This paper deals with efficient numerical representation and manipulation of differential and integral operators as symbols in phase-space, i.e., functions of space @math and frequency @math . The symbol smoothness conditions obeyed by many operators in connection to smooth linear partial differential equations allow to write fast-converging, non-asymptotic expansions in adequate systems of rational Chebyshev functions or hierarchical splines. The classical results of closedness of such symbol classes under multiplication, inversion and taking the square root translate into practical iterative algorithms for realizing these operations directly in the proposed expansions. Because symbol-based numerical methods handle operators and not functions, their complexity depends on the desired resolution @math very weakly, typically only through @math factors. We present three applications to computational problems related to wave propagation: 1) preconditioning the Helmholtz equation, 2) decomposing wavefields into one-way components and 3) depth-stepping in reflection seismology.
A different, competing approach to compressing operators is the partitioned separated'' method that consists in isolating off-diagonal squares of the kernel @math , and approximating each of them by a low-rank matrix. This also calls for an adapted notion of calculus, e.g., for composing and inverting operators. The first reference to this algorithmic framework is probably the partitioned SVD method described in @cite_20 . More recently, these ideas have been extensively developed under the name H-matrix, for hierarchical matrix; see @cite_29 @cite_4 and http: www.hlib.org.
@cite_29 @cite_4 @cite_20
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In this paper we develop a fast direct solver for large discretized linear systems using the supernodal multifrontal method together with low-rank approximations. For linear systems arising from certain partial differential equations such as elliptic equations, during the Gaussian elimination of the matrices with proper ordering, the fill-in has a low-rank property: all off-diagonal blocks have small numerical ranks with proper definition of off-diagonal blocks. Matrices with this low-rank property can be efficiently approximated with semiseparable structures called hierarchically semiseparable (HSS) representations. We reveal the above low-rank property by ordering the variables with nested dissection and eliminating them with the multifrontal method. All matrix operations in the multifrontal method are performed in HSS forms. We present efficient ways to organize the HSS structured operations along the elimination. Some fast HSS matrix operations using tree structures are proposed. This new structured multifrontal method has nearly linear complexity and a linear storage requirement. Thus, we call it a superfast multifrontal method. It is especially suitable for large sparse problems and also has natural adaptability to parallel computations and great potential to provide effective preconditioners. Numerical results demonstrate the efficiency. This paper presents a fast direct solver for 3D discretized linear systems using the supernodal multifrontal method together with low-rank approximations. For linear systems arising from certain partial differential equations (PDEs) such as elliptic equations, during the Gaussian elimination of the matrices with Nested Dissection ordering, the fill-in of L and U factors loses its sparsity and contains dense blocks with low-rank property. Off-diagonal blocks can be efficiently approximated with low-rank matrices; diagonal blocks approximated with semiseparable structures called hierarchically semiseparable (HSS) representations. Matrix operations in the multifrontal method are performed in low-rank arithmetic. We present efficient way to organize the HSS structured operations along the elimination. To compress dense blocks into low-rank or HSS structures, we use effective cross approximation (CA) approach. We also use idea of adaptive balancing between robust arithmetic for computing the small dense blocks and low-rank matrix operations for handling with compressed ones while performing the Gaussian elimination. This new proposed solver can be essentially parallelized both on architecture with shared and distributed memory and can be used as effective preconditioner. To check efficient of our solver we compare it with Intel MKL PARDISO - the high performance direct solver. Memory and performance tests demonstrate up to 3 times performance and memory gain for the 3D problems with more than (10^6 ) unknowns. Therefore, proposed multifrontal HSS solver can solve large problems, which cannot be resolved by direct solvers because of large memory consumptions. We give a short introduction to methods for the data-sparse approximation of matrices resulting from the discretisation of non-local operators occurring in boundary integral methods, as the inverses of partial differential operators or as solutions of control problems. The result of the approximation will be so-called hierarchical matrices (or short H-matrices). These matrices form a subset of the set of all matrices and have a data-sparse representation. The essential operations for these matrices (matrix-vector and matrix – matrix multiplication, addition and inversion) can be performed in, up to logarithmic factors, optimal complexity. We give a review of specialised variants of H-matrices, especially of H 2 -matrices, and finally consider applications of the different methods to problems from integral equations, partial differential equations and control theory. q 2003 Elsevier Science Ltd. All rights reserved. We propose a spectral partitioning approach for large-scale optimization problems, specifically structure from motion. In structure from motion, partitioning methods reduce the problem into smaller and better conditioned subproblems which can be efficiently optimized. Our partitioning method uses only the Hessian of the reprojection error and its eigenvector. We show that partitioned systems that preserve the eigenvectors corresponding to small eigenvalues result in lower residual error when optimized. We create partitions by clustering the entries of the eigenvectors of the Hessian corresponding to small eigenvalues. This is a more general technique than relying on domain knowledge and heuristics such as bottom-up structure from motion approaches. Simultaneously, it takes advantage of more information than generic matrix partitioning algorithms.