The Experts below are selected from a list of 360 Experts worldwide ranked by ideXlab platform
A. L. Sakhnovich - One of the best experts on this subject based on the ideXlab platform.
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nonlinear schrodinger equation in a semi strip evolution of the weyl titchmarsh function and recovery of the initial condition and Rectangular Matrix solutions from the boundary conditions
Journal of Mathematical Analysis and Applications, 2015Co-Authors: A. L. SakhnovichAbstract:Abstract Rectangular Matrix solutions of the defocusing nonlinear Schrodinger equation (dNLS) are studied in quarter-plane and semi-strip. Evolution of the corresponding Weyl–Titchmarsh (Weyl) function is described in terms of the initial Weyl function and boundary conditions. In the next step, the initial Weyl function is recovered (for the quarter-plane case) from the long-time asymptotics of the wave function considered at the boundary. Thus, it is shown that the evolution of the Weyl function is uniquely defined by the boundary conditions. Moreover, a procedure to recover solutions of dNLS (uniquely defined by the boundary conditions) is given. In a somewhat different way, the same boundary value problem is also dealt with in a semi-strip (for the case of a quasi-analytic initial condition).
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Skew-Self-Adjoint Dirac System with a Rectangular Matrix Potential: Weyl Theory, Direct and Inverse Problems
Integral Equations and Operator Theory, 2012Co-Authors: B. Fritzsche, B. Kirstein, I. Ya. Roitberg, A. L. SakhnovichAbstract:A non-classical Weyl theory is developed for skew-self-adjoint Dirac systems with Rectangular Matrix potentials. The notion of the Weyl function is introduced and direct and inverse problems are solved. A Borg–Marchenko type uniqueness result and the evolution of the Weyl function for the corresponding focusing nonlinear Schrödinger equation are also derived.
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recovery of the dirac system from the Rectangular weyl Matrix function
Inverse Problems, 2012Co-Authors: B. Fritzsche, B. Kirstein, Ya I Roitberg, A. L. SakhnovichAbstract:Weyl theory for Dirac systems with Rectangular Matrix potentials is non-classical. The corresponding Weyl functions are Rectangular Matrix functions. Furthermore, they are non-expansive in the upper semi-plane. Inverse problems are studied for such Weyl functions, and some results are new even for the square Weyl functions. High-energy asymptotics of Weyl functions and Borg–Marchenko-type uniqueness results are derived too.
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recovery of dirac system from the Rectangular weyl Matrix function
arXiv: Classical Analysis and ODEs, 2011Co-Authors: B. Fritzsche, B. Kirstein, Ya I Roitberg, A. L. SakhnovichAbstract:Weyl theory for Dirac systems with Rectangular Matrix potentials is non-classical. The corresponding Weyl functions are Rectangular Matrix functions. Furthermore, they are non-expansive in the upper semi-plane. Inverse problems are treated for such Weyl functions, and some results are new even for the square Weyl functions. High energy asymptotics of Weyl functions and Borg-Marchenko type uniqueness results are derived too.
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weyl theory and explicit solutions of direct and inverse problems for a dirac system with Rectangular Matrix potential
arXiv: Spectral Theory, 2011Co-Authors: B. Fritzsche, B. Kirstein, Ya I Roitberg, A. L. SakhnovichAbstract:A non-classical Weyl theory is developed for Dirac systems with Rectangular Matrix potentials. The notion of the Weyl function is introduced and the corresponding direct problem is treated. Furthermore, explicit solutions of the direct and inverse problems are obtained for the case of rational Weyl Matrix functions.
Francois Le Gall - One of the best experts on this subject based on the ideXlab platform.
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Barriers for Rectangular Matrix multiplication
2020Co-Authors: Christandl Matthias, Francois Le Gall, Lysikov Vladimir, Zuiddam JeroenAbstract:We study the algorithmic problem of multiplying large matrices that are Rectangular. We prove that the method that has been used to construct the fastest algorithms for Rectangular Matrix multiplication cannot give optimal algorithms. In fact, we prove a precise numerical barrier for this method. Our barrier improves the previously known barriers, both in the numerical sense, as well as in its generality. We prove our result using the asymptotic spectrum of tensors. More precisely, we crucially make use of two families of real tensor parameters with special algebraic properties: the quantum functionals and the support functionals. In particular, we prove that any lower bound on the dual exponent of Matrix multiplication $\alpha$ via the big Coppersmith-Winograd tensors cannot exceed 0.625
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improved Rectangular Matrix multiplication using powers of the coppersmith winograd tensor
Symposium on Discrete Algorithms, 2018Co-Authors: Francois Le Gall, Florent UrrutiaAbstract:In the past few years, successive improvements of the asymptotic complexity of square Matrix multiplication have been obtained by developing novel methods to analyze the powers of the Coppersmith-Winograd tensor, a basic construction introduced thirty years ago. In this paper we show how to generalize this approach to make progress on the complexity of Rectangular Matrix multiplication as well, by developing a framework to analyze powers of tensors in an asymmetric way. By applying this methodology to the fourth power of the Coppersmith-Winograd tensor, we succeed in improving the complexity of Rectangular Matrix multiplication. Let α denote the maximum value such that the product of an n × nα Matrix by an nα × n Matrix can be computed with O(n2+ϵ) arithmetic operations for any ϵ > 0. By analyzing the fourth power of the Coppersmith-Winograd tensor using our methods, we obtain the new lower bound α > 0.31389, which improves the previous lower bound α > 0.30298 obtained by Le Gall (FOCS'12) from the analysis of the second power of the Coppersmith-Winograd tensor. More generally, we give faster algorithms computing the product of an n × nk Matrix by an nk × n Matrix for any value k ≠ 1. (In the case k = 1, we recover the bounds recently obtained for square Matrix multiplication). These improvements immediately lead to improvements in the complexity of a multitude of fundamental problems for which the bottleneck is Rectangular Matrix multiplication, such as computing the all-pair shortest paths in directed graphs with bounded weights.
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improved Rectangular Matrix multiplication using powers of the coppersmith winograd tensor
arXiv: Data Structures and Algorithms, 2017Co-Authors: Francois Le Gall, Florent UrrutiaAbstract:In the past few years, successive improvements of the asymptotic complexity of square Matrix multiplication have been obtained by developing novel methods to analyze the powers of the Coppersmith-Winograd tensor, a basic construction introduced thirty years ago. In this paper we show how to generalize this approach to make progress on the complexity of Rectangular Matrix multiplication as well, by developing a framework to analyze powers of tensors in an asymmetric way. By applying this methodology to the fourth power of the Coppersmith-Winograd tensor, we succeed in improving the complexity of Rectangular Matrix multiplication. Let $\alpha$ denote the maximum value such that the product of an $n\times n^\alpha$ Matrix by an $n^\alpha\times n$ Matrix can be computed with $O(n^{2+\epsilon})$ arithmetic operations for any $\epsilon>0$. By analyzing the fourth power of the Coppersmith-Winograd tensor using our methods, we obtain the new lower bound $\alpha>0.31389$, which improves the previous lower bound $\alpha>0.30298$ obtained five years ago by Le Gall (FOCS'12) from the analysis of the second power of the Coppersmith-Winograd tensor. More generally, we give faster algorithms computing the product of an $n\times n^k$ Matrix by an $n^k\times n$ Matrix for any value $k\neq 1$. (In the case $k=1$, we recover the bounds recently obtained for square Matrix multiplication). These improvements immediately lead to improvements in the complexity of a multitude of fundamental problems for which the bottleneck is Rectangular Matrix multiplication, such as computing the all-pair shortest paths in directed graphs with bounded weights.
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faster algorithms for Rectangular Matrix multiplication
Foundations of Computer Science, 2012Co-Authors: Francois Le GallAbstract:Let $\alpha$ be the maximal value such that the product of an $n\times n^\alpha$ Matrix by an $n^\alpha\times n$ Matrix can be computed with $n^{2+o(1)}$ arithmetic operations. In this paper we show that $\alpha>0.30298$, which improves the previous record $\alpha>0.29462$ by Coppersmith (Journal of Complexity, 1997). More generally, we construct a new algorithm for multiplying an $n\times n^k$ Matrix by an $n^k\times n$ Matrix, for any value $k\neq 1$. The complexity of this algorithm is better than all known algorithms for Rectangular Matrix multiplication. In the case of square Matrix multiplication (i.e., for $k=1$), we recover exactly the complexity of the algorithm by Coppersmith and Wino grad (Journal of Symbolic Computation, 1990). These new upper bounds can be used to improve the time complexity of several known algorithms that rely on Rectangular Matrix multiplication. For example, we directly obtain a $O(n^{2.5302})$-time algorithm for the all-pairs shortest paths problem over directed graphs with small integer weights, where $n$ denotes the number of vertices, and also improve the time complexity of sparse square Matrix multiplication.
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faster algorithms for Rectangular Matrix multiplication
arXiv: Data Structures and Algorithms, 2012Co-Authors: Francois Le GallAbstract:Let {\alpha} be the maximal value such that the product of an n x n^{\alpha} Matrix by an n^{\alpha} x n Matrix can be computed with n^{2+o(1)} arithmetic operations. In this paper we show that \alpha>0.30298, which improves the previous record \alpha>0.29462 by Coppersmith (Journal of Complexity, 1997). More generally, we construct a new algorithm for multiplying an n x n^k Matrix by an n^k x n Matrix, for any value k\neq 1. The complexity of this algorithm is better than all known algorithms for Rectangular Matrix multiplication. In the case of square Matrix multiplication (i.e., for k=1), we recover exactly the complexity of the algorithm by Coppersmith and Winograd (Journal of Symbolic Computation, 1990). These new upper bounds can be used to improve the time complexity of several known algorithms that rely on Rectangular Matrix multiplication. For example, we directly obtain a O(n^{2.5302})-time algorithm for the all-pairs shortest paths problem over directed graphs with small integer weights, improving over the O(n^{2.575})-time algorithm by Zwick (JACM 2002), and also improve the time complexity of sparse square Matrix multiplication.
Roman Vershynin - One of the best experts on this subject based on the ideXlab platform.
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smallest singular value of a random Rectangular Matrix
Communications on Pure and Applied Mathematics, 2009Co-Authors: Mark Rudelson, Roman VershyninAbstract:We prove an optimal estimate of the smallest singular value of a random sub- Gaussian Matrix, valid for all dimensions. For an Nn Matrix A with inde- pendent and identically distributed sub-Gaussian entries, the smallest singular value of A is at least of the order p Np n � 1 with high probability. A sharp
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the smallest singular value of a random Rectangular Matrix
arXiv: Probability, 2008Co-Authors: Mark Rudelson, Roman VershyninAbstract:We prove an optimal estimate on the smallest singular value of a random subgaussian Matrix, valid for all fixed dimensions. For an N by n Matrix A with independent and identically distributed subgaussian entries, the smallest singular value of A is at least of the order \sqrt{N} - \sqrt{n-1} with high probability. A sharp estimate on the probability is also obtained.
Mark Rudelson - One of the best experts on this subject based on the ideXlab platform.
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smallest singular value of a random Rectangular Matrix
Communications on Pure and Applied Mathematics, 2009Co-Authors: Mark Rudelson, Roman VershyninAbstract:We prove an optimal estimate of the smallest singular value of a random sub- Gaussian Matrix, valid for all dimensions. For an Nn Matrix A with inde- pendent and identically distributed sub-Gaussian entries, the smallest singular value of A is at least of the order p Np n � 1 with high probability. A sharp
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the smallest singular value of a random Rectangular Matrix
arXiv: Probability, 2008Co-Authors: Mark Rudelson, Roman VershyninAbstract:We prove an optimal estimate on the smallest singular value of a random subgaussian Matrix, valid for all fixed dimensions. For an N by n Matrix A with independent and identically distributed subgaussian entries, the smallest singular value of A is at least of the order \sqrt{N} - \sqrt{n-1} with high probability. A sharp estimate on the probability is also obtained.
B. Fritzsche - One of the best experts on this subject based on the ideXlab platform.
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Skew-Self-Adjoint Dirac System with a Rectangular Matrix Potential: Weyl Theory, Direct and Inverse Problems
Integral Equations and Operator Theory, 2012Co-Authors: B. Fritzsche, B. Kirstein, I. Ya. Roitberg, A. L. SakhnovichAbstract:A non-classical Weyl theory is developed for skew-self-adjoint Dirac systems with Rectangular Matrix potentials. The notion of the Weyl function is introduced and direct and inverse problems are solved. A Borg–Marchenko type uniqueness result and the evolution of the Weyl function for the corresponding focusing nonlinear Schrödinger equation are also derived.
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recovery of the dirac system from the Rectangular weyl Matrix function
Inverse Problems, 2012Co-Authors: B. Fritzsche, B. Kirstein, Ya I Roitberg, A. L. SakhnovichAbstract:Weyl theory for Dirac systems with Rectangular Matrix potentials is non-classical. The corresponding Weyl functions are Rectangular Matrix functions. Furthermore, they are non-expansive in the upper semi-plane. Inverse problems are studied for such Weyl functions, and some results are new even for the square Weyl functions. High-energy asymptotics of Weyl functions and Borg–Marchenko-type uniqueness results are derived too.
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recovery of dirac system from the Rectangular weyl Matrix function
arXiv: Classical Analysis and ODEs, 2011Co-Authors: B. Fritzsche, B. Kirstein, Ya I Roitberg, A. L. SakhnovichAbstract:Weyl theory for Dirac systems with Rectangular Matrix potentials is non-classical. The corresponding Weyl functions are Rectangular Matrix functions. Furthermore, they are non-expansive in the upper semi-plane. Inverse problems are treated for such Weyl functions, and some results are new even for the square Weyl functions. High energy asymptotics of Weyl functions and Borg-Marchenko type uniqueness results are derived too.
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weyl theory and explicit solutions of direct and inverse problems for a dirac system with Rectangular Matrix potential
arXiv: Spectral Theory, 2011Co-Authors: B. Fritzsche, B. Kirstein, Ya I Roitberg, A. L. SakhnovichAbstract:A non-classical Weyl theory is developed for Dirac systems with Rectangular Matrix potentials. The notion of the Weyl function is introduced and the corresponding direct problem is treated. Furthermore, explicit solutions of the direct and inverse problems are obtained for the case of rational Weyl Matrix functions.
