On the Computational Cost and Complexity of Stochastic Inverse Solvers

The goal of this paper is to provide a starting point for investigations into a mainly underdeveloped area of research regarding the computational cost analysis of complex stochastic strategies for solving parametric inverse problems. This area has two main components: solving global optimization problems and solving forward problems (to evaluate the misfit function that we try to minimize). For the first component, we pay particular attention to genetic algorithms with heuristics and to multi-deme algorithms that can be modeled as ergodic Markov chains. We recall a simple method for evaluating the first hitting time for the single-deme algorithm and we extend it to the case of HGS, a multi-deme hierarchic strategy. We focus on the case in which at least the demes in the leaves are well tuned. Finally, we also express the problems of finding local and global optima in terms of a classic complexity theory. We formulate the natural result that finding a local optimum of a function is an NP-complete task, and we argue that finding a global optimum is a much harder, DP-complete, task. Furthermore, we argue that finding all global optima is, possibly, even harder (#P-hard) task. Regarding the second component of solving parametric inverse problems (i.e., regarding the forward problem solvers), we discuss the computational cost of hp-adaptive Finite Element solvers and their rates of convergence with respect to the increasing number of degrees of freedom. The presented results provide a useful taxonomy of problems and methods of studying the computational cost and complexity of various strategies for solving inverse parametric problems. Yet, we stress that our goal was not to deliver detailed evaluations for particular algorithms applied to particular inverse problems, but rather to try to identify possible ways of obtaining such results

Computer Science Research Institute 2005 annual report of activities.

This report summarizes the activities of the Computer Science Research Institute (CSRI) at Sandia National Laboratories during the period January 1, 2005 to December 31, 2005. During this period, the CSRI hosted 182 visitors representing 83 universities, companies and laboratories. Of these, 60 were summer students or faculty. The CSRI partially sponsored 2 workshops and also organized and was the primary host for 3 workshops. These 3 CSRI sponsored workshops had 105 participants, 78 from universities, companies and laboratories, and 27 from Sandia. Finally, the CSRI sponsored 12 long-term collaborative research projects and 3 Sabbaticals

Sensitivity technologies for large scale simulation.

Sensitivity analysis is critically important to numerous analysis algorithms, including large scale optimization, uncertainty quantification,reduced order modeling, and error estimation. Our research focused on developing tools, algorithms and standard interfaces to facilitate the implementation of sensitivity type analysis into existing code and equally important, the work was focused on ways to increase the visibility of sensitivity analysis. We attempt to accomplish the first objective through the development of hybrid automatic differentiation tools, standard linear algebra interfaces for numerical algorithms, time domain decomposition algorithms and two level Newton methods. We attempt to accomplish the second goal by presenting the results of several case studies in which direct sensitivities and adjoint methods have been effectively applied, in addition to an investigation of h-p adaptivity using adjoint based a posteriori error estimation. A mathematical overview is provided of direct sensitivities and adjoint methods for both steady state and transient simulations. Two case studies are presented to demonstrate the utility of these methods. A direct sensitivity method is implemented to solve a source inversion problem for steady state internal flows subject to convection diffusion. Real time performance is achieved using novel decomposition into offline and online calculations. Adjoint methods are used to reconstruct initial conditions of a contamination event in an external flow. We demonstrate an adjoint based transient solution. In addition, we investigated time domain decomposition algorithms in an attempt to improve the efficiency of transient simulations. Because derivative calculations are at the root of sensitivity calculations, we have developed hybrid automatic differentiation methods and implemented this approach for shape optimization for gas dynamics using the Euler equations. The hybrid automatic differentiation method was applied to a first order approximation of the Euler equations and used as a preconditioner. In comparison to other methods, the AD preconditioner showed better convergence behavior. Our ultimate target is to perform shape optimization and hp adaptivity using adjoint formulations in the Premo compressible fluid flow simulator. A mathematical formulation for mixed-level simulation algorithms has been developed where different physics interact at potentially different spatial resolutions in a single domain. To minimize the implementation effort, explicit solution methods can be considered, however, implicit methods are preferred if computational efficiency is of high priority. We present the use of a partial elimination nonlinear solver technique to solve these mixed level problems and show how these formulation are closely coupled to intrusive optimization approaches and sensitivity analyses. Production codes are typically not designed for sensitivity analysis or large scale optimization. The implementation of our optimization libraries into multiple production simulation codes in which each code has their own linear algebra interface becomes an intractable problem. In an attempt to streamline this task, we have developed a standard interface between the numerical algorithm (such as optimization) and the underlying linear algebra. These interfaces (TSFCore and TSFCoreNonlin) have been adopted by the Trilinos framework and the goal is to promote the use of these interfaces especially with new developments. Finally, an adjoint based a posteriori error estimator has been developed for discontinuous Galerkin discretization of Poisson's equation. The goal is to investigate other ways to leverage the adjoint calculations and we show how the convergence of the forward problem can be improved by adapting the grid using adjoint-based error estimates. Error estimation is usually conducted with continuous adjoints but if discrete adjoints are available it may be possible to reuse the discrete version for error estimation. We investigate the advantages and disadvantages of continuous and discrete adjoints through a simple example

SciCADE 95: International conference on scientific computation and differential equations

This report consists of abstracts from the conference. Topics include algorithms, computer codes, and numerical solutions for differential equations. Linear and nonlinear as well as boundary-value and initial-value problems are covered. Various applications of these problems are also included

Proceedings, MSVSCC 2015

The Virginia Modeling, Analysis and Simulation Center (VMASC) of Old Dominion University hosted the 2015 Modeling, Simulation, & Visualization Student capstone Conference on April 16th. The Capstone Conference features students in Modeling and Simulation, undergraduates and graduate degree programs, and fields from many colleges and/or universities. Students present their research to an audience of fellow students, faculty, judges, and other distinguished guests. For the students, these presentations afford them the opportunity to impart their innovative research to members of the M&S community from academic, industry, and government backgrounds. Also participating in the conference are faculty and judges who have volunteered their time to impart direct support to their students’ research, facilitate the various conference tracks, serve as judges for each of the tracks, and provide overall assistance to this conference. 2015 marks the ninth year of the VMASC Capstone Conference for Modeling, Simulation and Visualization. This year our conference attracted a number of fine student written papers and presentations, resulting in a total of 51 research works that were presented. This year’s conference had record attendance thanks to the support from the various different departments at Old Dominion University, other local Universities, and the United States Military Academy, at West Point. We greatly appreciated all of the work and energy that has gone into this year’s conference, it truly was a highly collaborative effort that has resulted in a very successful symposium for the M&S community and all of those involved. Below you will find a brief summary of the best papers and best presentations with some simple statistics of the overall conference contribution. Followed by that is a table of contents that breaks down by conference track category with a copy of each included body of work. Thank you again for your time and your contribution as this conference is designed to continuously evolve and adapt to better suit the authors and M&S supporters. Dr.Yuzhong Shen Graduate Program Director, MSVE Capstone Conference Chair John ShullGraduate Student, MSVE Capstone Conference Student Chai

Generalized averaged Gaussian quadrature and applications

A simple numerical method for constructing the optimal generalized averaged Gaussian quadrature formulas will be presented. These formulas exist in many cases in which real positive GaussKronrod formulas do not exist, and can be used as an adequate alternative in order to estimate the error of a Gaussian rule. We also investigate the conditions under which the optimal averaged Gaussian quadrature formulas and their truncated variants are internal

MS FT-2-2 7 Orthogonal polynomials and quadrature: Theory, computation, and applications

Quadrature rules find many applications in science and engineering. Their analysis is a classical area of applied mathematics and continues to attract considerable attention. This seminar brings together speakers with expertise in a large variety of quadrature rules. It is the aim of the seminar to provide an overview of recent developments in the analysis of quadrature rules. The computation of error estimates and novel applications also are described