Backward Linear Control Systems on Time Scales

We show how a linear control systems theory for the backward nabla differential operator on an arbitrary time scale can be obtained via Caputo's duality. More precisely, we consider linear control systems with outputs defined with respect to the backward jump operator. Kalman criteria of controllability and observability, as well as realizability conditions, are proved.Comment: Submitted November 11, 2009; Revised March 28, 2010; Accepted April 03, 2010; for publication in the International Journal of Control

Nabla derivatives associated with nonlinear control systems on homogeneous time scales

The backward shift and nabla derivative operators, defined by the control system on homogeneous time scale, in vector spaces of one-forms and vector fields are introduced and some of their properties are proven. In particular the formulas for components of the backward shift and nabla derivative of an arbitrary vector field are presented

Equivalence of finite dimensional input-output models of solute transport and diffusion in geosciences

We show that for a large class of finite dimensional input-output positive systems that represent networks of transport and diffusion of solute in geological media, there exist equivalent multi-rate mass transfer and multiple interacting continua representations, which are quite popular in geo-sciences. Moreover, we provide explicit methods to construct these equivalent representations. The proofs show that controllability property is playing a crucial role for obtaining equivalence. These results contribute to our fundamental understanding on the effect of fine-scale geological structures on the transfer and dispersion of solute, and, eventually, on their interaction with soil microbes and minerals

Algebraic Multigrid for Markov Chains and Tensor Decomposition

The majority of this thesis is concerned with the development of efficient and robust numerical methods based on adaptive algebraic multigrid to compute the stationary distribution of Markov chains. It is shown that classical algebraic multigrid techniques can be applied in an exact interpolation scheme framework to compute the stationary distribution of irreducible, homogeneous Markov chains. A quantitative analysis shows that algebraically smooth multiplicative error is locally constant along strong connections in a scaled system operator, which suggests that classical algebraic multigrid coarsening and interpolation can be applied to the class of nonsymmetric irreducible singular M-matrices with zero column sums. Acceleration schemes based on fine-level iterant recombination, and over-correction of the coarse-grid correction are developed to improve the rate of convergence and scalability of simple adaptive aggregation multigrid methods for Markov chains. Numerical tests over a wide range of challenging nonsymmetric test problems demonstrate the effectiveness of the proposed multilevel method and the acceleration schemes. This thesis also investigates the application of adaptive algebraic multigrid techniques for computing the canonical decomposition of higher-order tensors. The canonical decomposition is formulated as a least squares optimization problem, for which local minimizers are computed by solving the first-order optimality equations. The proposed multilevel method consists of two phases: an adaptive setup phase that uses a multiplicative correction scheme in conjunction with bootstrap algebraic multigrid interpolation to build the necessary operators on each level, and a solve phase that uses additive correction cycles based on the full approximation scheme to efficiently obtain an accurate solution. The alternating least squares method, which is a standard one-level iterative method for computing the canonical decomposition, is used as the relaxation scheme. Numerical tests show that for certain test problems arising from the discretization of high-dimensional partial differential equations on regular lattices the proposed multilevel method significantly outperforms the standard alternating least squares method when a high level of accuracy is required

Structure-Preserving Model Reduction of Physical Network Systems

This paper considers physical network systems where the energy storage is naturally associated to the nodes of the graph, while the edges of the graph correspond to static couplings. The first sections deal with the linear case, covering examples such as mass-damper and hydraulic systems, which have a structure that is similar to symmetric consensus dynamics. The last section is concerned with a specific class of nonlinear physical network systems; namely detailed-balanced chemical reaction networks governed by mass action kinetics. In both cases, linear and nonlinear, the structure of the dynamics is similar, and is based on a weighted Laplacian matrix, together with an energy function capturing the energy storage at the nodes. We discuss two methods for structure-preserving model reduction. The first one is clustering; aggregating the nodes of the underlying graph to obtain a reduced graph. The second approach is based on neglecting the energy storage at some of the nodes, and subsequently eliminating those nodes (called Kron reduction).</p

Application of non-linear system identification approaches to modelling, analysis, and control of fluid flows.

Flow control has become a topic of great importance for several applications, ranging from commercial aircraft, to intercontinental pipes and skyscrapers. In these applications, and many more, the interaction with a fluid flow can have a significant influence on the performance of the system. In many cases the fluids encountered are turbulent and detrimental to the latter. Several attempts have been made to solve this problem. However, due to the non-linearity and infinite dimensionality of fluid flows and their governing equations, a complete understanding of turbulent behaviour and a feasible control approach has not been obtained. In this thesis, model reduction approaches that exploit non-linear system identification are applied using data obtained from numerical simulations of turbulent three-dimensional channel flow, and two-dimensional flow over the backward facing step. A multiple-input multiple-output model, consisting of 27 sub-structures, is obtained for the fluctuations of the velocity components of the channel flow. A single-input single-output model for fluctuations of the pressure coefficient, and two multiple-input single-output models for fluctuations of the velocity magnitude are obtained in flow over the BFS. A non-linear model predictive control strategy is designed using identified one- and multi-step ahead predictors, with the inclusion of integral action for robustness. The proposed control approach incorporates a non-linear model without the need for expensive non-linear optimizations. Finally, a frequency domain analysis of unmanipulated turbulent flow is perfumed using five systems. Higher order generalized frequency response functions (GFRF) are computed to study the non-linear energy transfer phenomena. A more detailed investigation is performed using the output FRF (OFRF), which can elucidate the contribution of the n-th order frequency response to the output frequency response

Parameter and state model reduction for Bayesian statistical inverse problems

Thesis (S.M.)--Massachusetts Institute of Technology, Computation for Design and Optimization Program, 2009.This electronic version was submitted by the student author. The certified thesis is available in the Institute Archives and Special Collections.Cataloged from student submitted PDF version of thesis.Includes bibliographical references (p. 113-118).Decisions based on single-point estimates of uncertain parameters neglect regions of significant probability. We consider a paradigm based on decision-making under uncertainty including three steps: identification of parametric probability by solution of the statistical inverse problem, propagation of that uncertainty through complex models, and solution of the resulting stochastic or robust mathematical programs. In this thesis we consider the first of these steps, solution of the statistical inverse problem, for partial differential equations (PDEs) parameterized by field quantities. When these field variables and forward models are discretized, the resulting system is high-dimensional in both parameter and state space. The system is therefore expensive to solve. The statistical inverse problem is one of Bayesian inference. With assumption on prior belief about the form of the parameter and an assignment of normal error in sensor measurements, we derive the solution to the statistical inverse problem analytically, up to a constant of proportionality. The parametric probability density, or posterior, depends implicitly on the parameter through the forward model. In order to understand the distribution in parameter space, we must sample. Markov chain Monte Carlo (MCMC) sampling provides a method by which a random walk is constructed through parameter space. By following a few simple rules, the random walk converges to the posterior distribution and the resulting samples represent draws from that distribution. This set of samples from the posterior can be used to approximate its moments.(cont.) In the multi-query setting, it is computationally intractable to utilize the full-order forward model to perform the posterior evaluations required in the MCMC sampling process. Instead, we implement a novel reduced-order model which reduces in parameter and state. The reduced bases are generated by greedy sampling. We iteratively sample the field in parameter space which maximizes the error in full-order and current reduced-order model outputs. The parameter is added to its basis and then a high-fidelity forward model is solved for the state, which is then added to the state basis. The reduction in state accelerates posterior evaluation while the reduction in parameter allows the MCMC sampling to be conducted with a simpler, non-adaptive 3 Metropolis-Hastings algorithm. In contrast, the full-order parameter space is high-dimensional and requires more expensive adaptive methods. We demonstrate for the groundwater inverse problem in 1-D and 2-D that the reduced-order implementation produces accurate results with a factor of three speed up even for the model problems of dimension N ~~500. Our complexity analysis demonstrates that the same approach applied to the large-scale models of interest (e.g. N > 10⁴) results in a speed up of three orders of magnitude.by Chad Eric Lieberman.S.M

Q(sqrt(-3))-Integral Points on a Mordell Curve

We use an extension of quadratic Chabauty to number fields,recently developed by the author with Balakrishnan, Besser and M ̈uller,combined with a sieving technique, to determine the integral points overQ(√−3) on the Mordell curve y2 = x3 − 4