21. Master Equation Solvers Master equation solvers. General Information. Master equation solvers solve forthe time evolution of the density matrix/Wigner function of a quantum system. http://t8web.lanl.gov/people/salman/nersc/mastercodes.html

Master Equation Solvers General Information Master equation solvers solve for the time evolution of the density matrix/Wigner function of a quantum system. These codes are very memory intensive since, for a single degree of freedom, the memory requirements go as L^(2n) where n is the space dimensionality and L the number of points per dimension. In the case n=1, we have run system sizes as large as 8K X 8K. The basic method of solution is a split-operator spectral technique. We have implemented algorithms that offer second and fourth order accuracy in time. Code performance is controlled typically by the performance achieved by the FFT implementation in the spectral method. The Wigner representation is often advantageous since it allows the use of nonsymmetric grids to optimize accuracy. Atom-optical Master Equation The purpose of the code is to simulate experiments being performed at the University of Texas (Austin) and at the University of Auckland, on dynamical localization in the quantum delta-kicked rotor. Previous simulation codes have not explicitly treated the atomic internal-state dynamics, and so were incapable of directly addressing questions about the role of atomic spontaneous emission. In FY 98 we had focused on validating the F90/MPI code by comparing its output to those of a proven HPF code without internal dynamics, in a parameter regime where spontaneous emission can justifiably be neglected. We now plan to use it for detailed studies of the effects of decoherence on dynamical localization in atom-optical systems.

22. Schrodinger Solvers Schrodinger Solvers. General Information. Schrodinger equation solverssolve for the time evolution of the wave function of a quantum system. http://t8web.lanl.gov/people/salman/nersc/schrcodes.html

Schrodinger Solvers General Information Schrodinger equation solvers solve for the time evolution of the wave function of a quantum system. The memory requirements go as L^(mn) where n is the space dimensionality, L the number of points per dimension, and m the number of degrees of freedom. The basic method of solution is a split-operator technique which may be implemented either in Fourier or coordinate space. The algorithm is fully unitary and also applies to the nonlinear Schrodinger equation. Options for second or fourth order accuracy in time are available. Code performance is controlled typically by the performance achieved by the FFT implementation in the spectral method and by the implementation of shift operators in the coordinate space versions. We have recently added the capability to solve nonlinear stochastic Schrodinger equations used to model continuous quantum measurements via the so-called ``quantum trajectory'' technique. These codes use tracking methods to reduce memory usage for a given resolution and certain unique capabilities compared to solutions of the quantum Master equation. Split-operator Integrators This is an HPF and F90/MPI code designed for accurate, very high-resolution, long-term integration of the single-particle Schrodinger equation using a split-operator spectral method. (Unitary nonspectral split-operator solvers also exist.) We have used this code to evaluate various approximation strategies for tracking the dynamics of quantum field theories by testing these schemes in quantum mechanics. The code has proved to be of central importance in our recent studies of chaotic quantum systems where accuracy is an essential requirement. A stochastic version of this code has been used to study quantum decoherence.

23. Octave/Control Toolbox Reference Guide: Equation Solvers And Additional Math Fun Next Previous Contents 10. equation solvers and Additional Math Functions.10.1 are. x = are (a, b, c ,opt) Solves algebraic riccati http://www.ifr.ing.tu-bs.de/octave/csref-10.html

10. Equation Solvers and Additional Math Functions 10.1 are x = are (a, b, c [,opt]) Solves algebraic riccati equation a' x + x a - x b x + c = for identically dimensioned square matrices a, b, c. If b (c) is not square, then the function attempts to use b * b' (c' * c) instead. Solution method: apply Laub's Schur method (IEEE Trans. Auto. Contr, 1979) to the appropriate Hamiltonian matrix. opt is an option passed to the eigenvalue balancing routine default is "B". See also: balance 10.2 dlyap x = dlyap (a, b) Solve a x a' - x + b = (discrete Lyapunov equation) for square matrices a and b. If b is not square, then the function attempts to solve either a x a' - x + b b' = or a' x a - x + b' b = whichever is appropriate. Uses Schur decomposition as in Kitagawa (1977). 10.3 dare x = dare (a, b, c, r [,opt]) 10.4 lyap x = lyap (a, b, c) x = lyap (a, c) 10.5 pinv pinv ( [, tol]) Returns the pseudoinverse of X; singular values less than tol are ignored. 10.6 zgfmul y = zgfmul(a,b,c,d,x)

24. Talk Abstract: Combining Model-reduction And Integral-equation Solvers Institute for Mathematics and Its Applications. Talk abstract Combiningmodelreduction and integral-equation solvers. Joel R. Phillips, Cadence http://www.ima.umn.edu/dynsys/wkshp_abstracts/phillips1.html

Institute for Mathematics and Its Applications Talk abstract: Combining model-reduction and integral-equation solvers Joel R. Phillips , Cadence Development of efficient integral-equation based electromagnetic analysis tools has been hampered by the high computational complexity of dense matrix representations and difficulty in obtaining and utilizing the frequency-domain response. In this talk we show that an algorithm based on application of a novel model-order reduction scheme directly to the sparse model generated by a fast integral transform has advantages for frequency- and time-domain simulation. Back to Workshop Schedule

25. Parallel Multigrid Equation Solvers For Unstructured Finite Element Problems Parallel Multigrid equation solvers for Unstructured Finite Element Problems.Mark Adams (Professor James W. Demmel) (DOE) W7405-ENG-48 http://www.eecs.berkeley.edu/IPRO/Summary/98abstracts/madams.1.html

Parallel Multigrid Equation Solvers for Unstructured Finite Element Problems Mark Adams (Professor James W. Demmel) (DOE) W-7405-ENG-48 B. Smith, P. Bjorstad, and W. Gropp, Domain Decomposition, Cambridge University Press, 1996. H. Guillard, Node-Nested Multi-Grid with Delaunay Coarsening, Institute National de Recherche en Informatique et en Automatique, Technical Report No. 1898, 1992. More information (http://www.mcs.anl.gov/CCST/research/discipline/index.html) or Send mail to Mark Adams : (madams@ce.berkeley.edu) Edit this abstract

26. Mark Adams' Home Page Interests My research interests are in high performance finite element simulationsystems in particular, parallel multigrid equation solvers for large http://www.cs.berkeley.edu/~madams/

Mark Adams I received my Ph.D. in Civil Engineering, from U.C. Berkele y in 1998 and am a former postdoc with Jim Demmel in the Computer Science Division , at U.C. Berkeley. I am currently a John von Neumann research fellow in the Computational Sciences, Computer Sciences and Mathematics Center at Sandia National Laboratories Mark Adams Sandia National Laboratories PO Box 969 Livermore CA 94551-9217 925.294.4820 (phone) 925.294.2234 (fax) mfadams@ca.sandia.gov Research Interests: My research interests are in high performance finite element simulation systems - in particular, parallel multigrid equation solvers for large unstructured finite element problems in solid mechanics. I have developed Athena, a parallel finite element implementation, built on FEAP a serial finite element implementation. Athena uses my solver Prometheus , an unstructured multigrid equation solver for large scale ( degrees of freedom ) finite element problems, which was the focus of my dissertation . I use the PETSc - numerical libraries, from Argonne National Laboratory (ANL) , to provide high performance, cross platform, support for iterative solvers for discretized partial differential equations and ParMetis , from the University of Minnesota for parallel mesh partitioning.

27. Workload Characterization Of CFD Applications Using Partial Differential Equatio Workload Characterization of CFD Applications Using Partial Differentialequation solvers. Abdul Waheed and Jerry Yan. NAS Parallel http://www.nas.nasa.gov/Research/Reports/Techreports/1998/nas-98-011-abstract.ht

Workload Characterization of CFD Applications Using Partial Differential Equation Solvers Abdul Waheed and Jerry Yan NAS Parallel Tools Group Nasa Ames Research Center Mail Stop T27A-2 Moffett Field, CA 94035-1000 waheed@nas.nasa.gov yan@nas.nasa.gov NAS-98-011 March 1998 Abstract To view the full report: PDF Version To read this file you will need the free Adobe Acrobat Reader Curator: Jill Dunbar Last Update: July 5, 2002 NASA Official: Walt Brooks

28. Fast Banded Linear Equation Solvers Fast Banded Linear equation solvers. Russell K. Standish ANU Supercomputing FacilityThe Australian National University GPO Box 4, Canberra, 2601, Australia. http://parallel.hpc.unsw.edu.au/rks/docs/blockcr/blockcr.html

29. Topic [MATHCAD] Differential Equation Solvers Sender, Message. Robert Garner Wed, 9 Sep 1998 093035 +0100, MATHCADDifferential equation solvers MC7 has a good range of differential http://lists.adeptscience.co.uk/mathcad/mathcad_Sep_1998/thid_bfa2d9f27a74ee336c

30. Message Re [MATHCAD] Differential Equation Solvers Sender, Message. Byrge Birkeland Wed, 09 Sep 1998 123851 +0200,Re MATHCAD Differential equation solvers At 0930 09.09.98 http://lists.adeptscience.co.uk/mathcad/mathcad_Sep_1998/msg_573.html

31. Comparing Krylov Linear Equation Solvers Comparing Krylov Linear equation solvers. Subject Comparing KrylovLinear equation solvers; From Charles Broyden Broyden@tesco.net ; http://www.csc.fi/math_topics/Mail/NANET00-3/msg00001.html

Message Prev Message Next Message Index Comparing Krylov Linear Equation Solvers Subject : Comparing Krylov Linear Equation Solvers From Broyden@tesco.net Date : Thu, 29 Jun 2000 18:19:45 +0100 Prev by Message: - NA Digest V. 00, # 27 Next by Message: Generalized Saddle-point Problem Index(es): Message

32. HEAT EQUATION SOLVERS HEAT equation solvers. Backwards differencing with dirichlet boundaryconditions heat1d_dir.m and Neumann boundary conditions heat1d_neu http://www.math.sfu.ca/mast/people/faculty/mkropins/math922/lectures/heatsolvers

HEAT EQUATION SOLVERS Backwards differencing with dirichlet boundary conditions heat1d_dir.m and Neumann boundary conditions heat1d_neu.m . Both of the above require the routine heat1dmat.m that computes the tridiagonal matrix associated with this difference scheme. Radiative boundary conditions are incorporated in heat1d_farr.m . This requires the tridiagonal matrix heat1dradmat.m . Please download and review these routines for class on Feb. 21. One dimensional heat equation with non-constant coefficients: heat1d_DC.m . This requires the routine heat1dDCmat.m that assembles the tridiagonal matrix associated with this difference scheme. Heat conduction into a rod with D=0.01 on the left, D=1 on the right: Two dimensional heat equation on a square with Dirichlet boundary conditions: heat2d.m . You may also want to take a look at my_delsqdemo.m to see more on two dimensional finite difference problems in Matlab. Two dimensional heat equation on a square with Neumann boundary conditions: heat2dN.m

33. Fast Surface Integral Equation Solvers For Electromagnetic Scattering Fast surface integral equation solvers. Fast surface integral equationsolvers for electromagnetic scattering. Current methods rely http://www.ccem.uiuc.edu/reschew13.html

Fast surface integral equation solvers Fast surface integral equation solvers for electromagnetic scattering Current methods rely on iterative linear system solution methods such as Conjugate Gradients. For some types of scatterers, including inlets, with resonance-regime features, the iteration count required for the solution to converge becomes large, leading to high computational cost. Some researchers have proposed techniques to overcome this, such as multigrid or near neighbor preconditioning, which perform well for other classes of PDEs. Dr. Warnick's current research into the spectrum of surface integral operators shows that the difficulty in solving the scattering problem grows as the scatterer becomes large due to long-range coupling effects, which causes local methods such as near-neighbor preconditioning and multigrid to break down for Maxwell's equations. A new type of spectral multigrid is currently being developped. This multigrid preconditions the long-range interactions inherent in the surface integral operator, with the intent of developing an optimal surface integral equation solver for which computational cost grows linearly with problem size. The first Figure compares the convergence history of GMRES in work units to that of spectral multigrid for TM scattering from an inlet with a depth of about 20 wavelengths.

34. Citation ACM Transactions on Mathematical Software (TOMS) archive Volume 1 , Issue 4 (December1975) toc Parallel Tridiagonal equation solvers Author Harold S. Stone http://portal.acm.org/citation.cfm?id=355657&dl=ACM&coll=portal&CFID=11111111&CF

35. Citation archive Volume 3 , Issue 1 (March 1977) toc Stiffness and Nonstiff Differentialequation solvers, II Detecting Stiffness with RungeKutta Methods Author LF http://portal.acm.org/citation.cfm?id=355719.355722&coll=portal&dl=ACM&idx=J782&

36. Equation Solvers next up previous Next Eigenvalue and Eigenvector Extraction Up Finite Element Library Subject Listing Previous Standard Vector and Matrix equation solvers. http://www.mathsoft.cse.clrc.ac.uk/felib/Docs/html/Level-0/level0/level0-node3.h

Next: Eigenvalue and Eigenvector Extraction Up: Finite Element Library - Subject Listing Previous: Standard Vector and Matrix Equation Solvers CSYRDN - Complex symmetric decomposition into triangular matrices CSYSOL - Solves a complex symmetric linear system CSYSUB - Complex forward and backward substitution CHOBAK - Choleski backward substitution CHOFWD - Choleski forward substitution CHORDN - Choleski decomposition into triangular matrices CHOSOL - Solves a system of linear equations using Choleski reduction GAURDN - Gaussian decomposition into upper and lower triangles GAUSOL - Solves a system of linear equations using Gaussian reduction GAUSUB - Gaussian forward and backward substitution Chris Greenough (c.greenough@rl.ac.uk): October 2000

37. Ode_v1.10.tar.gz, RK Ordinary Differnential Equation Solvers ode_v1.10.tar.gz, RK ordinary differnential equation solvers. Tooctavesources@bevo.che.wisc.edu; Subject ode_v1.10.tar.gz, RK http://www.octave.org/octave-lists/archive/octave-sources.2001/msg00002.html

Marc Compere on Tue, 9 Jan 2001 18:26:08 -0600 Date Prev Date Next Thread Prev Thread Next ... Thread Index ode_v1.10.tar.gz, RK ordinary differnential equation solvers To octave-sources@bevo.che.wisc.edu Subject : ode_v1.10.tar.gz, RK ordinary differnential equation solvers From CompereM@asme.org Date : Tue, 9 Jan 2001 18:26:08 -0600 Organization : The University of Texas at Austin Reply-to CompereM@asme.org Sender octave-sources-request@bevo.che.wisc.edu http://nerdlab.me.utexas.edu/~compere http://www.octave.org How to fund new projects: http://www.octave.org/funding.html Subscription information: http://www.octave.org/archive.html Prev by Date: a couple of image processing tools Next by Date: improved/corrected dot.m Previous by thread: a couple of image processing tools Next by thread: improved/corrected dot.m Index(es): Date Thread

38. Efficient Poisson Equation Solvers For Large Scale 3D Simulations Chapter Numerics, Algorithms. Title Efficient Poisson equation solvers for LargeScale 3D Simulations. Author(s) G. Speyer, D. Vasileska and SM Goodnick. http://www.cr.org/publications/MSM2001/html/W31.1.html

Technical Proceedings of the 2001 International Conference on Modeling and Simulation of Microsystems MSM 2001 Hilton Oceanfront Resort, South Carolina, U.S.A. March 19-21, 2001 Chapter: Numerics, Algorithms Title: Efficient Poisson Equation Solvers for Large Scale 3D Simulations Author(s): G. Speyer, D. Vasileska and S.M. Goodnick Affiliation: Arizona State University, U.S.A. Pages: Keywords: semiconductor device modeling, poisson solver, multi-grid, BiCGSTAB Abstract: View paper ISBN: Back Boston * Geneva * San Francisco www.comppub.com In association with Applied Computational Research Society www.cr.org Nano Science and Technology Institute www.nsti.org

39. Linear-Equation Solvers 5: The summary for this Korean page contains characters that cannot be correctly displayed in this language/character set. http://www.act-e.co.kr/support/cfd/lecture/twophs/sld034.htm

40. Linear-Equation Solvers 3: The summary for this Korean page contains characters that cannot be correctly displayed in this language/character set. http://www.act-e.co.kr/support/cfd/lecture/twophs/sld032.htm

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