Gil Shklarski

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Publications

  • Partitioned Triangular Tridiagonalization

    ACM Transactions on Mathematical Software (TOMS)

    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…

    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.

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  • Computing the null space of finite element problems

    Computer Methods in Applied Mechanics and Engineering

    We present a method for computing the null space of finite element models, including models with equality constraints. The method is purely algebraic; it requires access to the element matrices, but not to the geometry or material properties of the model.

    Theoretical considerations show that under certain conditions, both the amount of computation and the amount of memory required by our method scale linearly with model size; memory scales linearly but computation scales quadratically…

    We present a method for computing the null space of finite element models, including models with equality constraints. The method is purely algebraic; it requires access to the element matrices, but not to the geometry or material properties of the model.

    Theoretical considerations show that under certain conditions, both the amount of computation and the amount of memory required by our method scale linearly with model size; memory scales linearly but computation scales quadratically with the dimension of the null space. Our experiments confirm this: the method scales extremely well on 3-dimensional model problems. In general, large industrial models do not satisfy all the conditions that the theoretical results assume; however, experimentally the method performs well and outperforms an established method on industrial models, including models with many equality constraints.

    The accuracy of the computed null vectors is acceptable, but the method is usually less accurate than a more naive (and computationally much more expensive) method.

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  • Combinatorial preconditioners for scalar elliptic finite-element problems

    SIAM Journal on Matrix Analysis and Applications

    We present a new preconditioner for linear systems arising from finite-element discretizations of scalar elliptic partial differential equations (PDE's). The solver splits the collection $\{K_{e}\}$ 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 $K_{e}$'s are approximated by diagonally dominant matrices $L_{e}$'s that are assembled to form a global diagonally dominant matrix…

    We present a new preconditioner for linear systems arising from finite-element discretizations of scalar elliptic partial differential equations (PDE's). The solver splits the collection $\{K_{e}\}$ 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 $K_{e}$'s are approximated by diagonally dominant matrices $L_{e}$'s that are assembled to form a global diagonally dominant matrix L. A combinatorial graph algorithm then approximates L by another diagonally dominant matrix M that is easier to factor. Finally, M is added to the inapproximable elements to form the preconditioner, which is then factored. When all the element matrices are approximable, which is often the case, the preconditioner is provably efficient. Approximating element matrices by diagonally dominant ones is not a new idea, but we present a new approximation method which is both efficient and provably good. The splitting idea is simple and natural in the context of combinatorial preconditioners, but hard to exploit in other preconditioning paradigms. Experimental results show that on problems in which some of the $K_{e}$'s are ill conditioned, our new preconditioner is more effective than an algebraic multigrid solver, than an incomplete-factorization preconditioner, and than a direct solver.

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  • Improving Content Relevance With Automated Word Of Mouth

    IP.com Prior Art Database Disclosure

    The present solution improves the relevance of search results for any content that is rated, shared, or personally filtered. The solution utilizes social networks, such as Facebook™ or MySpace™, to provide the computational equivalent of "trust" and build an automated "word of mouth" algorithm as a user specific extended personal trust network (EPTN) composed of a set of friends, friends-of-friends, and the like. In operation, when a user inputs a search query, the results are cross-referenced…

    The present solution improves the relevance of search results for any content that is rated, shared, or personally filtered. The solution utilizes social networks, such as Facebook™ or MySpace™, to provide the computational equivalent of "trust" and build an automated "word of mouth" algorithm as a user specific extended personal trust network (EPTN) composed of a set of friends, friends-of-friends, and the like. In operation, when a user inputs a search query, the results are cross-referenced with the user's EPTN to provide a display of the most personally relevant search results. In addition, learning events are provided as user feedback on search result items that alter a user's EPTN such that subsequent search results are improved.

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  • Asynchronous Active Browser Cache Management for Efficient and Secure Web Applications

    IP.com Prior Art Database Disclosure

    Two major concerns with browsing the Internet are the time it takes for web pages to load and security. The solution introduces a novel concept in loading and running online applications (e.g. dynamic pages). The general concept of the solution is to replace passive browser cache management with explicit application driven asynchronous cache management. The overall effect of the solution to the user appears as if the application is installed locally and while the application is running, it…

    Two major concerns with browsing the Internet are the time it takes for web pages to load and security. The solution introduces a novel concept in loading and running online applications (e.g. dynamic pages). The general concept of the solution is to replace passive browser cache management with explicit application driven asynchronous cache management. The overall effect of the solution to the user appears as if the application is installed locally and while the application is running, it checks for a new online version. If a new version is available, it is installed in the background and becomes effective the next time the user launches the application.

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  • Parallel unsymmetric-pattern multifrontal sparse LU with column preordering

    ACM Transactions on Mathematical Software (TOMS)

    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…

    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.

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

    SIAM Journal on Matrix Analysis and Applications

    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…

    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.

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  • Interactive topology-aware surface reconstruction

    Proceedings of ACM SIGGRAPH 2007

    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 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.

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  • Parallel and fully recursive multifrontal sparse Cholesky

    Future Generation Computer Systems

    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…

    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.

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Patents

  • Extracting Facts from Unstructured Data

    Issued US 10,783,448

    Methods, systems, and apparatus, including computer programs encoded on computer storage media, to present a video. One of the methods includes obtaining one or more unstructured documents. The method includes obtaining, by a computer system, a data model, the data model identifying a type of fact that can be determined from the one or more unstructured documents. The method includes determining, by the computer system, a channel to extract facts from the document based on the type of fact. The…

    Methods, systems, and apparatus, including computer programs encoded on computer storage media, to present a video. One of the methods includes obtaining one or more unstructured documents. The method includes obtaining, by a computer system, a data model, the data model identifying a type of fact that can be determined from the one or more unstructured documents. The method includes determining, by the computer system, a channel to extract facts from the document based on the type of fact. The method includes distributing, by the computer system, the one or more unstructured documents to the channel. The method includes extracting, by the channel, facts from the one or more unstructured documents. The method also includes storing the facts in a data model.

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  • Statistical security for anonymous mesh-up oriented online services

    Issued US US 9160737

    Web pages and applications commonly consume functionality provided by services to provide users with a rich experience. For example, a backend mapping service may provide access to these services. However, the users and application consuming the services may be anonymous and unverified. Accordingly, a two ticket validation technique is provided to validate service execution requests from anonymous applications. In particular, a user is provided with a client ticket comprising a reputation. The…

    Web pages and applications commonly consume functionality provided by services to provide users with a rich experience. For example, a backend mapping service may provide access to these services. However, the users and application consuming the services may be anonymous and unverified. Accordingly, a two ticket validation technique is provided to validate service execution requests from anonymous applications. In particular, a user is provided with a client ticket comprising a reputation. The reputation may be adjusted over time based upon how the user consumes services. An application may request access to a service by providing the client ticket and an application ticket for validation. The reputation of the user may be used to determine an access level at which the application may access the service. Users with a high reputation may receive high quality access to the service, while users with a low reputation may receive lower quality access

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  • Systems and methods for event tracking using time-windowed counters

    Issued US US 9031883

    To allow for tracking events and classifying assets within a social networking system. A time series of occurrences of an event type associated with at least one asset is generated. A first signal value and a second signal value is determined based on the time series. The at least one asset is classified based on comparison of the first signal value and the second signal value. In an embodiment, the time series is based on at least one time window including time intervals. In an embodiment…

    To allow for tracking events and classifying assets within a social networking system. A time series of occurrences of an event type associated with at least one asset is generated. A first signal value and a second signal value is determined based on the time series. The at least one asset is classified based on comparison of the first signal value and the second signal value. In an embodiment, the time series is based on at least one time window including time intervals. In an embodiment, counters to determine a number of occurrences of an event type are associated with the time intervals. In an embodiment, each of the counters are incremented upon occurrence of the event type associated with the at least one asset during an associated time interval.

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  • Depersonalizing location traces

    Issued US US 8463289

    Positional information is provided while minimizing the possibility that personally identifiable information can be derived therefrom. Positional information is received in the form of trails that can be aggregated. Individual cells of a grid reflect a quantity of aggregated trails through those cells, an average intensity and direction of movement through those cells, or a more detailed distribution thereof. Alternatively, individual trails are aggregated to an aggregated trail in the form of…

    Positional information is provided while minimizing the possibility that personally identifiable information can be derived therefrom. Positional information is received in the form of trails that can be aggregated. Individual cells of a grid reflect a quantity of aggregated trails through those cells, an average intensity and direction of movement through those cells, or a more detailed distribution thereof. Alternatively, individual trails are aggregated to an aggregated trail in the form of a line. Further obfuscation of personally identifiable information occurs by resampling aggregated positional information, by introducing false positional information, or by falsely modifying existing positional information, in a manner that does not impact the overall aggregations, and by pruning, or deleting, positional information, especially around sensitive locations, such as a user's home, place of business, or other location that users typically would seek to keep private. Provision of positional information is delayed until a sufficient amount is received.

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