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University of Colorado
1.
Heavner, Nathan.
Building Rank-Revealing Factorizations with Randomization.
Degree: PhD, 2019, University of Colorado
URL: https://scholar.colorado.edu/appm_gradetds/155 
► This thesis describes a set of randomized algorithms for computing rank revealing factorizations of matrices. These algorithms are designed specifically to minimize the amount…
(more)
▼ This thesis describes a set of randomized algorithms for computing rank revealing
factorizations of matrices. These algorithms are designed specifically to minimize the amount of data movement required, which is essential to high practical performance on modern computing hardware. The work presented builds on existing randomized algorithms for computing low-rank approximations to matrices, but essentially ex- tends the range of applicability of these methods by allowing for the efficient decomposition of matrices of any numerical rank, including full rank matrices. In contrast, existing methods worked well only when the numerical rank was substantially smaller than the dimensions of the
matrix. The thesis describes algorithms for computing two of the most popular rank-revealing
matrix decom- positions: the column pivoted QR (CPQR) decomposition, and the so called UTV decomposition that factors a given
matrix A as A = UTV∗, where U and V have orthonormal columns and T is triangular. For each algorithm, the thesis presents algorithms that are tailored for different computing environments, including multicore shared memory processors, GPUs, distributed memory machines, and matrices that are stored on hard drives (“out of core”). The first chapter of the thesis consists of an introduction that provides context, reviews previous work in the field, and summarizes the key contributions. Beside the introduction, the thesis contains six additional chapters: Chapter 2 introduces a fully blocked algorithm HQRRP for computing a QR factorization with col- umn pivoting. The key to the full blocking of the algorithm lies in using randomized projections to create a low dimensional sketch of the data, where multiple good pivot columns may be cheaply computed. Nu- merical experiments show that HQRRP is several times faster than the classical algorithm for computing a column pivoted QR on a multicore machine, and the acceleration factor increases with the number of cores. Chapter 3 introduces randUTV, a randomized algorithm for computing a rank-revealing factorization of the form A = UTV∗, where U and V are orthogonal and T is upper triangular. RandUTV uses random- ized methods to efficiently build U and V as approximations of the column and row spaces of A. The result is an algorithm that reveals rank nearly as well as the SVD and costs at most as much as a column pivoted QR. Chapter 4 provides optimized implementations for shared and distributed memory architectures. For shared memory, we show that formulating randUTV as an algorithm-by-blocks increases its efficiency in parallel. The fifth chapter implements randUTV on the GPU and augments the algorithm with an over- sampling technique to further increase the low rank approximation properties of the resulting factorization. Chapter 6 implements both randUTV and HQRRP for use with matrices stored out of core. It is shown that reorganizing HQRRP as a left-looking algorithm to reduce the number of writes to the drive is in the tested cases…
Advisors/Committee Members: Per-Gunnar Martinsson, Stephen Becker, Gregory Beylkin, Gregorio Quintana-Ortí, Christian Ketelsen.
Subjects/Keywords: linear algebra; matrix factorizations; randomization; rank-revealing factorizations; Applied Mathematics
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APA (6th Edition):
Heavner, N. (2019). Building Rank-Revealing Factorizations with Randomization. (Doctoral Dissertation). University of Colorado. Retrieved from https://scholar.colorado.edu/appm_gradetds/155
Chicago Manual of Style (16th Edition):
Heavner, Nathan. “Building Rank-Revealing Factorizations with Randomization.” 2019. Doctoral Dissertation, University of Colorado. Accessed April 13, 2021.
https://scholar.colorado.edu/appm_gradetds/155.
MLA Handbook (7th Edition):
Heavner, Nathan. “Building Rank-Revealing Factorizations with Randomization.” 2019. Web. 13 Apr 2021.
Vancouver:
Heavner N. Building Rank-Revealing Factorizations with Randomization. [Internet] [Doctoral dissertation]. University of Colorado; 2019. [cited 2021 Apr 13].
Available from: https://scholar.colorado.edu/appm_gradetds/155.
Council of Science Editors:
Heavner N. Building Rank-Revealing Factorizations with Randomization. [Doctoral Dissertation]. University of Colorado; 2019. Available from: https://scholar.colorado.edu/appm_gradetds/155
University of Saskatchewan
2.
Vlahu, Izabela 1986-.
The Orthogonal Band Decomposition of the Finite Dirichlet Matrix and its Applications.
Degree: 2018, University of Saskatchewan
URL: http://hdl.handle.net/10388/8523
► In my work I establish and extend the theory of finite D-matrices for the purposes of signal processing applications in the finite, digital setting. Finite…
(more)
▼ In my work I establish and extend the theory of finite D-matrices for the purposes of signal processing applications in the finite, digital setting. Finite D-matrices are obtained by truncating infinite D-matrices to upper-left corners. I show that finite D-matrices are furnished with a number-theoretical structure that is not present in their infinite counterparts. In particular, I show that the columns of every finite D-
matrix of size N × N admits a natural, non-trivial, Orthogonal Band Decomposition, induced by the Floor Band Decomposition on the finite set {1,2,\dots,N}. When the D-
matrix is invertible, its Orthogonal Band Decomposition induces a non-trivial resolution of the identity. Furthermore, for every finite D-
matrix A, I show that the sum P of the orthogonal projections corresponding to each band of A admits the following sparse representation P=AΛ
-1 A^*, where Λ is a special diagonal
matrix and A^\star is the Hermitian adjoint of A. I also show that the
matrix P and its inverse induce another non-trivial resolution of the identity. Being a sum of projection matrices, I call the
matrix P the associated P-
matrix of A.
Both the finite D-matrices and their associated P-matrices can be applied in the processing of digital signals. For example, given a D-
matrix A, its associated P-
matrix allows us to pass from a signal representation in the Fourier basis to a representation, as a sum of projections, in the basis induced by the Orthogonal Band Decomposition of A. Preliminary experiments suggest that the error of approximating signals with partial sums of projections might offer a more suitable metric to choose D-
matrix representations in specific applications. Significantly, computations with finite D-matrices and P-matrices can be carried out via fast algorithms, which makes these transforms computationally competitive.
Advisors/Committee Members: Sowa, Artur, Babyn, Paul, Bui, Francis, Frank, Cameron, Soteros, Christine.
Subjects/Keywords: Finite Dirichlet Matrix; Non-trivial factorizations; resolutions of identity
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APA (6th Edition):
Vlahu, I. 1. (2018). The Orthogonal Band Decomposition of the Finite Dirichlet Matrix and its Applications. (Thesis). University of Saskatchewan. Retrieved from http://hdl.handle.net/10388/8523
Note: this citation may be lacking information needed for this citation format:
Not specified: Masters Thesis or Doctoral Dissertation
Chicago Manual of Style (16th Edition):
Vlahu, Izabela 1986-. “The Orthogonal Band Decomposition of the Finite Dirichlet Matrix and its Applications.” 2018. Thesis, University of Saskatchewan. Accessed April 13, 2021.
http://hdl.handle.net/10388/8523.
Note: this citation may be lacking information needed for this citation format:
Not specified: Masters Thesis or Doctoral Dissertation
MLA Handbook (7th Edition):
Vlahu, Izabela 1986-. “The Orthogonal Band Decomposition of the Finite Dirichlet Matrix and its Applications.” 2018. Web. 13 Apr 2021.
Vancouver:
Vlahu I1. The Orthogonal Band Decomposition of the Finite Dirichlet Matrix and its Applications. [Internet] [Thesis]. University of Saskatchewan; 2018. [cited 2021 Apr 13].
Available from: http://hdl.handle.net/10388/8523.
Note: this citation may be lacking information needed for this citation format:
Not specified: Masters Thesis or Doctoral Dissertation
Council of Science Editors:
Vlahu I1. The Orthogonal Band Decomposition of the Finite Dirichlet Matrix and its Applications. [Thesis]. University of Saskatchewan; 2018. Available from: http://hdl.handle.net/10388/8523
Note: this citation may be lacking information needed for this citation format:
Not specified: Masters Thesis or Doctoral Dissertation
University of Illinois – Urbana-Champaign
3.
Mastroeni, Matthew N.
Betti numbers of Koszul algebras and codimension two matrix factorizations.
Degree: PhD, Mathematics, 2018, University of Illinois – Urbana-Champaign
URL: http://hdl.handle.net/2142/101658
► This thesis consists of two projects on the structure of free resolutions in commutative algebra. After developing some necessary background, we prove a structure theorem…
(more)
▼ This thesis consists of two projects on the structure of free resolutions in commutative algebra. After developing some necessary background, we prove a structure theorem in Chapter 3 for the defining ideals of Koszul almost complete intersections and, in the process, give an affirmative answer for all such rings to a question of Avramov, Conca, and Iyengar about the Betti numbers of Koszul algebras. In Chapter 4, we study the codimension two
matrix factorizations of Eisenbud and Peeva. Each
matrix factorization compactly encodes the data of a free resolution of its corresponding
matrix factorization module. By showing that each
matrix factorization also encodes a canonical system of higher homotopies on this free resolution, we are able to construct a functor from codimension two
matrix factorizations to the singularity category of the corresponding complete intersection. This represents the first step towards reconciling higher codimension
matrix factorizations with known generalizations of a theorem of Buchweitz and Orlov in the hypersurface case.
Advisors/Committee Members: Schenck, Hal (advisor), Katz, Sheldon (Committee Chair), Dutta, Sankar (committee member), Griffith, Phil (committee member).
Subjects/Keywords: Koszul algebras; almost complete intersections; Betti numbers; free resolutions; matrix factorizations
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APA (6th Edition):
Mastroeni, M. N. (2018). Betti numbers of Koszul algebras and codimension two matrix factorizations. (Doctoral Dissertation). University of Illinois – Urbana-Champaign. Retrieved from http://hdl.handle.net/2142/101658
Chicago Manual of Style (16th Edition):
Mastroeni, Matthew N. “Betti numbers of Koszul algebras and codimension two matrix factorizations.” 2018. Doctoral Dissertation, University of Illinois – Urbana-Champaign. Accessed April 13, 2021.
http://hdl.handle.net/2142/101658.
MLA Handbook (7th Edition):
Mastroeni, Matthew N. “Betti numbers of Koszul algebras and codimension two matrix factorizations.” 2018. Web. 13 Apr 2021.
Vancouver:
Mastroeni MN. Betti numbers of Koszul algebras and codimension two matrix factorizations. [Internet] [Doctoral dissertation]. University of Illinois – Urbana-Champaign; 2018. [cited 2021 Apr 13].
Available from: http://hdl.handle.net/2142/101658.
Council of Science Editors:
Mastroeni MN. Betti numbers of Koszul algebras and codimension two matrix factorizations. [Doctoral Dissertation]. University of Illinois – Urbana-Champaign; 2018. Available from: http://hdl.handle.net/2142/101658
University of Texas – Austin
4.
Wong, Michael Andrew.
Dimer models and Hochschild cohomology.
Degree: PhD, Mathematics, 2018, University of Texas – Austin
URL: http://hdl.handle.net/2152/68467
► Dimer models have appeared in the context of noncommutative crepant resolutions and homological mirror symmetry for punctured Riemann surfaces. For a zigzag consistent dimer embedded…
(more)
▼ Dimer models have appeared in the context of noncommutative crepant resolutions and homological mirror symmetry for punctured Riemann surfaces. For a zigzag consistent dimer embedded in a torus, we explicitly describe the Hochschild cohomology of its Jacobi algebra in terms of dimer combinatorics. We then compute the compactly supported Hochschild cohomology of the category of
matrix factorizations for the Jacobi algebra with its canonical potential.
Advisors/Committee Members: Ben-Zvi, David, 1974- (advisor), Schedler, Travis (advisor), Neitzke, Andrew (committee member), Perutz, Timothy (committee member).
Subjects/Keywords: Dimer models; Matrix factorizations; Hochschild cohomology; Mirror symmetry; Noncommutative geometry
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APA (6th Edition):
Wong, M. A. (2018). Dimer models and Hochschild cohomology. (Doctoral Dissertation). University of Texas – Austin. Retrieved from http://hdl.handle.net/2152/68467
Chicago Manual of Style (16th Edition):
Wong, Michael Andrew. “Dimer models and Hochschild cohomology.” 2018. Doctoral Dissertation, University of Texas – Austin. Accessed April 13, 2021.
http://hdl.handle.net/2152/68467.
MLA Handbook (7th Edition):
Wong, Michael Andrew. “Dimer models and Hochschild cohomology.” 2018. Web. 13 Apr 2021.
Vancouver:
Wong MA. Dimer models and Hochschild cohomology. [Internet] [Doctoral dissertation]. University of Texas – Austin; 2018. [cited 2021 Apr 13].
Available from: http://hdl.handle.net/2152/68467.
Council of Science Editors:
Wong MA. Dimer models and Hochschild cohomology. [Doctoral Dissertation]. University of Texas – Austin; 2018. Available from: http://hdl.handle.net/2152/68467
Texas A&M University
5.
Escobedo, Adolfo Raphael.
Foundational Factorization Algorithms for the Efficient Roundoff-Error-Free Solution of Optimization Problems.
Degree: PhD, Industrial Engineering, 2016, Texas A&M University
URL: http://hdl.handle.net/1969.1/157772
► LU and Cholesky factorizations play a central role in solving linear and mixed-integer programs. In many documented cases, the round-off errors accrued during the construction…
(more)
▼ LU and Cholesky
factorizations play a central role in solving linear and mixed-integer programs. In many documented cases, the round-off errors accrued during the construction and implementation of these
factorizations cause the misclassification of suboptimal solutions as optimal and infeasible problems as feasible and vice versa.
Such erroneous outputs bring the reliability of optimization solvers into question and, therefore, it is imperative to eliminate these round off errors altogether and to do so efficiently to ensure practicality.
Firstly, this work introduces two round off-error-free
factorizations (REF) constructed exclusively in integer arithmetic: the REF LU and Cholesky
factorizations.
Additionally, it develops supplementary integer-preserving substitution algorithms, thereby providing a complete tool set for solving systems of linear equations (SLEs) exactly and efficiently. An inherent property of the REF factorization algorithms is that their entries' bit-length – i.e., the number of bits required for expression – is bounded polynomially. Unlike the exact rational arithmetic methods used in practice, however, the algorithms herein presented do not require any greatest common divisor operations to guarantee this pivotal property.
Secondly, this work derives various useful theoretical results and details computational tests to demonstrate that the REF factorization framework is considerably superior to the rational arithmetic LU factorization approach in computational performance and storage requirements. This is significant because the latter approach is the solution validation tool of choice of state-of-the-art exact linear programming solvers due to its ability to handle both numerically difficult and intricate problems. An additional theoretical contribution and further computational tests also demonstrate the predominance of the featured framework over Q-matrices, which comprise an alternative integer-preserving approach relying on the basis adjunct
matrix.
Thirdly, this work develops special algorithms for updating the REF
factorizations.
This is necessary because applying the traditional approach to the REF
factorizations is inefficient in terms of entry growth and computational effort. In fact, these inefficiencies virtually wipe out all the computational savings commonly expected of factorization updates. Hence, the current work develops REF update algorithms that differ significantly from their traditional counterparts. The featured REF updates are column/row addition, deletion, and replacement.
Advisors/Committee Members: Moreno-Centeno, Erick (advisor), Butenko, Sergiy (committee member), Wilhelm, Wilbert E (committee member), Yan, Catherine (committee member).
Subjects/Keywords: Exact mathematical programming; exact algorithms; matrix factorizations; roundoff errors; solving linear systems; factorization update algorithms.
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APA ·
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APA (6th Edition):
Escobedo, A. R. (2016). Foundational Factorization Algorithms for the Efficient Roundoff-Error-Free Solution of Optimization Problems. (Doctoral Dissertation). Texas A&M University. Retrieved from http://hdl.handle.net/1969.1/157772
Chicago Manual of Style (16th Edition):
Escobedo, Adolfo Raphael. “Foundational Factorization Algorithms for the Efficient Roundoff-Error-Free Solution of Optimization Problems.” 2016. Doctoral Dissertation, Texas A&M University. Accessed April 13, 2021.
http://hdl.handle.net/1969.1/157772.
MLA Handbook (7th Edition):
Escobedo, Adolfo Raphael. “Foundational Factorization Algorithms for the Efficient Roundoff-Error-Free Solution of Optimization Problems.” 2016. Web. 13 Apr 2021.
Vancouver:
Escobedo AR. Foundational Factorization Algorithms for the Efficient Roundoff-Error-Free Solution of Optimization Problems. [Internet] [Doctoral dissertation]. Texas A&M University; 2016. [cited 2021 Apr 13].
Available from: http://hdl.handle.net/1969.1/157772.
Council of Science Editors:
Escobedo AR. Foundational Factorization Algorithms for the Efficient Roundoff-Error-Free Solution of Optimization Problems. [Doctoral Dissertation]. Texas A&M University; 2016. Available from: http://hdl.handle.net/1969.1/157772
6.
Platt, David.
Chern Character for Global Matrix Factorizations.
Degree: PhD, Department of Mathematics, 2013, University of Oregon
URL: http://hdl.handle.net/1794/13244
► We give a formula for the Chern character on the DG category of global matrix factorizations on a smooth scheme X with superpotential w∈ Γ(\OX).…
(more)
▼ We give a formula for the Chern character on the DG category of global
matrix factorizations on a smooth scheme X with superpotential w∈ Γ(\O
X). Our formula takes values in a Cech model for Hochschild homology. Our methods may also be adapted to get an explicit formula for the Chern character for perfect complexes of sheaves on X taking values in right derived global sections of the De-Rham algebra. Along the way we prove that the DG version of the Chern Character coincides with the classical one for perfect complexes.
Advisors/Committee Members: Polishchuk, Alexander (advisor).
Subjects/Keywords: Chern Character; Matrix Factorizations; Noncommutative Geometry
…Factorizations . . . . . . . . . . . . . . . . . . . . . . .
28
III. HOMOTOPY THEORY OF MATRIX… …Homotopy Theory of Matrix Factorizations . . . . . . . . . . .
49
IV. HOCHSCHILD HOMOLOGY… …version of matrix factorizations (taken from [PV1]). Section III
contains… …contains results on the homotopy theory of matrix factorizations. Much of
the work therein is… …between modules over matrix factorizations on one
hand and quasi-coherent curved modules (…
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APA (6th Edition):
Platt, D. (2013). Chern Character for Global Matrix Factorizations. (Doctoral Dissertation). University of Oregon. Retrieved from http://hdl.handle.net/1794/13244
Chicago Manual of Style (16th Edition):
Platt, David. “Chern Character for Global Matrix Factorizations.” 2013. Doctoral Dissertation, University of Oregon. Accessed April 13, 2021.
http://hdl.handle.net/1794/13244.
MLA Handbook (7th Edition):
Platt, David. “Chern Character for Global Matrix Factorizations.” 2013. Web. 13 Apr 2021.
Vancouver:
Platt D. Chern Character for Global Matrix Factorizations. [Internet] [Doctoral dissertation]. University of Oregon; 2013. [cited 2021 Apr 13].
Available from: http://hdl.handle.net/1794/13244.
Council of Science Editors:
Platt D. Chern Character for Global Matrix Factorizations. [Doctoral Dissertation]. University of Oregon; 2013. Available from: http://hdl.handle.net/1794/13244
University of Toronto
7.
Hovinen, Bradford.
Matrix Factorizations of the Classical Discriminant.
Degree: 2009, University of Toronto
URL: http://hdl.handle.net/1807/17466
► The classical discriminant Dn of degree n polynomials detects whether a given univariate polynomial f has a repeated root. It is itself a polynomial in…
(more)
▼ The classical discriminant Dn of degree n polynomials detects whether a given univariate polynomial f has a repeated root. It is itself a polynomial in the coefficients of f which, according to several classical results by Bézout, Sylvester, Cayley, and others, may be expressed as the determinant of a matrix whose entries are much simpler polynomials in the coefficients of f. This thesis is concerned with the construction and classification of such determinantal formulae for Dn.
In particular, all of the formulae for Dn appearing in the classical literature are equivalent in the sense that the cokernels of their associated matrices are isomorphic as modules over the associated polynomial ring. This begs the question of whether there exist formulae which are not equivalent to the classical formulae and not trivial in the sense of having the same cokernel as the 1 x 1 matrix (Dn).
The results of this thesis lie in two directions. First, we construct an explicit non-classical formula: the presentation matrix of the open swallowtail first studied by Arnol'd and Givental. We study the properties of this formula, contrasting them with the properties of the classical formulae.
Second, for the discriminant of the polynomial x4+a2x2+a3x+a4 we embark on a classification of determinantal formulae which are homogeneous in the sense that the cokernels of their associated matrices are graded modules with respect to the grading deg ai=i. To this end, we use deformation theory: a moduli space of such modules arises from the base spaces of versal deformations of certain modules over the E6 singularity {x4-y3}. The method developed here can in principle be used to classify determinantal formulae for all discriminants, and, indeed, for all singularities.
PhD
Advisors/Committee Members: Ragnar-Olaf, Buchweitz, Mathematics.
Subjects/Keywords: commutative algebra; algebraic geometry; discriminants; singularities; matrix factorizations; homological algebra; 0405
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APA ·
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APA (6th Edition):
Hovinen, B. (2009). Matrix Factorizations of the Classical Discriminant. (Doctoral Dissertation). University of Toronto. Retrieved from http://hdl.handle.net/1807/17466
Chicago Manual of Style (16th Edition):
Hovinen, Bradford. “Matrix Factorizations of the Classical Discriminant.” 2009. Doctoral Dissertation, University of Toronto. Accessed April 13, 2021.
http://hdl.handle.net/1807/17466.
MLA Handbook (7th Edition):
Hovinen, Bradford. “Matrix Factorizations of the Classical Discriminant.” 2009. Web. 13 Apr 2021.
Vancouver:
Hovinen B. Matrix Factorizations of the Classical Discriminant. [Internet] [Doctoral dissertation]. University of Toronto; 2009. [cited 2021 Apr 13].
Available from: http://hdl.handle.net/1807/17466.
Council of Science Editors:
Hovinen B. Matrix Factorizations of the Classical Discriminant. [Doctoral Dissertation]. University of Toronto; 2009. Available from: http://hdl.handle.net/1807/17466
8.
Pippi, Massimo.
Catégories des singularités, factorisations matricielles et cycles évanescents : Categories of singularities, matrix factorizations and vanishing cycles.
Degree: Docteur es, Mathématiques et Applications, 2020, Université Toulouse III – Paul Sabatier
URL: http://www.theses.fr/2020TOU30049
► Le but de cette thèse est d'étudier les dg-catégories de singularités Sing(X, s), associées à des couples (X, s), où X est un schéma et…
(more)
▼ Le but de cette thèse est d'étudier les dg-catégories de singularités Sing(X, s), associées à des couples (X, s), où X est un schéma et s est une section d'un fibré vectoriel sur X. La dg-catégorie Sing(X, s) est définie comme le noyau du dg foncteur de Sing(X0) vers Sing(X) induit par l'image directe le long de l'inclusion du lieu de zéros (dérivé) X0 de s dans X. Dans une première partie, nous supposons que le fibré vectoriel est trivial de rang n. On démontre alors un théorème de structure pour Sing(X, s) dans le cas où X = Spec(B) est affine. Cet énoncé affirme que tout objet de Sing(X, s) est représenté par un complexe de B-modules concentré dans n+1 degrés. Lorsque n = 1, cet énoncé généralise l'équivalence d'Orlov , qui identifie Sing(X, s) avec la dg-catégorie des factorisations matricielles MF(X, s), au cas où s epsilon OX(X) n'est pas nécessairement plat. Dans une seconde partie, nous étudions la cohomologie l-adique de Sing(X, s) (définie par A. Blanc - M. Robalo - B. Toën and G. Vezzosi), où s est une section globale d'un fibré en droites. Pour cela, on introduit le faisceau l-adique des cycles évanescents invariantes par monodromie. En utilisant un théorème de D. Orlov généralisé par J. Burke et M. Walker, on calcule la réalisation l-adique de Sing(Spec(B), (f1 ,..., fn)) pour (f1 ,..., fn) epsilon Bn. Dans le dernier chapitre, nous introduisons les faisceaux l-adiques des cycles évanescents itérés pour un schéma sur un anneau de valuation discrète de rang 2. On relie ces faisceaux l-adiques à la réalisation l-adique des dg catégories de singularités des fibres prises sur certains sous-schémas fermés de la base.
The aim of this thesis is to study the dg categories of singularities Sing(X, s) of pairs (X, s), where X is a scheme and s is a global section of some vector bundle over X. Sing(X, s) is defined as the kernel of the dg functor from Sing(X0) to Sing(X) induced by the pushforward along the inclusion of the (derived) zero locus X0 of s in X. In the first part, we restrict ourselves to the case where the vector bundle is trivial. We prove a structure theorem for Sing(X, s) when X = Spec(B) is affine. Roughly, it tells us that every object in Sing(X, s) is represented by a complex of B-modules concentrated in n + 1 consecutive degrees (if s epsilon Bn). By specializing to the case n = 1, we generalize Orlov's theorem, which identifies Sing(X, s) with the dg category of matrix factorizations MF(X, s), to the case where s epsilon OX(X) is not flat. In the second part, we study the l-adic cohomology of Sing(X, s) (as defined by A. Blanc - M. Robalo - B. Toën and G. Vezzosi) when s is a global section of a line bundle. In order to do so, we introduce the l-adic sheaf of monodromy-invariant vanishing cycles. Using a theorem of D. Orlov generalized by J. Burke and M. Walker, we compute the l-adic realization of Sing(Spec(B), (f1 ,..., fn)) for (f1 ,..., fn) epsilon Bn. In the last chapter, we introduce the l-adic sheaves of iterated vanishing cycles of a scheme over a discrete valuation ring of rank 2.…
Advisors/Committee Members: Toën, Bertrand (thesis director), Vezzosi, Gabriele (thesis director).
Subjects/Keywords: Géométrie algébrique dérivée; Géométrie non-commutative; Cycles évanescents; Dg-catégories des singularités; Factorisations matricielles; Réalisations motivique et l-adique des dg-catégories; Derived algebraic geometry; Non-commutative geometry; Vanishing cycles; Dg categories of singularitie; Matrix factorizations; Motivic and`-adic realizationsof dg categories
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APA ·
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MLA ·
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Manager
APA (6th Edition):
Pippi, M. (2020). Catégories des singularités, factorisations matricielles et cycles évanescents : Categories of singularities, matrix factorizations and vanishing cycles. (Doctoral Dissertation). Université Toulouse III – Paul Sabatier. Retrieved from http://www.theses.fr/2020TOU30049
Chicago Manual of Style (16th Edition):
Pippi, Massimo. “Catégories des singularités, factorisations matricielles et cycles évanescents : Categories of singularities, matrix factorizations and vanishing cycles.” 2020. Doctoral Dissertation, Université Toulouse III – Paul Sabatier. Accessed April 13, 2021.
http://www.theses.fr/2020TOU30049.
MLA Handbook (7th Edition):
Pippi, Massimo. “Catégories des singularités, factorisations matricielles et cycles évanescents : Categories of singularities, matrix factorizations and vanishing cycles.” 2020. Web. 13 Apr 2021.
Vancouver:
Pippi M. Catégories des singularités, factorisations matricielles et cycles évanescents : Categories of singularities, matrix factorizations and vanishing cycles. [Internet] [Doctoral dissertation]. Université Toulouse III – Paul Sabatier; 2020. [cited 2021 Apr 13].
Available from: http://www.theses.fr/2020TOU30049.
Council of Science Editors:
Pippi M. Catégories des singularités, factorisations matricielles et cycles évanescents : Categories of singularities, matrix factorizations and vanishing cycles. [Doctoral Dissertation]. Université Toulouse III – Paul Sabatier; 2020. Available from: http://www.theses.fr/2020TOU30049
9.
Jia, Yulu.
Algorithm-Based Fault Tolerance for Two-Sided Dense Matrix Factorizations.
Degree: 2015, University of Tennessee – Knoxville
URL: https://trace.tennessee.edu/utk_graddiss/3588
► The mean time between failure (MTBF) of large supercomputers is decreasing, and future exascale computers are expected to have a MTBF of around 30 minutes.…
(more)
▼ The mean time between failure (MTBF) of large supercomputers is decreasing, and future exascale computers are expected to have a MTBF of around 30 minutes. Therefore, it is urgent to prepare important algorithms for future machines with such a short MTBF. Eigenvalue problems (EVP) and singular value problems (SVP) are common in engineering and scientific research. Solving EVP and SVP numerically involves two-sided matrix factorizations: the Hessenberg reduction, the tridiagonal reduction, and the bidiagonal reduction. These three factorizations are computation intensive, and have long running times. They are prone to suffer from computer failures.
We designed algorithm-based fault tolerant (ABFT) algorithms for the parallel Hessenberg reduction and the parallel tridiagonal reduction. The ABFT algorithms target fail-stop errors. These two fault tolerant algorithms use a combination of ABFT and diskless checkpointing. ABFT is used to protect frequently modified data . We carefully design the ABFT algorithm so the checksums are valid at the end of each iterative cycle. Diskless checkpointing is used for rarely modified data. These checkpoints are in the form of checksums, which are small in size, so the time and storage cost to store them in main memory is small. Also, there are intermediate results which need to be protected for a short time window. We store a copy of this data on the neighboring process in the process grid.
We also designed algorithm-based fault tolerant algorithms for the CPU-GPU hybrid Hessenberg reduction algorithm and the CPU-GPU hybrid bidiagonal reduction algorithm. These two fault tolerant algorithms target silent errors. Our design employs both ABFT and diskless checkpointing to provide data redundancy. The low cost error detection uses two dot products and an equality test. The recovery protocol uses reverse computation to roll back the state of the matrix to a point where it is easy to locate and correct errors.
We provided theoretical analysis and experimental verification on the correctness and efficiency of our fault tolerant algorithm design. We also provided mathematical proof on the numerical stability of the factorization results after fault recovery. Experimental results corroborate with the mathematical proof that the impact is mild.
Subjects/Keywords: ABFT; fault tolerance; dense linear algebra; two-sided matrix factorizations; Hessenberg; checksum; Numerical Analysis and Scientific Computing
…explore methods to provide fault resilience for two-sided
dense matrix factorizations, namely… …dense matrix.
The common characteristics of these three factorizations are that their… …version for soft error resilience of
the two-side matrix factorizations for CPU-GPU hybrid… …errors in two-sided dense matrix
factorizations: hard errors and soft errors. We define hard… …checkpointing for the panel factorization result.
In the
two-sided matrix factorizations, the…
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APA (6th Edition):
Jia, Y. (2015). Algorithm-Based Fault Tolerance for Two-Sided Dense Matrix Factorizations. (Doctoral Dissertation). University of Tennessee – Knoxville. Retrieved from https://trace.tennessee.edu/utk_graddiss/3588
Chicago Manual of Style (16th Edition):
Jia, Yulu. “Algorithm-Based Fault Tolerance for Two-Sided Dense Matrix Factorizations.” 2015. Doctoral Dissertation, University of Tennessee – Knoxville. Accessed April 13, 2021.
https://trace.tennessee.edu/utk_graddiss/3588.
MLA Handbook (7th Edition):
Jia, Yulu. “Algorithm-Based Fault Tolerance for Two-Sided Dense Matrix Factorizations.” 2015. Web. 13 Apr 2021.
Vancouver:
Jia Y. Algorithm-Based Fault Tolerance for Two-Sided Dense Matrix Factorizations. [Internet] [Doctoral dissertation]. University of Tennessee – Knoxville; 2015. [cited 2021 Apr 13].
Available from: https://trace.tennessee.edu/utk_graddiss/3588.
Council of Science Editors:
Jia Y. Algorithm-Based Fault Tolerance for Two-Sided Dense Matrix Factorizations. [Doctoral Dissertation]. University of Tennessee – Knoxville; 2015. Available from: https://trace.tennessee.edu/utk_graddiss/3588
. 
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