Gil Shklarski

Interactive topology-aware surface reconstruction

The reconstruction of a complete watertight model from scan data is still a difficult process. In particular, since scanned data is often incomplete, the reconstruction of the expected shape is an ill-posed problem. Techniques that reconstruct poorly-sampled areas without any user intervention fail in many cases to faithfully reconstruct the topology of the model. The method that we introduce in this paper is topology-aware: it uses minimal user input to make correct decisions at regions where the topology of the model cannot be automatically induced with a reasonable degree of confidence. We first construct a continuous function over a three-dimensional domain. This function is constructed by minimizing a penalty function combining the data points, user constraints, and a regularization term. The optimization problem is formulated in a mesh-independent manner, and mapped onto a specific mesh using the finite-element method. The zero level-set of this function is a first approximation of the reconstructed surface. At complex under-sampled regions, the constraints might be insufficient. Hence, we analyze the local topological stability of the zero level-set to detect weak regions of the surface. These regions are suggested to the user for adding local inside/outside constraints by merely scribbling over a 2D tablet. Each new user constraint modifies the minimization problem, which is solved incrementally. The process is repeated, converging to a topology-stable reconstruction. Reconstructions of models acquired by a structured-light scanner with a small number of scribbles demonstrate the effectiveness of the method.

Parallel and fully recursive multifrontal sparse Cholesky

We describe the design, implementation, and performance of a new parallel sparse Cholesky factorization code. The code uses a multifrontal factorization strategy. Operations on small dense submatrices are performed using new dense matrix subroutines that are part of the code, although the code can also use the BLAS and LAPACK. The new code is recursive at both the sparse and the dense levels, it uses a novel recursive data layout for dense submatrices, and it is parallelized using Cilk, an extension of C specifically designed to parallelize recursive codes. We demonstrate that the new code performs well and scales well on SMPs. In particular, on up to 16 processors, the code outperforms two state-of-the-art message-passing codes. The scalability and high performance that the code achieves imply that recursive schedules, blocked data layouts, and dynamic scheduling are effective in the implementation of sparse factorization codes.

Parallel and Fully Recursive Multifrontal Supernodal Sparse Cholesky

We describe the design, implementation, and performance of a new parallel sparse Cholesky factorization code. The code uses a supernodal multifrontal factorization strategy. Operations on small dense submatrices are performed using new dense-matrix subroutines that are part of the code, although the code can also use the BLAS and LAPACK. The new code is recursive at both the sparse and the dense levels, it uses a novel recursive data layout for dense submatrices, and it is parallelized using Cilk, an extension of C specifically designed to parallelize recursive codes. We demonstrate that the new code performs well and scales well on SMPs.

Partitioned Triangular Tridiagonalization

We present a partitioned algorithm for reducing a symmetric matrix to a tridiagonal form, with partial pivoting. That is, the algorithm computes a factorization PAPT = LTLT, where, P is a permutation matrix, L is lower triangular with a unit diagonal and entries’ magnitudes bounded by 1, and T is symmetric and tridiagonal. The algorithm is based on the basic (nonpartitioned) methods of Parlett and Reid and of Aasen. We show that our factorization algorithm is componentwise backward stable (provided that the growth factor is not too large), with a similar behavior to that of Aasen’s basic algorithm. Our implementation also computes the QR factorization of T and solves linear systems of equations using the computed factorization. The partitioning allows our algorithm to exploit modern computer architectures (in particular, cache memories and high-performance blas libraries). Experimental results demonstrate that our algorithms achieve approximately the same level of performance as the partitioned Bunch-Kaufman factor and solve routines in lapack.

Rigidity in Finite-Element Matrices: Sufficient Conditions for the Rigidity of Structures and Substructures

We present an algebraic theory of rigidity for finite-element matrices. The theory provides a formal algebraic definition of finite-element matrices; notions of rigidity of finite-element matrices and of mutual rigidity between two such matrices; and sufficient conditions for rigidity and mutual rigidity. We also present a novel sparsification technique, called fretsaw extension, for finite-element matrices. We show that this sparsification technique generates matrices that are mutually rigid with the original matrix. We also show that one particular construction algorithm for fretsaw extensions generates matrices that can be factored with essentially no fill. This algorithm can be used to construct preconditioners for finite-element matrices. Both our theory and our algorithms are applicable to a wide range of finite-element matrices, including matrices arising from finite-element discretizations of both scalar and vector partial differential equations (e.g., electrostatics and linear elasticity). Both the theory and the algorithms are purely algebraic-combinatorial. They manipulate only the element matrices and are oblivious to the geometry, the material properties, and the discretization details of the underlying continuous problem.

Parallel unsymmetric-pattern multifrontal sparse LU with column preordering

We present a new parallel sparse LU factorization algorithm and code. The algorithm uses a column-preordering partial-pivoting unsymmetric-pattern multifrontal approach. Our baseline sequential algorithm is based on UMFPACK 4, but is somewhat simpler and is often somewhat faster than UMFPACK version 4.0. Our parallel algorithm is designed for shared-memory machines with a small or moderate number of processors (we tested it on up to 32 processors). We experimentally compare our algorithm with SuperLU_MT, an existing shared-memory sparse LU factorization with partial pivoting. SuperLU_MT scales better than our new algorithm, but our algorithm is more reliable and is usually faster. More specifically, on matrices that are costly to factor, our algorithm is usually faster on up to 4 processors, and is usually faster on 8 and 16. We were not able to run SuperLU_MT on 32. The main contribution of this article is showing that the column-preordering partial-pivoting unsymmetric-pattern multifrontal approach, developed as a sequential algorithm by Davis in several recent versions of UMFPACK, can be effectively parallelized.

Combinatorial Preconditioners for Scalar Elliptic Finite-Element Problems

We present a new preconditioner for linear systems arising from finite-element discretizations of scalar elliptic partial differential equations (PDEs). The solver splits the collection

Depersonalizing location traces

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STATISTICAL SECURITY FOR ANONYMOUS MESH-UP ORIENTED ONLINE SERVICES

of element matrices into a subset of matrices that are approximable by diagonally dominant matrices and a subset of matrices that are not approximable. The approximable

Computing the null space of finite element problems

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