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.

Using Recurrent Networks for Dimensionality Reduction

This report explores how recurrent neural networks can be exploited for learning high-dimensional mappings. Since recurrent networks are as powerful as Turing machines, an interesting question is how recurrent networks can be used to simplify the problem of learning from examples. The main problem with learning high-dimensional functions is the curse of dimensionality which roughly states that the number of examples needed to learn a function increases exponentially with input dimension. This thesis proposes a way of avoiding this problem by using a recurrent network to decompose a high-dimensional function into many lower dimensional functions connected in a feedback loop.

Perceptually-based Comparison of Image Similarity Metrics

The image comparison operation ??sessing how well one image matches another ??rms a critical component of many image analysis systems and models of human visual processing. Two norms used commonly for this purpose are L1 and L2, which are specific instances of the Minkowski metric. However, there is often not a principled reason for selecting one norm over the other. One way to address this problem is by examining whether one metric better captures the perceptual notion of image similarity than the other. With this goal, we examined perceptual preferences for images retrieved on the basis of the L1 versus the L2 norm. These images were either small fragments without recognizable content, or larger patterns with recognizable content created via vector quantization. In both conditions the subjects showed a consistent preference for images matched using the L1 metric. These results suggest that, in the domain of natural images of the kind we have used, the L1 metric may better capture human notions of image similarity.

Natural Language Input for a Computer Problem Solving System

The STUDENT problem solving system, programmed in LISP, accepts as input a comfortable but restricted subset of English which can express a wide variety of algebra story problems. STUDENT finds the solution to a large class of these problems. STUDENT can utilize a store of global information not specific to any one problem, and may make assumptions about the interpretation of ambiguities in the wording of the problem being solved. If it uses such information or makes any assumptions, STUDENT communicates this fact to the user. The thesis includes a summary of other English language questions-answering systems. All these systems, and STUDENT, are evaluated according to four standard criteria. The linguistic analysis in STUDENT is a first approximation to the analytic portion of a semantic theory of discourse outlined in the thesis. STUDENT finds the set of kernel sentences which are the base of the input discourse, and transforms this sequence of kernel sentences into a set of simultaneous equations which form the semantic base of the STUDENT system. STUDENT then tries to solve this set of equations for the values of requested unknowns. If it is successful it gives the answers in English. If not, STUDENT asks the user for more information, and indicates the nature of the desired information. The STUDENT system is a first step toward natural language communication with computers. Further work on the semantic theory proposed should result in much more sophisticated systems.

A Formulation for Active Learning with Applications to Object Detection

We discuss a formulation for active example selection for function learning problems. This formulation is obtained by adapting Fedorov's optimal experiment design to the learning problem. We specifically show how to analytically derive example selection algorithms for certain well defined function classes. We then explore the behavior and sample complexity of such active learning algorithms. Finally, we view object detection as a special case of function learning and show how our formulation reduces to a useful heuristic to choose examples to reduce the generalization error.