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.

Multivariate Density Estimation: An SVM Approach

We formulate density estimation as an inverse operator problem. We then use convergence results of empirical distribution functions to true distribution functions to develop an algorithm for multivariate density estimation. The algorithm is based upon a Support Vector Machine (SVM) approach to solving inverse operator problems. The algorithm is implemented and tested on simulated data from different distributions and different dimensionalities, gaussians and laplacians in $R^2$ and $R^{12}$. A comparison in performance is made with Gaussian Mixture Models (GMMs). Our algorithm does as well or better than the GMMs for the simulations tested and has the added advantage of being automated with respect to parameters.

Trajectory and Force Control of a Direct Drive Arm

Using the MIT Serial Link Direct Drive Arm as the main experimental device, various issues in trajectory and force control of manipulators were studied in this thesis. Since accurate modeling is important for any controller, issues of estimating the dynamic model of a manipulator and its load were addressed first. Practical and effective algorithms were developed fro the Newton-Euler equations to estimate the inertial parameters of manipulator rigid-body loads and links. Load estimation was implemented both on PUMA 600 robot and on the MIT Serial Link Direct Drive Arm. With the link estimation algorithm, the inertial parameters of the direct drive arm were obtained. For both load and link estimation results, the estimated parameters are good models of the actual system for control purposes since torques and forces can be predicted accurately from these estimated parameters. The estimated model of the direct drive arm was them used to evaluate trajectory following performance by feedforward and computed torque control algorithms. The experimental evaluations showed that the dynamic compensation can greatly improve trajectory following accuracy. Various stability issues of force control were studied next. It was determined that there are two types of instability in force control. Dynamic instability, present in all of the previous force control algorithms discussed in this thesis, is caused by the interaction of a manipulator with a stiff environment. Kinematics instability is present only in the hybrid control algorithm of Raibert and Craig, and is caused by the interaction of the inertia matrix with the Jacobian inverse coordinate transformation in the feedback path. Several methods were suggested and demonstrated experimentally to solve these stability problems. The result of the stability analyses were then incorporated in implementing a stable force/position controller on the direct drive arm by the modified resolved acceleration method using both joint torque and wrist force sensor feedbacks.

Inverse Metabolic Engineering of Synechocystis PCC 6803 for Improved Growth Rate and Poly-3-hydroxybutyrate Production

Synechocystis PCC 6803 is a photosynthetic bacterium that has the potential to make bioproducts from carbon dioxide and light. Biochemical production from photosynthetic organisms is attractive because it replaces the typical bioprocessing steps of crop growth, milling, and fermentation, with a one-step photosynthetic process. However, low yields and slow growth rates limit the economic potential of such endeavors. Rational metabolic engineering methods are hindered by limited cellular knowledge and inadequate models of Synechocystis. Instead, inverse metabolic engineering, a scheme based on combinatorial gene searches which does not require detailed cellular models, but can exploit sequence data and existing molecular biological techniques, was used to find genes that (1) improve the production of the biopolymer poly-3-hydroxybutyrate (PHB) and (2) increase the growth rate. A fluorescence activated cell sorting assay was developed to screen for high PHB producing clones. Separately, serial sub-culturing was used to select clones that improve growth rate. Novel gene knock-outs were identified that increase PHB production and others that increase the specific growth rate. These improvements make this system more attractive for industrial use and demonstrate the power of inverse metabolic engineering to identify novel phenotype-associated genes in poorly understood systems.

Behavioral Measures and their Correlation with IPM Iteration Counts on Semi-Definite Programming Problems

We study four measures of problem instance behavior that might account for the observed differences in interior-point method (IPM) iterations when these methods are used to solve semidefinite programming (SDP) problem instances: (i) an aggregate geometry measure related to the primal and dual feasible regions (aspect ratios) and norms of the optimal solutions, (ii) the (Renegar-) condition measure C(d) of the data instance, (iii) a measure of the near-absence of strict complementarity of the optimal solution, and (iv) the level of degeneracy of the optimal solution. We compute these measures for the SDPLIB suite problem instances and measure the correlation between these measures and IPM iteration counts (solved using the software SDPT3) when the measures have finite values. Our conclusions are roughly as follows: the aggregate geometry measure is highly correlated with IPM iterations (CORR = 0.896), and is a very good predictor of IPM iterations, particularly for problem instances with solutions of small norm and aspect ratio. The condition measure C(d) is also correlated with IPM iterations, but less so than the aggregate geometry measure (CORR = 0.630). The near-absence of strict complementarity is weakly correlated with IPM iterations (CORR = 0.423). The level of degeneracy of the optimal solution is essentially uncorrelated with IPM iterations.