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Hierarchical interpolative factorization for elliptic operators: differential equations
 Comm. Pure Appl. Math
"... This paper introduces the hierarchical interpolative factorization for integral equations (HIFIE) associated with elliptic problems in two and three dimensions. This factorization takes the form of an approximate generalized LU decomposition that permits the efficient application of the discretiz ..."
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This paper introduces the hierarchical interpolative factorization for integral equations (HIFIE) associated with elliptic problems in two and three dimensions. This factorization takes the form of an approximate generalized LU decomposition that permits the efficient application of the discretized operator and its inverse. HIFIE is based on the recursive skeletonization algorithm but incorporates a novel combination of two key features: (1) a matrix factorization framework for sparsifying structured dense matrices and (2) a recursive dimensional reduction strategy to decrease the cost. Thus, higherdimensional problems are effectively mapped to one dimension, and we conjecture that constructing, applying, and inverting the factorization all have linear or quasilinear complexity. Numerical experiments support this claim and further demonstrate the performance of our algorithm as a generalized fast multipole method, direct solver, and preconditioner. HIFIE is compatible with geometric adaptivity and can handle both boundary and volume problems. MATLAB ® codes are freely available.
A FAST SEMIDIRECT LEAST SQUARES ALGORITHM FOR HIERARCHICALLY BLOCK SEPARABLE MATRICES∗
"... Abstract. We present a fast algorithm for linear least squares problems governed by hierarchically block separable (HBS) matrices. Such matrices are generally dense but data sparse and can describe many important operators including those derived from asymptotically smooth radial kernels that are ..."
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Abstract. We present a fast algorithm for linear least squares problems governed by hierarchically block separable (HBS) matrices. Such matrices are generally dense but data sparse and can describe many important operators including those derived from asymptotically smooth radial kernels that are not too oscillatory. The algorithm is based on a recursive skeletonization procedure that exposes this sparsity and solves the dense least squares problem as a larger, equalityconstrained, sparse one. It relies on a sparse QR factorization coupled with iterative weighted least squares methods. In essence, our scheme consists of a direct component, comprised of matrix compression and factorization, followed by an iterative component to enforce certain equality constraints. At most two iterations are typically required for problems that are not too illconditioned. For an M×N HBS matrix with M ≥ N having bounded offdiagonal block rank, the algorithm has optimal O(M +N) complexity. If the rank increases with the spatial dimension as is common for operators that are singular at the origin, then this becomes O(M +N) in one dimension, O(M +N3/2) in two dimensions, and O(M + N2) in three dimensions. We illustrate the performance of the method on both overdetermined and underdetermined systems in a variety of settings, with an emphasis on radial basis function approximation and efficient updating and downdating.
Sparsifying Preconditioner for the LippmannSchwinger Equation, ArXiv eprints (August 2014), available at 1408.4495
"... Abstract. The LippmannSchwinger equation is an integral equation formulation for acoustic and electromagnetic scattering from an inhomogeneous media and quantum scattering from a localized potential. We present the sparsifying preconditioner for accelerating the iterative solution of the Lippmann ..."
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Abstract. The LippmannSchwinger equation is an integral equation formulation for acoustic and electromagnetic scattering from an inhomogeneous media and quantum scattering from a localized potential. We present the sparsifying preconditioner for accelerating the iterative solution of the LippmannSchwinger equation. This new preconditioner transforms the discretized LippmannSchwinger equation into sparse form and leverages the efficient sparse linear algebra algorithms for computing an approximate inverse. This preconditioner is efficient and easy to implement. When combined with standard iterative methods, it results in almost frequencyindependent iteration counts. We provide 2D and 3D numerical results to demonstrate the effectiveness of this new preconditioner. 1.
Progress toward fast algorithms for protein design
"... I Structurefunction relationship is central to biochemistry I “Theorem”: structure = ⇒ function I Examples: ligandreceptor binding, DNA replication ..."
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I Structurefunction relationship is central to biochemistry I “Theorem”: structure = ⇒ function I Examples: ligandreceptor binding, DNA replication
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"... Abstract. The Lippmann–Schwinger equation is an integral equation formulation for acoustic and electromagnetic scattering from an inhomogeneous medium and quantum scattering from a localized potential. We present the sparsifying preconditioner for accelerating the iterative solution of the Lippman ..."
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Abstract. The Lippmann–Schwinger equation is an integral equation formulation for acoustic and electromagnetic scattering from an inhomogeneous medium and quantum scattering from a localized potential. We present the sparsifying preconditioner for accelerating the iterative solution of the Lippmann–Schwinger equation. This new preconditioner transforms the discretized Lippmann–Schwinger equation into sparse form and leverages the efficient sparse linear algebra algorithms for computing an approximate inverse. This preconditioner is efficient and easy to implement. When combined with standard iterative methods, it results in almost frequencyindependent iteration counts. We provide two and threedimensional numerical results to demonstrate the effectiveness of this new preconditioner. Key words. Lippmann–Schwinger equation, acoustic and electromagnetic scattering, quantum scattering, preconditioner, sparse linear algebra
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"... Abstract. The boundary integral method is an efficient approach for solving timeharmonic obstacle scattering problems from bounded scatterers. This paper presents the directional preconditioner for the linear systems of the boundary integral method in two dimensions. This new preconditioner build ..."
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Abstract. The boundary integral method is an efficient approach for solving timeharmonic obstacle scattering problems from bounded scatterers. This paper presents the directional preconditioner for the linear systems of the boundary integral method in two dimensions. This new preconditioner builds a datasparse approximation of the integral operator, transforms it into a sparse linear system, and computes an approximate inverse with efficient sparse linear algebra algorithms. This preconditioner is efficient and results in small and almost frequencyindependent iteration counts for nonresonant scatterers when combined with standard iterative solvers. Numerical results are provided to demonstrate the effectiveness of the new preconditioner.
A TECHNIQUE FOR UPDATING HIERARCHICAL SKELETONIZATIONBASED FACTORIZATIONS OF INTEGRAL OPERATORS
"... Abstract. We present a method for updating certain hierarchical factorizations for solving linear integral equations with elliptic kernels. In particular, given a factorization corresponding to some initial geometry or material parameters, we can locally perturb the geometry or coefficients and upda ..."
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Abstract. We present a method for updating certain hierarchical factorizations for solving linear integral equations with elliptic kernels. In particular, given a factorization corresponding to some initial geometry or material parameters, we can locally perturb the geometry or coefficients and update the initial factorization to reflect this change with asymptotic complexity that is polylogarithmic in the total number of unknowns and linear in the number of perturbed unknowns. We apply our method to the recursive skeletonization factorization and hierarchical interpolative factorization and demonstrate scaling results for a number of different 2D problem setups.