Handbook of Numerical Analysis : Special Volume: Foundations of Computational Mathematics (Handbook of Numerical Analysis)

The book offers a panorama of a part of mathematics known as "Foundations of Computational Mathematics." This is achieved through five survey papers describing some foundational issues in different branches of computational mathematics. In addition, a sixth paper gives an account of what "Foundations of Computational Mathematics" is. 0002.tif......Page 1 0001.tif......Page 0 Contents of Volume XI......Page 2 On the place of numerical analysis in the mathematical universe......Page 4 Approximation theory......Page 7 Functional analysis......Page 11 Complexity theory......Page 15 Probability theory and statistics......Page 17 Nonlinear dynamical systems......Page 20 Topology and algebraic geometry......Page 21 Differential geometry......Page 23 Abstract algebra......Page 25 ...and beyond......Page 27 Adaptivity......Page 28 Conditioning......Page 29 Structure......Page 31 Acknowledgements......Page 32 References......Page 33 Geometric Integration and its Applications......Page 36 Introduction......Page 37 Qualitative properties......Page 39 A partial listing......Page 40 Why preserve structure?......Page 43 The theory of Hamiltonian methods......Page 44 Outline......Page 46 Symplectic Runge-Kutta methods......Page 50 Splitting and composition methods......Page 52 Example 1. The Harmonic oscillator......Page 55 Example 2. The pendulum......Page 58 Example 4. Stellar dynamics and the Kepler problem......Page 60 Error analysis - a brief overview......Page 62 The backward error analysis of symplectic methods......Page 63 An analysis of splitting methods......Page 65 Outline......Page 68 Serendipitous and enforced conservation laws......Page 69 Discrete gradient methods......Page 71 Lie group methods......Page 72 Example 1. Evolution on the sphere......Page 77 Example 2. Rigid body motion......Page 78 Symmetries and reversing symmetries......Page 80 Outline......Page 82 Numerical methods for scale invariant problems......Page 84 Examples......Page 87 A comparison of methods......Page 90 Overview......Page 95 Basic theory......Page 96 Examples......Page 97 Lagrangian and action based methods for PDEs......Page 100 Methods for partial differential equations that preserve all symmetries......Page 105 Noether's theorem for discrete schemes......Page 110 Scaling invariance of partial differential equations......Page 113 Spatial adaptivity......Page 115 Singularities in the nonlinear Schrödinger equation (NLS)......Page 116 The nonlinear diffusion equation......Page 119 The Euler equation......Page 123 The Arakawa Jacobian......Page 124 Sine bracket type truncation......Page 125 The semi-geostrophic equations, frontogenesis and prediction ensembles......Page 126 Hamiltonian formulations......Page 129 Links with moving mesh theory......Page 130 Calculating Lyapunov exponents......Page 131 Conclusion......Page 133 Acknowledgements......Page 134 References......Page 135 A few words on linear programming......Page 142 A few words on complexity theory......Page 144 System of linear equations......Page 146 Linear least-squares problem......Page 147 System of linear inequalities (linear conic systems)......Page 148 Linear programming (LP)......Page 149 Semi-definite programming (SDP)......Page 152 Linear programming and complexity theory......Page 154 Finite precision and round-off errors......Page 155 Linear systems......Page 156 Condition-based complexity......Page 158 C(A)......Page 159 C(A)......Page 160 chiA......Page 162 sigma(A)......Page 163 Relations between condition measures for linear programming......Page 164 Interior point algorithms......Page 165 An example of complexity analysis: the Vavasis-Ye method......Page 171 Layered least squares......Page 174 One step of the algorithm......Page 178 Crossover events and LIP's complexity......Page 182 An example of round-off (and complexity) analysis......Page 184 Introduction......Page 191 Proof of Theorem 6.3(i)......Page 192 Proof of Theorem 6.2......Page 194 Proof of Theorem 6.2 for sigma(A)......Page 195 Semidefinite programming algorithms and analyses......Page 196 Potential reduction algorithm......Page 197 Primal-dual algorithm......Page 202 Notes......Page 204 References......Page 205 Introduction......Page 210 Random product homotopy......Page 213 m-homogeneous structure......Page 216 Cheater's homotopy......Page 225 Bernshteín's theorem......Page 229 Mixed volume and mixed subdivisions......Page 235 Polyhedral homotopy method......Page 239 The polyhedral homotopy......Page 240 Solutions of binomial systems in (C*)n......Page 243 Polyhedral homotopy procedure......Page 246 Mixed volume computation......Page 248 A basic linear programming algorithm......Page 250 Finding all lower edges of a lifted point set......Page 252 Extending level-k subfaces......Page 257 Finding all mixed cells......Page 261 Semi-mixed polynomial systems......Page 262 The relation table......Page 265 Level-1 subfaces of S=(S(1),..., S(r))......Page 270 The extension of level-xi subfaces......Page 272 Finding all cells of type (k1,...,kr)......Page 273 Stable mixed volume......Page 274 An alternative algorithm......Page 276 A revision......Page 278 Solutions of positive dimension......Page 281 Fundamental procedure for following paths......Page 286 Projective coordinates and the projective Newton method......Page 289 Balancing the lifting values in polyhedral homotopies......Page 291 The end game......Page 299 Softwares......Page 300 References......Page 301 Further reading......Page 304 Preface......Page 306 Pioneers in chaotic dynamical system......Page 308 From order to chaos......Page 310 Li-Yorke theorem......Page 314 Snap-back repeller......Page 316 Euler's difference scheme......Page 318 Yamaguti-Matano theorem......Page 319 Walrasian general equilibrium theory......Page 322 A mathematical model of an exchange economy......Page 323 Chaotic tatonnement......Page 324 O.D.E. with globally asymptotical stability......Page 327 Three types of O.D.E.'s and chaos with large time steps......Page 329 Three types of O.D.E.'s and chaos with small time step......Page 336 Lipschitz continuity at the equilibrium point......Page 340 Necessary condition for chaos......Page 343 Existence of stable periodic orbits......Page 347 Preliminary study......Page 348 Modified Euler scheme......Page 351 Central difference scheme......Page 356 Eigenvalues and eigenvectors at the fixed points......Page 358 Discretization of a system of ordinary differential equation......Page 359 Discretization of O.D.E. with one equilibrium......Page 363 Introduction......Page 373 A threshold model of collective behavior for a hairstyle fashion......Page 374 Multigrid difference scheme......Page 378 Lebesgue's singular function and the Takagi function......Page 379 References......Page 381 Introduction......Page 384 Geodesic curves and minimal surfaces......Page 386 Energy based active contours.......Page 387 The geodesic curve flow.......Page 389 The level-sets geodesic flow: Derivation.......Page 393 The level-sets geodesic flow: Boundary detection.......Page 394 Existence, uniqueness, stability, and consistency of the geodesic model......Page 397 Experimental results......Page 400 Three-dimensional minimal surfaces......Page 401 Vector-valued edges......Page 403 Color snakes......Page 405 Remark on level-lines of vector valued images......Page 406 Finding the minimal geodesic......Page 407 Computing the minimal geodesic......Page 408 Equal distance contours computation.......Page 409 Affine invariant active contours......Page 410 Affine invariant gradient......Page 411 Affine invariant gradient snakes......Page 414 Additional extensions and modifications......Page 417 Tracking and morphing active contours......Page 419 Gaussian filtering and linear scale-spaces......Page 424 Edge stopping diffusion......Page 426 Perona-Malik formulation......Page 427 Robust estimation.......Page 428 Robust statistics and anisotropic diffusion......Page 431 Exploiting the relationship......Page 432 Robust estimation and line processes......Page 436 Directional diffusion......Page 439 Introducing prior knowledge......Page 440 The general technique......Page 442 Contrast enhancement......Page 445 Histogram modification......Page 447 Existence and uniqueness of the flow.......Page 448 Variational interpretation of the histogram flow.......Page 449 Experimental results......Page 451 Shape preserving contrast enhancement......Page 452 Acknowledgements......Page 454 References......Page 456 Further reading......Page 462 Subject Index......Page 464 v. 1. Finite difference methods (pt. 1). Solutions of equations in Rn (pt. 1) v. 2. Finite element methods (pt. 1) v. 3. Techniques of scientific computing (pt. 1). Numerical methods for solids (pt. 1). Solution of equations in Rn (pt. 2) v. 4. Finite element methods (pt. 2). Numerical methods for solids (pt. 2) v. 6. Numerical methods for solids (pt. 3). Numerical methods for fluids (pt. 1) v. 7. Solution of equations in Rn (pt. 3). Techniques of scientific computing (pt. 3) v. 8. Solution of equations in Rn (pt. 4). Techniques of scientific computing (pt. 4). Numerical methods for fluids (pt. 2) v. 9. Numerical methods for fluids (pt. 3) v. 10. Special volume computational chemistry / guest editor, C. Le Bris v. 11. Special volume foundations of computational mathematics / guest editor, F. Cucker Mathematical Finance is a prolific scientific domain in which there exists a particular characteristic of developing both advanced theories and practical techniques simultaneously. This book addresses the three aspects in the field: mathematical models, computational methods, and applications. A long time ago, when younger and rasher mathematicians, we both momentarily harboured the ambition that one day, older and wiser, we might write a multivolume treatise titled "On the Mathematical Foundations of Numerical Analysis".