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**Solving Large-scale Eigenvalue Problems in SciDAC Applications**

Chao Yang Lawrence Berkeley National Laboratory June 27, 2005

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People Involved LBNL: W. Gao, P. Husbands, X. S. Li, E. Ng, C. Yang (TOPS) J. Meza, L. W. Wang, C. Yang (Nano-science) SLAC: L. Lee, K. Ko Stanford: G. Golub UC-Davis Z. Bai

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SciDAC Applications Accelerator Modeling Nano-science

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Algorithms Krylov Subspace Method Alternatives Optimization based approach non-linear solver based approach Multi-level Sub-structuring Non-linear Eigenvalue Problems Structure preserving methods Optimization based method

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**Krylov Subspace Method**

Widely used, relatively well understood (Polynomial approximation theory): Convergence of KSM: Well separated, large eigenvalues converge rapidly the starting vector

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**Acceleration Techniques**

Implicit Restart Spectral transformation ARPACK filter out unwanted spectral components from v0

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**Using KSM in accelerator modeling**

the spectrum of the problem Example: H60VG3 structure, linear element, N=30M, nnz=484M 1024 CPUs, 738GB Ordering time: 4143s Numerical Factorization: 133s Total: 5068s for 12 eigenvalues Software: PARPACK (implicit restart) + SuperLU, WSMP (spectral transformation)

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Limitations of the KSM High degree polynomial needed for computing small clustered eigenvalues many matrix vector multiplications Spectral transformation can be expensive memory limitation scalability Not easy to introduce a preconditioner eigenvectors of P-1A are different from eigenvectors of A

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**Alternative algorithms**

Optimization based approach Minimizing Rayleigh Quotient Minimizing Residual (Wood & Zunger 85, Jia 97) Nonlinear equation solver based approach (Jacobi-Davidson) Newton correction Preconditioner stopping criteria for the inner iteration (Notay 2002, Stathopoulos 2005) Allows us to solve problems with more than 90M DOF

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**Multi-level Sub-structuring (for computing many eigenpairs)**

Domain Decomposition concept Multi-level extension of the Component Mode Synthesis (CMS) method (Bennighof 92) Decomposition can be done algebraically (Lehoucq & Bennighof 2002) Success story in structure engineering.... Error analysis Extend to accelerator modeling

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**Single-level Sub-structuring**

Matrix Partition Block elimination Sub-structure calculation (mode selection) Subspace assembling

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Mode Selection

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**Implementation & cost attractive when: 1) the problem is large enough**

Flops: more than a single sparse Cholesky factorization Storage: Block Cholesky factor + Projected matrix + some other stuff NO triangular solves (involving the original K and M), NO orthogonalization attractive when: 1) the problem is large enough 2) a large number of eigenvalues are needed

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**AMLS vs. Shift-invert Lanczos (SIL)**

DOF=65K, 3 levels of partition

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**Cavity with External Coupling**

Open Cavity Waveguide BC Vector wave equation with waveguide boundary conditions can be modeled by a non-linear eigenvalue problem Closed cavity is an approximation for the real model. Usually the cavity has some sort of openings. For example, in accelearator cavity, we need put power into cavity and damp high-order-modes. These external coupling requires waveguide connected to cavity. Waveguide shape can be circular, rectangular or coax. To model the open cavtity with waveguides, we can introduce a virtual BC on waveguide as shown. The problem is emerged from a cavity with external coupling through waveguides. We can put waveguide boundary conditions on the virtual boundaries. Curl-curl equation along with waveguide boundary conditions will be discretized using finite-element methods (Electric field E is expanded by a set of vector basis functions.) and becomes a non-linear eigenvalue problem, where matrix K corresponds to curl-curl operator. M is mass matrix. Matrix W is waveguide matrix and has nonzero entries only on waveguide boundary. Note that k_cj is a physical known quantity called cutoff. With

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**Quadratic Eigenvalue Problem**

Consider only one mode propagating in the waveguides Algorithms Linearize then solve by KSM (does not preserve the structure of the problem) Second Order Arnoldi Iteration (Bai & Su 2005) project the QEP into 2nd order Krylov Subspace

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**Second-Order Krylov Space (Bai)**

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**SOAR is faster and more accurate (than linearization)**

Accelerating cavity model for international linear collider (ILC) 9-cell superconducting cavity coupled to one input coupler and two Higher-Order-Mode couplers. NDOFs=3.2million, NCPUs=768, Memory=300GB 18 eigenpairs in 2634 seconds (linearization took more than 1 hour)

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**Electronic Structure Calculation**

Etotal(X) = Ekinetic + Eionic + EHartree + Exc wave function n – real space grid size, e.g. 323~32000 k – number of occupied states, 1~10% of n Charge density Ekinetic = Eionic = EHartree= Exc =

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**Non-linear Eigenvalue Problem**

Total energy minimization KKT condition

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**The Self Consistent Field Iteration**

Input: initial guess and Output: Major steps For i=1,2,…,until converged Form Compute k smallest eigpairs of Consistent (try tex point)

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**Direct Constrained Minimization (DCM)**

For i=1,2,… until convergence Form Compute If (i>1) then set else Solve Be consistent with subscripts etc K is the preconditioner

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**DCM vs. SCF Atomic system: SiH4**

Discretization: spectral method with plane wave basis: n=323 in real space, N=2103 (# of basis functions) in frequency space Number of occupied states: k = 4 PETOT version of SCF uses 10 PCG steps (inner iterations) per outer iteration DCM: 3 inner iterations

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Concluding Remarks Krylov Subspace Method (with appropriate acceleration strategies) continues to play an important role in solving SciDAC eigenvalue problems Steady progress has been made in alternative approaches that can make better use of preconditioners Multi-level sub-structuring is promising for computing many eigenpairs Significant progress made in solving QEP Non-linear eigenvalue problems remain challenging

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