CS 267 Applications of Parallel Processors Lecture 9: Computational Electromagnetics - Large Dense Linear SystemsSlide 2Slide 3Slide 4Slide 5Slide 6Slide 7Slide 8Slide 9Slide 10Slide 11Slide 12Slide 13Slide 14Slide 15Slide 16Slide 17Slide 18Slide 19Slide 20Slide 21Slide 22Slide 23Slide 24Slide 25Slide 26Slide 27Slide 28Slide 29Slide 30Slide 31Slide 32CS 267 Applications of Parallel Processors Lecture 10: Large Dense Linear Systems -Distributed ImplementationsSlide 34Slide 35Slide 36Slide 37Slide 38Slide 39Slide 40Slide 41Slide 42Slide 43Slide 44Slide 45Slide 46Slide 47Slide 48Slide 49Slide 50Slide 51Slide 52Slide 53Slide 54Slide 55Slide 56Slide 57Slide 58Slide 59Slide 60Slide 61Slide 62Slide 63Slide 64Slide 65H. Simon - CS267 - L801/14/19 1CS 267 Applications of Parallel ProcessorsLecture 9: Computational Electromagnetics -Large Dense Linear Systems2/19/97Horst D. Simonhttp://www.cs.berkeley.edu/cs267H. Simon - CS267 - L801/14/19 2Outline - Lecture 9- Computational Electromagnetics - Sources of large dense linear systems- Review of solution of linear systems with Gaussian elimination- BLAS and memory hierarchy for linear algebra kernelsH. Simon - CS267 - L801/14/19 3Outline - Lecture 10- Layout of matrices on distributed memory machines - Distributed Gaussian elimination- Speeding up with advanced algorithms- LINPACK and LAPACK- LINPACK benchmark- Tflops resultH. Simon - CS267 - L801/14/19 4Outline - Lecture 11- Designing portable libraries for parallel machines- BLACS- ScaLAPACK for dense linear systems- other linear algebra algorithms in ScaLAPACKH. Simon - CS267 - L801/14/19 5Computational Electromagnetics- developed during 1980s, driven by defense applications- determine the RCS (radar cross section) of airplane- reduce signature of plane (stealth technology)- other applications are antenna design, medical equipment- two fundamental numerical approaches: MOM methods of moments ( frequency domain), and finite differences (time domain)H. Simon - CS267 - L801/14/19 6Computational Electromagnetics image: NW Univ. Comp. Electromagnetics Laboratory http://nueml.ece.nwu.edu/- discretize surface into triangular facets using standard modeling tools- amplitude of currents on surface are unknowns - integral equation is discretized into a set of linear equationsH. Simon - CS267 - L801/14/19 7Computational Electromagnetics (MOM)After discretization the integral equation has the form Z J = VwhereZ is the impedance matrix, J is the unknown vector of amplitudes, and V is the excitation vector.Z is given as a four dimensional integral.(see Cwik, Patterson, and Scott, Electromagnetic Scattering on the Intel Touchstone Delta, IEEE Supercomputing ‘92, pp 538 - 542)H. Simon - CS267 - L801/14/19 8The main steps in the solution process areA) computing the matrix elementsB) factoring the dense matrixC) solving for one or more excitationsD) computing the fields scattered from the objectComputational Electromagnetics (MOM)H. Simon - CS267 - L801/14/19 9Analysis of MOM for Parallel ImplementationTask Work Parallelism Parallel SpeedFill O(n**2) embarrassing lowFactor O(n**3) moderately diff. very highSolve O(n**2) moderately diff. highField Calc. O(n) embarrassing highFor most scientific applications the biggest gain in performance can be obtained by focusing on one tasks.H. Simon - CS267 - L801/14/19 10Results for Parallel Implementation on DeltaTask Time (hours) Performance (Gflop/s)Fill 9.20 ~ 1.0Factor 8.25 10.35Solve 2.17 -Field Calc. 0.12 3.0The problem solved was for a matrix of size 48,672. (The world record in 1991.)H. Simon - CS267 - L801/14/19 11Current Records for Solving Dense SystemsYear System Size Machine1950's O(100) 1991 55,296 CM-2 1992 75,264 Intel 1993 75,264 Intel 1994 76,800 CM-5 1995 128,600 Paragon XP1996 215,000 ASCI Red source: Alan Edelman http://www-math.mit.edu/~edelman/records.htmlH. Simon - CS267 - L801/14/19 12Sources for large dense linear systems- not many outside CEM- even within CEM community alternatives such FD-TD are heavily debatedIn many instances choices for algorithms or methods in existing scientific codes or applications are not the resultof careful planning and design. At best they are reflecting the start-of-the-art at the time, at worst they are purelycoincidental.H. Simon - CS267 - L801/14/19 13Review of Gaussian Elimination see Demmel http://HTTP.CS.Berkeley.EDU/~demmel/cs267/lecture12/lecture12.htmlGaussian elimination to solve Ax=b - start with a dense matrix - add multiples of each row to subsequent rows in order to create zeros below the diagonal- ending up with an upper triangular matrix U. Solve a linear system with U by substitution, startingwith the last variable.H. Simon - CS267 - L801/14/19 14 ... for each column i, ... zero it out below the diagonal by ... adding multiples of row i to later rows for i = 1 to n-1 ... each row j below row i for j = i+1 to n ... add a multiple of row i to row j for k = i to n A(j,k) = A(j,k) - (A(j,i)/A(i,i)) * A(i,k)Review of Gaussian Elimination (cont.)H. Simon - CS267 - L801/14/19 15Review of Gaussian Elimination (cont.)H. Simon - CS267 - L801/14/19 16 ... for each column i, ... zero it out below the diagonal by ... adding multiples of row i to later rows for i = 1 to n-1 ... each row j below row i for j = i+1 to n ... add a multiple of row i to row j for k = i to n A(j,k) = A(j,k) - (A(j,i)/A(i,i)) * A(i,k)Review of Gaussian Elimination (cont.)= mH. Simon - CS267 - L801/14/19 17Review of Gaussian Elimination (cont.) for i = 1 to n-1 for j = i+1 to n m = A(j,i)/A(i,i) for k = i+1 to n A(j,k) = A(j,k) - m * A(i,k)avoid computation of known matrix entryH. Simon - CS267 - L801/14/19 18Review of Gaussian Elimination (cont.)It will be convenient to store the multipliers m in the implicitly created zeros below the diagonal, so we can use them later to transform the right hand side b: for i = 1 to n-1 for j = i+1 to n A(j,i) = A(j,i)/A(i,i) for j = i+1 to n for k = i+1 to n A(j,k) = A(j,k) - A(j,i) * A(i,k)H.
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