
Inverse Problems in Earth Sciences
The problems described here resulted from a collaboration with Don Vasco, (LBNL/Earth Sciences Division). In this collaboration, we investigated tools for the solution of large inverse problems from applications in Geophysics; namely in the study of a physical model for the internal structure of the Earth. In this study, a linear model for the Earth structure is obtained by discretizing the Earth into layers, and the layers into cells. The velocity of wave propagation in each cell is represented by a set of parameters that account for anisotropy, location (and correction) of sources and receivers, etc. These parameters are written in matrix form and the goal is to fit the model by means of some known data. Travel times of sound waves generated by earthquakes are used to construct the model. The validation of the model requires the computation of an approximate generalized inverse and a (partial) singular value decomposition (SVD) is used for that purpose. The SVD is also used to estimate resolution and uncertainties.
The following table lists some of the numerical models that
have been studied, where m is number of travel
times (number of rows in the matrix),
n is the number of parameters in the model (number
of columns in the matrix)
and nnz is the number of nonzero entries in the
corresponding matrix.
A block Lanczos algorithm was used to compute a rankk
SVD approximation
for the models. Although sparse, such problems are difficult to deal
with
because several thousand singular values and singular vectors may be
required
for a proper estimation of the model parameters and resolution. For
details,
see Solving
Large Linear Inverse Problems
in Geophysics by means of Eigenvalue Calculations
and also BLZPACK.

model m n nnz

1 846968 96300 28587210
2 1433102 307134 51506866
3 1433102 307134 47968477

Gallery/Applications:
Conformational Changes of Proteins
This was a collaboration with YvesHenri
Sanejouand (Laboratoire de Physique, ENS Lyon). The goal was
to study conformational changes of proteins by means of linear
combinations
of normal modes coordinates. These coordinates are obtained from
solutions of an eigenvalue problem Ax=µx,
where A is a symmetric matrix computed from the
potential energy and the atomic masses of the protein. The following
table lists some cases that have been studied, where n
is the dimension of the matrix A (three times the
number of atoms) and nnz is the number of nonzero
entries in the
upper triangle of A. For details, see Protein Motions through
Eigenanalyses:
A Set of Study Cases and also BLZPACK.
Later,
YvesHenri and collaborators proposed a simplified method,
RTB (rotationstranslations of blocks), to study conformation changes
of proteins. For most cases, this method can significantly speed up the
calculations of the lowfrequency normal modes that are usually
required
in the analyses, without loss of accuracy. The software can be obtained
from YvesHenri.

protein n nnz

crambin 1188 134217
lysozyme 3795 490602
ras_p21A 4986 669060
ras_p21B 4998 677832
ras_p21C 4998 698973
ras_p21D 4986 691776
arabinose 8592 1161360
citrate synthase 25584 3691020

Gallery/Applications:

In 19971998 I collaborated with Mike Palmer, from Inktomi Inc, in the investigation of matrix methods for information retrieval on sets of documents obtained from the World Wide Web. The dimension of one of the termdocument matrices examined was 100,000 (number of terms or words) by 2,559,430 (number of documents or home pages), which contained 421 million nonzero entries. One of our eigenvalue solvers (BLZPACK) was used to compute singular values and singular vectors of this matrix. To the best of our knowledge, at the time it was the largest matrix for which a (partial) singular value decomposition had been computed. The computation of 10 singular values and left singular vector required 710 seconds of wall clock time on 64 processors of a Cray T3E900 (900 Mflops/PE, 256 MB/PE): 99.5% of that time was spent with matrixvectors multiplications (for the generation of Lanczos vectors). For details about information retrieval and related topics see Michael Berry's LSI Web Site, the LSA Web Site at CU Boulder, and also some of the papers written by Jon Kleinberg and Inderjit Dhillon.
Gallery/Applications:
Tools for Spectral Portrait Computations
This work was carried out at in collaboration with the Qualitative Computing Group (led by Prof. Francoise ChaitinChatelin) of the Parallel Algorithms Project (led by Prof. Iain Duff) at CERFACS. The goal was to develop tools for the computation of pseudospectra of linear operators. For details, see Spectral Portrait Computation by a Lanczos Method (augmented and normal matrix versions). See also Pseudospectra of Linear Operators by Nick Trefethen, SIAM Review, 39:383406, 1997, and the Matrix Market page on spectral portraits.
Gallery/Applications:
Spectral portrait of the La Rose matrix (65536 grid points).  Spectral portrait of the Tolosa matrix (18225 grid points). 
Structural Engineering Problems
In my Ph.D. work at COPPE, Federal University of Rio de Janeiro, Brazil, I studied reduced basis techniques for the solution of systems of partial differential equations, and algorithms for the solution of large sparse eigenvalue problems. These problems arise very often in Structural Engineering Finite Element analyses. During the thesis I was a Research Associate with PETROBRAS (The Brazilian Petroleum Company) and also worked as a consultant for INPE (Brazilian Institute for Spatial Research), using the MSC/NASTRAN code.
Gallery/Applications:
Doublecross with frequencies
distributed in clusters (see Selected
Results from the NAFEMS, Dynamics Working Group Free Vibrations,
Benchmarks,
Part 1, by A. Morris, Benchmarks, April 1987). The finite
element model
used comprises 321 nodes, 320 2D beam elements and 947 degrees of
freedom.


Selfelevating drilling unit designed
for operation up to 100m
water depth. The finite element model used comprises 90 nodes, 96 3D
beam
elements and 531 degrees of freedom.


Steel jacket
platform designed for 170m water depth.
The finite element model comprises 437 nodes, 1097 3D beam elements and
2463 degrees of freedom.

Lawrence Berkeley National Laboratory  Computational Research Division  Resume 