Optical flow cannot be computed locally, since only one independent measurement is available from the image sequence at a point, while the flow velocity has two components. A second constraint is needed. A method for finding the optical flow pattern is presented which assumes that the apparent velocity of the brightness pattern varies smoothly almost everywhere in the image. An iterative implementation is shown which successfully computes the optical flow for a number of synthetic image sequences. The algorithm is robust in that it can handle image sequences that are quantized rather coarsely in space and time. It is also insensitive to quantization of brightness levels and additive noise. Examples are included where the assumption of smoothness is violated at singular points or along lines in the image.

An iterative method for computing shape from shading using occluding boundary information is proposed. Some applications of this method are shown. We employ the stereographic plane to express the orientations of surface patches, rather than the more commonly.used gradient space. Use of the stereographic plane makes it possible to incorporate occluding boundary information, but forces us to employ a smoothness constraint different from the one previously proposed. The new constraint follows directly from a particular definition of surface smoothness. We solve the set of equations arising from the smoothness constraints and the image-irradiance equation iteratively, using occluding boundary information to supply boundary conditions. Good initial values are found at certain points to help reduce the number of iterations required to reach a reasonable solution. Numerical experiments show that the method is effective and robust. Finally, we analyze scanning electron microscope (SEM) pictures using this method. Other applications are also proposed. 1.

Abstract. If a matrix or linear operator A is far from normal, its eigenvalues or, more generally, its spectrum may have little to do with its behavior as measured by quantities such as ‖An ‖ or ‖exp(tA)‖. More may be learned by examining the sets in the complex plane known as the pseudospectra of A, defined by level curves of the norm of the resolvent, ‖(zI − A) −1‖. Five years ago, the author published a paper that presented computed pseudospectra of thirteen highly nonnormal matrices arising in various applications. Since that time, analogous computations have been carried out for differential and integral operators. This paper, a companion to the earlier one, presents ten examples, each chosen to illustrate one or more mathematical or physical principles.

This work is directed toward approximating the evolution of forecast error covariances for data assimilation. We study the performance of different algorithms based on simplification of the standard Kalman filter (KF). These are suboptimal schemes (SOS's) when compared to the KF, which is optimal for linear problems with known statistics. The SOS's considered here are several versions of optimal interpolation (OI), a scheme for height error variance advection, and a simplified KF in which the full height error covariance is advected. In order to employ a methodology for exact comparison among these schemes we maintain a linear environment, choosing a beta--plane shallow water model linearized about a constant zonal flow for the testbed dynamics. Our results show that constructing dynamically--balanced forecast error covariances, rather than using conventional geostrophically--balanced ones, is essential for successful performance of any SOS. A posteriori initialization of SOS's to comp...

We introduce the Banach space S 0 # L which has a variety of properties making it a useful tool in Gabor analysis. S 0 can be characterized as the smallest time-frequency homogeneous Banach space of (continuous) functions. We also present other characterizations of S 0 turning it into a very flexible tool for Gabor analysis and allowing for simplifications of various proofs. A careful

. Let fv " (x; t)g "?0 be a family of approximate solutions for the nonlinear scalar conservation law u t + f(u)x = 0 with C 1 0 -initial data. Assume that fv " (x; t)g are Lip + -stable in the sense that they satisfy Oleinik's E-entropy condition. It is shown that if these approximate solutions are Lip 0 -consistent, i.e., if kv " (\Delta; 0) \Gamma u(\Delta; 0)k Lip 0 (x) + kv " t + f(v " )x k Lip 0 (x;t) = O("), then they converge to the entropy solution, and the convergence rate estimate kv " (\Delta; t) \Gamma u(\Delta; t)k Lip 0 (x) = O(") holds. Consequently, the familiar L p -type and new pointwise error estimates are derived. These convergence rate results are demonstrated in the context of entropy satisfying finite-difference and Glimm's schemes. Key Words. Conservation laws, entropy stability, weak consistency, error estimates,post-processing, finite-difference approximations, Glimm scheme AMS(MOS) subject classification. 35L65, 65M10,65M15. 1. Intro...

An implicit method is developed for the numerical solution of option pricing models where it is assumed that the underlying process is a jump diffusion. This method can be applied to a variety of contingent claim valuations, including American options, various kinds of exotic options, and models with uncertain volatility or transaction costs. Proofs of timestepping stability and convergence of a fixed point iteration scheme are presented. For typical model parameters, it is shown that the fixed point iteration reduces the error by two orders of magnitude at each iteration. The correlation integral is computed using a fast Fourier transform (FFT) method. Techniques are developed for avoiding wrap-around effects. Numerical tests of convergence for a variety of options are presented.

Introduction In this chapter we will discuss some practical and technical aspects of numerical methods that can be used to solve the equations that neuronal modelers frequently encounter. We will consider numerical methods for ordinary differential equations (ODEs) and for partial differential equations (PDEs) through examples. A typical case where ODEs arise in neuronal modeling is when one uses a single lumped-soma compartmental model to describe a neuron. Arguably the most famous PDE system in neuronal modeling is the phenomenological model of the squid giant axon due to Hodgkin and Huxley. The difference between ODEs and PDEs is that ODEs are equations in which the rate of change of an unknown function of a single variable is prescribed, usually the derivative with respect to time. In contrast, PDEs involve the rates of change of the solution with respect to two or more independent variables, such as time and space. The numerical methods we will discuss for both ODEs and

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In most models of reacting gas dynamics, the characteristic time scales of chemical reactions are much shorter than the hydrodynamic and di#usive time scales, rendering the reaction part of the model equations sti#. Moreover, nonlinear forcings may introduce into the solutions sharp gradients or shocks, the robust behavior and correct propagation of which require the use of specialized spatial discretization procedures. This study presents high-order conservative methods for the temporal integration of model equations of reacting flows. By means of a method of lines discretization on the flux di#erence form of the equations, these methods compute approximations to the cell-averaged or finite-volume solution. The temporal discretization is based on a multi-implicit generalization of spectral deferred correction methods. The advection term is integrated explicitly, and the di#usion and reaction terms are treated implicitly but independently, with the splitting errors reduced via the spectral deferred correction procedure. To reduce computational cost, di#erent time steps may be used to integrate processes with widely-di#ering time scales. Numerical results show that the conservative nature of the methods allows a robust representation of discontinuities and sharp gradients; the results also demonstrate the expected convergence rates for the methods of orders three, four, and five for smooth problems.