Certified Rapid Solution of Parametrized Linear Elliptic Equations: Application to Parameter Estimation

We present a technique for the rapid and reliable evaluation of linear-functional output of elliptic partial differential equations with affine parameter dependence. The essential components are (i) rapidly uniformly convergent reduced-basis approximations — Galerkin projection onto a space WN spanned by solutions of the governing partial differential equation at N (optimally) selected points in parameter space; (ii) a posteriori error estimation — relaxations of the residual equation that provide inexpensive yet sharp and rigorous bounds for the error in the outputs; and (iii) offline/online computational procedures — stratagems that exploit affine parameter dependence to de-couple the generation and projection stages of the approximation process. The operation count for the online stage — in which, given a new parameter value, we calculate the output and associated error bound — depends only on N (typically small) and the parametric complexity of the problem. The method is thus ideally suited to the many-query and real-time contexts. In this paper, based on the technique we develop a robust inverse computational method for very fast solution of inverse problems characterized by parametrized partial differential equations. The essential ideas are in three-fold: first, we apply the technique to the forward problem for the rapid certified evaluation of PDE input-output relations and associated rigorous error bounds; second, we incorporate the reduced-basis approximation and error bounds into the inverse problem formulation; and third, rather than regularize the goodness-of-fit objective, we may instead identify all (or almost all, in the probabilistic sense) system configurations consistent with the available experimental data — well-posedness is reflected in a bounded "possibility region" that furthermore shrinks as the experimental error is decreased.

On Degeneracy of Linear Reconstruction from Three Views: Linear Line Complex and Applications

This paper investigates the linear degeneracies of projective structure estimation from point and line features across three views. We show that the rank of the linear system of equations for recovering the trilinear tensor of three views reduces to 23 (instead of 26) in the case when the scene is a Linear Line Complex (set of lines in space intersecting at a common line) and is 21 when the scene is planar. The LLC situation is only linearly degenerate, and we show that one can obtain a unique solution when the admissibility constraints of the tensor are accounted for. The line configuration described by an LLC, rather than being some obscure case, is in fact quite typical. It includes, as a particular example, the case of a camera moving down a hallway in an office environment or down an urban street. Furthermore, an LLC situation may occur as an artifact such as in direct estimation from spatio-temporal derivatives of image brightness. Therefore, an investigation into degeneracies and their remedy is important also in practice.

Model-Based Matching by Linear Combinations of Prototypes

We describe a method for modeling object classes (such as faces) using 2D example images and an algorithm for matching a model to a novel image. The object class models are "learned'' from example images that we call prototypes. In addition to the images, the pixelwise correspondences between a reference prototype and each of the other prototypes must also be provided. Thus a model consists of a linear combination of prototypical shapes and textures. A stochastic gradient descent algorithm is used to match a model to a novel image by minimizing the error between the model and the novel image. Example models are shown as well as example matches to novel images. The robustness of the matching algorithm is also evaluated. The technique can be used for a number of applications including the computation of correspondence between novel images of a certain known class, object recognition, image synthesis and image compression.

Model-Based Matching of Line Drawings by Linear Combinations of Prototypes

We describe a technique for finding pixelwise correspondences between two images by using models of objects of the same class to guide the search. The object models are 'learned' from example images (also called prototypes) of an object class. The models consist of a linear combination ofsprototypes. The flow fields giving pixelwise correspondences between a base prototype and each of the other prototypes must be given. A novel image of an object of the same class is matched to a model by minimizing an error between the novel image and the current guess for the closest modelsimage. Currently, the algorithm applies to line drawings of objects. An extension to real grey level images is discussed.

Implementing Distributed Systems Using Linear Naming

Linear graph reduction is a simple computational model in which the cost of naming things is explicitly represented. The key idea is the notion of "linearity". A name is linear if it is only used once, so with linear naming you cannot create more than one outstanding reference to an entity. As a result, linear naming is cheap to support and easy to reason about. Programs can be translated into the linear graph reduction model such that linear names in the program are implemented directly as linear names in the model. Nonlinear names are supported by constructing them out of linear names. The translation thus exposes those places where the program uses names in expensive, nonlinear ways. Two applications demonstrate the utility of using linear graph reduction: First, in the area of distributed computing, linear naming makes it easy to support cheap cross-network references and highly portable data structures, Linear naming also facilitates demand driven migration of tasks and data around the network without requiring explicit guidance from the programmer. Second, linear graph reduction reveals a new characterization of the phenomenon of state. Systems in which state appears are those which depend on certain -global- system properties. State is not a localizable phenomenon, which suggests that our usual object oriented metaphor for state is flawed.

Exact Solution of the Nonlinear Dynamics of Recurrent Neural Mechanisms for Direction Selectivity

Different theoretical models have tried to investigate the feasibility of recurrent neural mechanisms for achieving direction selectivity in the visual cortex. The mathematical analysis of such models has been restricted so far to the case of purely linear networks. We present an exact analytical solution of the nonlinear dynamics of a class of direction selective recurrent neural models with threshold nonlinearity. Our mathematical analysis shows that such networks have form-stable stimulus-locked traveling pulse solutions that are appropriate for modeling the responses of direction selective cortical neurons. Our analysis shows also that the stability of such solutions can break down giving raise to a different class of solutions ("lurching activity waves") that are characterized by a specific spatio-temporal periodicity. These solutions cannot arise in models for direction selectivity with purely linear spatio-temporal filtering.

Thermal Noise Behavior of the Bridge Circuit

This paper considers a connection between the deterministic and noisy behavior of nonlinear networks. Specifically, a particular bridge circuit is examined which has two possibly nonlinear energy storage elements. By proper choice of the constitutive relations for the network elements, the deterministic terminal behavior reduces to that of a single linear resistor. This reduction of the deterministic terminal behavior, in which a natural frequency of a linear circuit does not appear in the driving-point impedance, has been shown in classical circuit theory books (e.g. [1, 2]). The paper shows that, in addition to the reduction of the deterministic behavior, the thermal noise at the terminals of the network, arising from the usual Nyquist-Johnson noise model associated with each resistor in the network, is also exactly that of a single linear resistor. While this result for the linear time-invariant (LTI) case is a direct consequence of a well-known result for RLC circuits, the nonlinear result is novel. We show that the terminal noise current is precisely that predicted by the Nyquist-Johnson model for R if the driving voltage is zero or constant, but not if the driving voltage is time-dependent or the inductor and capacitor are time-varying