SPATIAL UNDERSTANDING OF MATRIX INVERSION FOR INVERSE FORCE ESTIMATION

A basic approach to study a NVH problem is to break down the system in three basic elements – source, path and receiver. While the receiver (response) and the transfer path can be measured, it is difficult to measure the source (forces) acting on the system. It becomes necessary to predict these forces to know how they influence the responses. This requires inverting the transfer path. Singular Value Decomposition (SVD) method is used to decompose the transfer path matrix into its principle components which is required for the inversion. The usual approach to force prediction requires rejecting the small singular values obtained during SVD by setting a threshold, as these small values dominate the inverse matrix. This assumption of the threshold may be subjected to rejecting important singular values severely affecting force prediction. The new approach discussed in this report looks at the column space of the transfer path matrix which is the basis for the predicted response. The response participation is an indication of how the small singular values influence the force participation. The ability to accurately reconstruct the response vector is important to establish a confidence in force vector prediction. The goal of this report is to suggest a solution that is mathematically feasible, physically meaningful, and numerically more efficient through examples. This understanding adds new insight to the effects of current code and how to apply algorithms and understanding to new codes.

Sparse matrix inversion,

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Parallel Hybrid Monte Carlo Algorithms for Matrix Computations

In this paper we consider hybrid (fast stochastic approximation and deterministic refinement) algorithms for Matrix Inversion (MI) and Solving Systems of Linear Equations (SLAE). Monte Carlo methods are used for the stochastic approximation, since it is known that they are very efficient in finding a quick rough approximation of the element or a row of the inverse matrix or finding a component of the solution vector. We show how the stochastic approximation of the MI can be combined with a deterministic refinement procedure to obtain MI with the required precision and further solve the SLAE using MI. We employ a splitting A = D – C of a given non-singular matrix A, where D is a diagonal dominant matrix and matrix C is a diagonal matrix. In our algorithm for solving SLAE and MI different choices of D can be considered in order to control the norm of matrix T = D –1C, of the resulting SLAE and to minimize the number of the Markov Chains required to reach given precision. Further we run the algorithms on a mini-Grid and investigate their efficiency depending on the granularity. Corresponding experimental results are presented.

A sparse parallel hybrid Monte Carlo algorithm for matrix computations

In this paper we introduce a new algorithm, based on the successful work of Fathi and Alexandrov, on hybrid Monte Carlo algorithms for matrix inversion and solving systems of linear algebraic equations. This algorithm consists of two parts, approximate inversion by Monte Carlo and iterative refinement using a deterministic method. Here we present a parallel hybrid Monte Carlo algorithm, which uses Monte Carlo to generate an approximate inverse and that improves the accuracy of the inverse with an iterative refinement. The new algorithm is applied efficiently to sparse non-singular matrices. When we are solving a system of linear algebraic equations, Bx = b, the inverse matrix is used to compute the solution vector x = B(-1)b. We present results that show the efficiency of the parallel hybrid Monte Carlo algorithm in the case of sparse matrices.

Comparison of the computational cost of a Monte Carlo and deterministic algorithm for computing bilinear forms of matrix powers

In this paper we consider bilinear forms of matrix polynomials and show that these polynomials can be used to construct solutions for the problems of solving systems of linear algebraic equations, matrix inversion and finding extremal eigenvalues. An almost Optimal Monte Carlo (MAO) algorithm for computing bilinear forms of matrix polynomials is presented. Results for the computational costs of a balanced algorithm for computing the bilinear form of a matrix power is presented, i.e., an algorithm for which probability and systematic errors are of the same order, and this is compared with the computational cost for a corresponding deterministic method.

Development of the optimal estimation method for inversion of stratospheric emission spectra and validation of HNO3 profiles

This doctoral thesis focuses on ground-based measurements of stratospheric nitric acid (HNO3)concentrations obtained by means of the Ground-Based Millimeter-wave Spectrometer (GBMS). Pressure broadened HNO3 emission spectra are analyzed using a new inversion algorithm developed as part of this thesis work and the retrieved vertical profiles are extensively compared to satellite-based data. This comparison effort I carried out has a key role in establishing a long-term (1991-2010), global data record of stratospheric HNO3, with an expected impact on studies concerning ozone decline and recovery. The first part of this work is focused on the development of an ad hoc version of the Optimal Estimation Method (Rodgers, 2000) in order to retrieve HNO3 spectra observed by means of GBMS. I also performed a comparison between HNO3 vertical profiles retrieved with the OEM and those obtained with the old iterative Matrix Inversion method. Results show no significant differences in retrieved profiles and error estimates, with the OEM providing however additional information needed to better characterize the retrievals. A final section of this first part of the work is dedicated to a brief review on the application of the OEM to other trace gases observed by GBMS, namely O3 and N2O. The second part of this study deals with the validation of HNO3 profiles obtained with the new inversion method. The first step has been the validation of GBMS measurements of tropospheric opacity, which is a necessary tool in the calibration of any GBMS spectra. This was achieved by means of comparisons among correlative measurements of water vapor column content (or Precipitable Water Vapor, PWV) since, in the spectral region observed by GBMS, the tropospheric opacity is almost entirely due to water vapor absorption. In particular, I compared GBMS PWV measurements collected during the primary field campaign of the ECOWAR project (Bhawar et al., 2008) with simultaneous PWV observations obtained with Vaisala RS92k radiosondes, a Raman lidar, and an IR Fourier transform spectrometer. I found that GBMS PWV measurements are in good agreement with the other three data sets exhibiting a mean difference between observations of ~9%. After this initial validation, GBMS HNO3 retrievals have been compared to two sets of satellite data produced by the two NASA/JPL Microwave Limb Sounder (MLS) experiments (aboard the Upper Atmosphere Research Satellite (UARS) from 1991 to 1999, and on the Earth Observing System (EOS) Aura mission from 2004 to date). This part of my thesis is inserted in GOZCARDS (Global Ozone Chemistry and Related Trace gas Data Records for the Stratosphere), a multi-year project, aimed at developing a long-term data record of stratospheric constituents relevant to the issues of ozone decline and expected recovery. This data record will be based mainly on satellite-derived measurements but ground-based observations will be pivotal for assessing offsets between satellite data sets. Since the GBMS has been operated for more than 15 years, its nitric acid data record offers a unique opportunity for cross-calibrating HNO3 measurements from the two MLS experiments. I compare GBMS HNO3 measurements obtained from the Italian Alpine station of Testa Grigia (45.9° N, 7.7° E, elev. 3500 m), during the period February 2004 - March 2007, and from Thule Air Base, Greenland (76.5°N 68.8°W), during polar winter 2008/09, and Aura MLS observations. A similar intercomparison is made between UARS MLS HNO3 measurements with those carried out from the GBMS at South Pole, Antarctica (90°S), during the most part of 1993 and 1995. I assess systematic differences between GBMS and both UARS and Aura HNO3 data sets at seven potential temperature levels. Results show that, except for measurements carried out at Thule, ground based and satellite data sets are consistent within the errors, at all potential temperature levels.

The dynamics of matrix momentum

We analyse the matrix momentum algorithm, which provides an efficient approximation to on-line Newton's method, by extending a recent statistical mechanics framework to include second order algorithms. We study the efficacy of this method when the Hessian is available and also consider a practical implementation which uses a single example estimate of the Hessian. The method is shown to provide excellent asymptotic performance, although the single example implementation is sensitive to the choice of training parameters. We conjecture that matrix momentum could provide efficient matrix inversion for other second order algorithms.

Dielectric formulation for Many-Particle Systems with General Interactions

An extension of the self-consistent field approach formulation by Cohen in the preceding paper is proposed in order to include the most general kind of two-body interactions, i.e., interactions depending on position, momenta, spin, isotopic spin, etc. The dielectric function is replaced by a dielectric matrix. The evaluation of the energies involves the computation of a matrix inversion and trace.

Large-scale clustering through functional embedding

We present a new framework for large-scale data clustering. The main idea is to modify functional dimensionality reduction techniques to directly optimize over discrete labels using stochastic gradient descent. Compared to methods like spectral clustering our approach solves a single optimization problem, rather than an ad-hoc two-stage optimization approach, does not require a matrix inversion, can easily encode prior knowledge in the set of implementable functions, and does not have an ?out-of-sample? problem. Experimental results on both artificial and real-world datasets show the usefulness of our approach.

Enhanced MDIR receiver for space-time dispersive channels

A particular property of the matched desiredimpulse response receiver is introduced in this paper, namely,the fact that full exploitation of the diversity is obtained withmultiple beamformers when the channel is spatially and timelydispersive. This particularity makes the receiver specially suitablefor mobile and underwater communications. The new structureprovides better performance than conventional and weightedVRAKE receivers, and a diversity gain with no needs of additionalradio frequency equipment. The baseband hardware neededfor this new receiver may be obtained through reconfigurabilityof the RAKE architectures available at the base station. Theproposed receiver is tested through simulations assuming UTRAfrequency-division-duplexing mode.

A Practical Micro Mechanical Model Based Local Approach Methodology for the Analysis of Ductile Fracture of Welded T-Joints

This thesis concentrates on developing a practical local approach methodology based on micro mechanical models for the analysis of ductile fracture of welded joints. Two major problems involved in the local approach, namely the dilational constitutive relation reflecting the softening behaviour of material, and the failure criterion associated with the constitutive equation, have been studied in detail. Firstly, considerable efforts were made on the numerical integration and computer implementation for the non trivial dilational Gurson Tvergaard model. Considering the weaknesses of the widely used Euler forward integration algorithms, a family of generalized mid point algorithms is proposed for the Gurson Tvergaard model. Correspondingly, based on the decomposition of stresses into hydrostatic and deviatoric parts, an explicit seven parameter expression for the consistent tangent moduli of the algorithms is presented. This explicit formula avoids any matrix inversion during numerical iteration and thus greatly facilitates the computer implementation of the algorithms and increase the efficiency of the code. The accuracy of the proposed algorithms and other conventional algorithms has been assessed in a systematic manner in order to highlight the best algorithm for this study. The accurate and efficient performance of present finite element implementation of the proposed algorithms has been demonstrated by various numerical examples. It has been found that the true mid point algorithm (a = 0.5) is the most accurate one when the deviatoric strain increment is radial to the yield surface and it is very important to use the consistent tangent moduli in the Newton iteration procedure. Secondly, an assessment of the consistency of current local failure criteria for ductile fracture, the critical void growth criterion, the constant critical void volume fraction criterion and Thomason's plastic limit load failure criterion, has been made. Significant differences in the predictions of ductility by the three criteria were found. By assuming the void grows spherically and using the void volume fraction from the Gurson Tvergaard model to calculate the current void matrix geometry, Thomason's failure criterion has been modified and a new failure criterion for the Gurson Tvergaard model is presented. Comparison with Koplik and Needleman's finite element results shows that the new failure criterion is fairly accurate indeed. A novel feature of the new failure criterion is that a mechanism for void coalescence is incorporated into the constitutive model. Hence the material failure is a natural result of the development of macroscopic plastic flow and the microscopic internal necking mechanism. By the new failure criterion, the critical void volume fraction is not a material constant and the initial void volume fraction and/or void nucleation parameters essentially control the material failure. This feature is very desirable and makes the numerical calibration of void nucleation parameters(s) possible and physically sound. Thirdly, a local approach methodology based on the above two major contributions has been built up in ABAQUS via the user material subroutine UMAT and applied to welded T joints. By using the void nucleation parameters calibrated from simple smooth and notched specimens, it was found that the fracture behaviour of the welded T joints can be well predicted using present methodology. This application has shown how the damage parameters of both base material and heat affected zone (HAZ) material can be obtained in a step by step manner and how useful and capable the local approach methodology is in the analysis of fracture behaviour and crack development as well as structural integrity assessment of practical problems where non homogeneous materials are involved. Finally, a procedure for the possible engineering application of the present methodology is suggested and discussed.

Quantitative analysis of remote gas temperatures and concentrations from their infrared emission spectra

Techniques for obtaining quantitative values of the temperatures and concentrations of remote hot gaseous effluents from their measured passive emission spectra have been examined in laboratory experiments and on field trials. These emission spectra were obtained using an adapted FTIR spectrometer with 0.25 cm-1 spectral resolution. The CO2 and H2O vapour content in the plume from a 55 m smoke stack and the temperature of these gases were obtained by comparing the measured emission spectra with those modelled using the HITRAN atmospheric transmission database. The spatial distributions of CO2, CO and unburnt CH4 in a laboratory methane flame were reconstructed tomographically using a matrix inversion technique.

Remote gas detection and quantitative analysis from infrared emission spectra obtained by Fourier transform infrared spectroscopy

Techniques for obtaining quantitative values of the temperatures and concentrations of remote hot gaseous effluents from their measured passive emission spectra have been examined in laboratory experiments. The high sensitivity of the spectrometer in the vicinity of the 2397 cm-1 band head region of CO2 has allowed the gas temperature to be calculated from the relative intensity of the observed rotational lines. The spatial distribution of the CO2 in a methane flame has been reconstructed tomographically using a matrix inversion technique. The spectrometer has been calibrated against a black body source at different temperatures and a self absorption correction has been applied to the data avoiding the need to measure the transmission directly. Reconstruction artifacts have been reduced by applying a smoothing routine to the inversion matrix.