LOCA swMATH ID: 4717 Software Authors: Andy Salinger; Sandia National Laboratories Description: LOCA is a new software library for performing bifurcation analysis of large-scale applications. The source code has been available for download since April 19, 2002. The algorithms in LOCA are being designed as part of an ongoing research project at Sandia National Laboratories in Albuquerque into scalable stability analysis algorithms. When implemented with an application code, LOCA enables the tracking of solution branches as a function of system parameters and the direct tracking of bifurcation points. LOCA (which is written in ”C”) is designed to drive application codes that use Newton’s method to locate steady-state solutions to nonlinear problems. The algorithms are chosen to work for large problems, such as those that arise from discretizations of partial differential equations, and to run on distributed memory parallel machines. The approach in LOCA for locating and tracking bifurcations begins with augmenting the residual equations defining a steady state with additional equations that describe the bifurcation. A Newton method is then formulated for this augmented system; however, instead of loading up the Jacobian matrix for the entire augmented system (a task that involved second derivatives and dense matrix rows), bordering algorithms are used to decompose the linear solve into several solves with smaller matrices. Almost all of the algorithms just require multiple solves of the Jacobian matrix for the steady state problem to calculate the Newton updates for the augmented system. This greatly simplifies the implementation, since this is the same linear system that an application code using Newton’s method will already have invested in. Only the Hopf tracking algorithm requires the solution of a larger matrix, which is the complex matrix involving the Jacobian matrix and an imaginary multiple of the mass matrix. Homepage: http://www.cs.sandia.gov/loca/ Programming Languages: C Keywords: bifurcation analysis Related Software: AUTO; ARPACK; MATCONT; Trilinos; AUTO-86; Aztec; PyNCT; PETSc; MUMPS; pde2path; COCO; DDE-BIFTOOL; MPSalsa; PLTMG; AztecOO; P - ARPACK; Python; KELLEY; PITCON; NumPy Cited in: 33 Publications all top 5 Cited by 66 Authors 5 Salinger, Andrew G. 3 Kelley, Carl T. 3 Net, Marta 3 Pawlowski, Roger P. 3 Romero, Louis A. 3 Vanroose, Wim 2 Burroughs, Elizabeth A. 2 Canton, Jacopo 2 Juncu, Gheorghe 2 Lasater, M. S. 2 Lehoucq, Richard B. 2 Shadid, John N. 2 Woolard, Dwight L. 2 Wouters, Michiel 2 Zhao, Peiji 1 Auteri, Franco 1 Avitabile, Daniele 1 Bader, Brett W. 1 Bou-Rabee, Nawaf M. 1 Broeckhove, Jan 1 Carey, Graham Francis 1 Caulfield, C. P. 1 Ciani, Antonia 1 Coetzee, Etienne B. 1 Coppola, Gennaro 1 Coutinho, Alvaro L. G. A. 1 Deremble, Bruno 1 Dhillon, Daljit Singh J. 1 Douglas, Christopher M. 1 Draelants, Delphine 1 Elias, Renato N. 1 Emerson, Benjamin L. 1 Frouzakis, Christos Emmanouil 1 García-Archilla, Bosco 1 Groh, Rainer M. J. 1 Hennig, Rebekka-Sarah 1 Kerswell, Richard Rodney 1 Kłosiewicz, Przemysław 1 Krauskopf, Bernd 1 Kreutner, W. 1 Laing, Carlo R. 1 Lieuwen, Timothy C. 1 López, Vanessa 1 Lowenberg, Mark H. 1 Milinkovitch, Michel C. 1 Mosekilde, Erik 1 Mountziaris, T. J. 1 Neild, Simon A. 1 Örlü, Ramis 1 Parker, Jeremy P. 1 Phipps, Eric T. 1 Pirrera, Alberto 1 Popa, Constantin 1 Radziunas, Mindaugas 1 Recine, Greg 1 Sanchez-Umbria, Juan 1 Schlatter, Philipp 1 Schnabel, Robert B. 1 Sharma, Sanjiv 1 Sieber, Jan 1 Simó, Carles 1 Thies, Jonas 1 Thompson, John Michael T. 1 Uecker, Hannes 1 Valli, Andrea M. P. 1 Zwicker, Matthias all top 5 Cited in 20 Serials 6 Journal of Fluid Mechanics 4 SIAM Journal on Applied Dynamical Systems 3 International Journal of Bifurcation and Chaos in Applied Sciences and Engineering 2 International Journal for Numerical Methods in Fluids 1 Computers and Fluids 1 Computers & Mathematics with Applications 1 Computer Methods in Applied Mechanics and Engineering 1 International Journal of Heat and Mass Transfer 1 Jahresbericht der Deutschen Mathematiker-Vereinigung (DMV) 1 Bulletin of Mathematical Biology 1 BIT 1 Physica D 1 Applied Numerical Mathematics 1 SIAM Journal on Scientific Computing 1 International Journal of Numerical Methods for Heat & Fluid Flow 1 Communications in Nonlinear Science and Numerical Simulation 1 International Journal of Pure and Applied Mathematics 1 Acta Numerica 1 The Journal of Mathematical Neuroscience 1 Philosophical Transactions of the Royal Society of London. A. Mathematical, Physical and Engineering Sciences all top 5 Cited in 15 Fields 18 Numerical analysis (65-XX) 13 Fluid mechanics (76-XX) 9 Dynamical systems and ergodic theory (37-XX) 5 Partial differential equations (35-XX) 3 Ordinary differential equations (34-XX) 3 Classical thermodynamics, heat transfer (80-XX) 3 Biology and other natural sciences (92-XX) 2 Mechanics of deformable solids (74-XX) 2 Geophysics (86-XX) 2 Systems theory; control (93-XX) 1 Integral equations (45-XX) 1 Mechanics of particles and systems (70-XX) 1 Optics, electromagnetic theory (78-XX) 1 Quantum theory (81-XX) 1 Statistical mechanics, structure of matter (82-XX) Citations by Year