence relations in special high-order Maxwell solvers and for the construction of sparsity optimized high-order finite element basis functions on simplices, a symbolic approach to finite difference schemes, cylindrical algebraic decomposition and symbolic local Fourier analysis of multigrid methods, and white noise analysis for stochastic PDEs. The scope of other numerical-symbolic topics range from applied and computational geometry (approximate implicitization of space curves, symbolic–numeric genus computation, automated discovery in geometry) to computer algebra methods used for total variation energy minimization. One chapter deals with verification conditions in connection with functional recursive programs.Approximate Implicitization of Space Curves
Sparsity Optimized High Order Finite Element Functions on Simplices
Fast Solvers and A Posteriori Error Estimates in Elastoplasticity
A Symbolic-Numeric Algorithm for Genus Computation
The Seven Dwarfs of Symbolic Computation
Computer Algebra Meets Finite Elements: An Efficient Implementation for Maxwell’s Equations
A Symbolic Approach to Generation and Analysis of Finite Difference Schemes of Partial Differential Equations
White Noise Analysis for Stochastic Partial Differential Equations
Smoothing Analysis of an All-at-Once Multigrid Approach for Optimal Control Problems Using Symbolic Computation
Analytical Evaluations of Double Integral Expressions Related to Total Variation
Sound and Complete Verification Condition Generator for Functional Recursive Programs
An Introduction to Automated Discovery in Geometry through Symbolic Computation
Symbolic Analysis for Boundary Problems: From Rewriting to Parametrized Grӧbner Bases
Linear Partial Differential Equations and Linear Partial Differential Operators in Computer Algebra.