Tim Rees

Bio: Tim Rees is an academic researcher from University of Waterloo. The author has contributed to research in topics: Interior point method & Eigenvalues and eigenvectors. The author has an hindex of 1, co-authored 1 publications receiving 49 citations.

A Preconditioner for Linear Systems Arising From Interior Point Optimization Methods

Abstract: We explore a preconditioning technique applied to the problem of solving linear systems arising from primal-dual interior point algorithms in linear and quadratic programming. The preconditioner has the attractive property of improved eigenvalue clustering with increased ill-conditioning of the (1,1) block of the saddle point matrix. It fits well into the optimization framework since the interior point iterates yield increasingly ill-conditioned linear systems as the solution is approached. We analyze the spectral characteristics of the preconditioner, utilizing projections onto the null space of the constraint matrix, and demonstrate performance on problems from the NETLIB and CUTEr test suites. The numerical experiments include results based on inexact inner iterations.

Regularization-robust preconditioners for time-dependent pde-constrained optimization problems ∗

Abstract: In this article, we motivate, derive, and test effective preconditioners to be used with the Minres algorithm for solving a number of saddle point systems which arise in PDE-constrained optimization problems. We consider the distributed control problem involving the heat equation and the Neumann boundary control problem involving Poisson's equation and the heat equation. Crucial to the effectiveness of our preconditioners in each case is an effective approximation of the Schur complement of the matrix system. In each case, we state the problem being solved, propose the preconditioning approach, prove relevant eigenvalue bounds, and provide numerical results which demonstrate that our solvers are effective for a wide range of regularization parameter values, as well as mesh sizes and time-steps.

Practical Compressed Sensing: Modern data acquisition and signal processing

Abstract: Since 2004, the field of compressed sensing has grown quickly and seen tremendous interest because it provides a theoretically sound and computationally tractable method to stably recover signals by sampling at the information rate This thesis presents in detail the design of one of the world's first compressed sensing hardware devices, the random modulation pre-integrator (RMPI) The RMPI is an analog-to-digital converter (ADC) that bypasses a current limitation in ADC technology and achieves an unprecedented 8 effective number of bits over a bandwidth of 25 GHz Subtle but important design considerations are discussed, and state-of-the-art reconstruction techniques are presented Inspired by the need for a fast method to solve reconstruction problems for the RMPI, we develop two efficient large-scale optimization methods, NESTA and TFOCS, that are applicable to a wide range of other problems, such as image denoising and deblurring, MRI reconstruction, and matrix completion (including the famous Netflix problem) While many algorithms solve unconstrained l1 problems, NESTA and TFOCS can solve the constrained form of l1 minimization, and allow weighted norms In addition to l1 minimization problems such as the LASSO, both NESTA and TFOCS solve total-variation minimization problem TFOCS also solves the Dantzig selector and most variants of the nuclear norm minimization problem A common theme in both NESTA and TFOCS is the use of smoothing techniques, which make the problem tractable, and the use of optimal first-order methods that have an accelerated convergence rate yet have the same cost per iteration as gradient descent The conic dual methodology is introduced in TFOCS and proves to be extremely flexible, covering such generic problems as linear programming, quadratic programming, and semi-definite programming A novel continuation scheme is presented, and it is shown that the Dantzig selector benefits from an exact-penalty property Both NESTA and TFOCS are released as software packages available freely for academic use

On simulations of discrete fracture network flows with an optimization-based extended finite element method

Abstract: Following the approach introduced in [SIAM J Sci Comput, 35 (2013), pp B487--B510], we consider the formulation of the problem of fluid flow in a system of fractures as a PDE constrained optimization problem, with discretization performed using suitable extended finite elements; the method allows independent meshes on each fracture, thus completely circumventing meshing problems usually related to the discrete fracture network (DFN) approach The application of the method to DFNs of medium complexity is fully analyzed here, accounting for several issues related to viable and reliable implementations of the method in complex problems

A splitting preconditioner for saddle point problems

Abstract: For large sparse systems of linear equations iterative techniques are attractive. In this paper, we study a splitting method for an important class of symmetric and indefinite system. Theoretical analyses show that this method converges to the unique solution of the system of linear equations for all t>0 (t is the parameter). Moreover, all the eigenvalues of the iteration matrix are real and nonnegative and the spectral radius of the iteration matrix is decreasing with respect to the parameter t. Besides, a preconditioning strategy based on the splitting of the symmetric and indefinite coefficient matrices is proposed. The eigensolution of the preconditioned matrix is described and an upper bound of the degree of the minimal polynomials for the preconditioned matrix is obtained. Numerical experiments of a model Stokes problem and a least-squares problem with linear constraints presented to illustrate the effectiveness of the method. Copyright © 2011 John Wiley & Sons, Ltd.