IITG Mathematics Seminar Series

Abstract:

Abstract: Motivated by Riemann's $R_1$ summation method for independent and identically distributed (i.i.d.)random variables $ X_1, X_2, \cdots,$ in this talk we discuss random Fourier series of the form $\sum\limits_1^{\infty} a_nX_n\sin(nt+\Phi _n)$ where $\{a_n\}$ is sequence of constants and $\{\Phi_n\}$ is a sequence of independent random variables which are independent of $\{X_n\}$.

Abstract: Canonical commutation relation" in quantum mechanics the describes the non-commutativity between the position "x" and momentum "px" (by notation) in the "x" direction of a point particle in one dimension "R" (group of real numbers). In this talk we shall follow the natural generalization from "R" to any locally compact group "G". Finally we shall discuss it's further generalization to locally compact quantum groups.

Abstract: Science and engineering problems frequently require solving a sequence of single linear systems (A_kx_k = b_k, k = 2:N) or a sequence of dual linear systems (A_k x_k = b_k and A H_k y_k = c_k, k = 2:N). I have developed algorithms that recycle Krylov subspaces and preconditioners from one system (or pair of systems) in the sequence to the next, leading to efficient solutions. Besides the benefit of only having to store few Lanczos vectors, using BiConjugate Gradients (BiCG) to solve dual linear systems may have application specific advantages. For example, using BiCG to solve the dual linear systems arising in interpolatory model reduction provides a backward error formulation in the model reduction framework. In this talk I will introduce recycling BiCG, a BiCG method that recycles two Krylov subspaces from one pair of dual linear systems to the next pair. For a model reduction problem, I will show that by using recycling BiCG one can save up to 70% in iterations, and that solving the problem without recycling leads to (about) a 50% increase in runtime. Framework for new recycling solvers for sequences of single linear systems will be described as well. I will also introduce an algorithm that recycles preconditioners, leading to a dramatic reduction in the cost of variational Monte Carlo (VMC) for large(r) systems. The main cost of the VMC method is in constructing a sequence of so called Slater matrices and computing the ratios of determinants for successive Slater matrices. Recent work has improved the scaling of constructing Slater matrices for insulators, so that the cost of constructing Slater matrices in these systems is now linear. However, the cost of computing determinant ratios remains cubic. With the long term aim of simulating much larger systems, I will demonstrate improvement in the scaling of computing determinant ratios in the VMC method for simulating insulators by using preconditioned iterative solvers. I will show that by using the new algorithm, the scaling of the VMC algorithm is reduced from O(n^3) to roughly O(n^2), where n is the system size.

Acknowledgments: Eric de Sturler (Virginia Tech), David Ceperley (Univ. of Illinois at Urbana-Champaign), Serkan Gugercin (Virginia Tech), and Bryan Clark (Princeton).

Abstract: This talk deals with quadratic inverse eigenvalue problems that arise in mechanical vibration and structural dynamics. The problems arise in controlling dangerous vibrations in mechanical structures, such as buildings, bridges, highways, automobiles, air and space crafts, and others. The problems are to find two feedback matrices such that a small amount of the eigenvalues of the associated quadratic matrix pencil are reassigned to suitably chosen ones while keeping the remaining large number of eigenvalues and the associated eigenvectors unchanged. For robust and economic control design, these feedback matrices must be found in such a way that (i) they have norms as small as possible, and (ii) the condition number of the modified quadratic inverse problem is minimized. These considerations give rise to two nonlinear unconstrained optimization problems, known respectively, as the Robust Quadratic Partial Eigenvalue Assignment Problem and Minimum Norm Quadratic Partial Eigenvalue Assignment Problem. We will give an overview of the recent developments on computational methods for these difficult nonlinear optimization problems and discuss directions of future research. The talk is interdisciplinary in nature and will be of interests to mathematicians, computational and applied mathematicians, and control and vibration engineers.