The Experts below are selected from a list of 108 Experts worldwide ranked by ideXlab platform
Houman Owhadi - One of the best experts on this subject based on the ideXlab platform.
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Fast Eigenpairs Computation with Operator Adapted Wavelets and Hierarchical Subspace Correction
SIAM Journal on Numerical Analysis, 2019Co-Authors: Lei Zhang, Houman OwhadiAbstract:We present a method for the fast computation of the eigenpairs of a bijective positive Symmetric Linear Operator $\mathcal{L}$. The method is based on a combination of Operator adapted wavelets (ga...
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Fast eigenpairs computation with Operator adapted wavelets and hierarchical subspace correction
arXiv: Numerical Analysis, 2018Co-Authors: Lei Zhang, Houman OwhadiAbstract:We present a method for the fast computation of the eigenpairs of a bijective positive Symmetric Linear Operator $\mathcal{L}$. The method is based on a combination of Operator adapted wavelets (gamblets) with hierarchical subspace correction.First, gamblets provide a raw but fast approximation of the eigensubspaces of $\mathcal{L}$ by block-diagonalizing $\mathcal{L}$ into sparse and well-conditioned blocks. Next, the hierarchical subspace correction method, computes the eigenpairs associated with the Galerkin restriction of $\mathcal{L}$ to a coarse (low dimensional) gamblet subspace, and then, corrects those eigenpairs by solving a hierarchy of Linear problems in the finer gamblet subspaces (from coarse to fine, using multigrid iteration). The proposed algorithm is robust for the presence of multiple (a continuum of) scales and is shown to be of near-Linear complexity when $\mathcal{L}$ is an (arbitrary local, e.g.~differential) Operator mapping $\mathcal{H}^s_0(\Omega)$ to $\mathcal{H}^{-s}(\Omega)$ (e.g.~an elliptic PDE with rough coefficients).
Libor Veselý - One of the best experts on this subject based on the ideXlab platform.
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Delta-semidefinite and Delta-convex Quadratic Forms in Banach Spaces
Positivity, 2008Co-Authors: Nigel Kalton, Sergei V. Konyagin, Libor VeselýAbstract:A continuous quadratic form (“quadratic form”, in short) on a Banach space X is: (a) delta-semidefinite (i.e., representable as a difference of two nonnegative quadratic forms) if and only if the corresponding Symmetric Linear Operator $$T: X\rightarrow X^*$$ factors through a Hilbert space; (b) delta-convex (i.e., representable as a difference of two continuous convex functions) if and only if T is a UMD-Operator. It follows, for instance, that each quadratic form on an infinite-dimensional L _ p ( μ ) space (1 ≤ p ≤ ∞) is: (a) delta-semidefinite iff p ≥ 2; (b) delta-convex iff p > 1. Some other related results concerning delta-convexity are proved and some open probms are stated.
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Delta-semidefinite and delta-convex quadratic forms in Banach spaces
arXiv: Functional Analysis, 2006Co-Authors: Nigel Kalton, Sergei V. Konyagin, Libor VeselýAbstract:A continuous quadratic form ("quadratic form", in short) on a Banach space $X$ is: (a) delta-semidefinite (i.e., representable as a difference of two nonnegative quadratic forms) if and only if the corresponding Symmetric Linear Operator $T\colon X\to X^*$ factors through a Hilbert space; (b) delta-convex (i.e., representable as a difference of two continuous convex functions) if and only if $T$ is a UMD-Operator. It follows, for instance, that each quadratic form on an infinite-dimensional $L_p(\mu)$ space ($1\le p \le\infty$) is: (a) delta-semidefinite iff $p \ge 2$; (b) delta-convex iff $p>1$. Some other related results concerning delta-convexity are proved and some open problems are stated.
Lei Zhang - One of the best experts on this subject based on the ideXlab platform.
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Fast Eigenpairs Computation with Operator Adapted Wavelets and Hierarchical Subspace Correction
SIAM Journal on Numerical Analysis, 2019Co-Authors: Lei Zhang, Houman OwhadiAbstract:We present a method for the fast computation of the eigenpairs of a bijective positive Symmetric Linear Operator $\mathcal{L}$. The method is based on a combination of Operator adapted wavelets (ga...
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Fast eigenpairs computation with Operator adapted wavelets and hierarchical subspace correction
arXiv: Numerical Analysis, 2018Co-Authors: Lei Zhang, Houman OwhadiAbstract:We present a method for the fast computation of the eigenpairs of a bijective positive Symmetric Linear Operator $\mathcal{L}$. The method is based on a combination of Operator adapted wavelets (gamblets) with hierarchical subspace correction.First, gamblets provide a raw but fast approximation of the eigensubspaces of $\mathcal{L}$ by block-diagonalizing $\mathcal{L}$ into sparse and well-conditioned blocks. Next, the hierarchical subspace correction method, computes the eigenpairs associated with the Galerkin restriction of $\mathcal{L}$ to a coarse (low dimensional) gamblet subspace, and then, corrects those eigenpairs by solving a hierarchy of Linear problems in the finer gamblet subspaces (from coarse to fine, using multigrid iteration). The proposed algorithm is robust for the presence of multiple (a continuum of) scales and is shown to be of near-Linear complexity when $\mathcal{L}$ is an (arbitrary local, e.g.~differential) Operator mapping $\mathcal{H}^s_0(\Omega)$ to $\mathcal{H}^{-s}(\Omega)$ (e.g.~an elliptic PDE with rough coefficients).
Alexander Varchenko - One of the best experts on this subject based on the ideXlab platform.
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The B. and M. Shapiro conjecture in real algebraic geometry and the Bethe ansatz
Annals of Mathematics, 2009Co-Authors: Evgeny Mukhin, Vitaly Tarasov, Alexander VarchenkoAbstract:We prove the B. and M. Shapiro conjecture that if the Wronskian of a set of polynomials has real roots only, then the complex span of this set of polynomials has a basis consisting of polynomials with real coefficients. This, in particular, implies the following result: If all ramification points of a parametrized rational curve φ: ℂℙ 1 → ℂℙ r lie on a circle in the Riemann sphere ℂℙ 1 , then φ maps this circle into a suitable real subspace ℝℙ r ⊂ ℂℙ r . The proof is based on the Bethe ansatz method in the Gaudin model. The key observation is that a Symmetric Linear Operator on a Euclidean space has real spectrum. In Appendix A, we discuss properties of differential Operators associated with Bethe vectors in the Gaudin model. In particular, we prove a statement, which may be useful in complex algebraic geometry; it claims that certain Schubert cycles in a Grassmannian intersect transversally if the spectrum of the corresponding Gaudin Hamiltonians is simple. In Appendix B, we formulate a conjecture on reality of orbits of critical points of master functions and prove this conjecture for master functions associated with Lie algebras of types A r , B r and C r .
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The B. and M. Shapiro conjecture in real algebraic geometry and the Bethe ansatz
arXiv: Algebraic Geometry, 2005Co-Authors: Evgeny Mukhin, Vitaly Tarasov, Alexander VarchenkoAbstract:We prove the B. and M. Shapiro conjecture that says that if the Wronskian of a set of polynomials has real roots only, then the complex span of this set of polynomials has a basis consisting of polynomials with real coefficients. This in particular implies the following result: If all ramification points of a parametrized rational curve $ f : CP^1 \to CP^r $ lie on a circle in the Riemann sphere $ CP^1 $, then $f$ maps this circle into a suitable real subspace $ RP^r \subset CP^r $. The proof is based on the Bethe ansatz method in the Gaudin model. The key observation is that a Symmetric Linear Operator on a Euclidean space has a real spectrum. In Appendix we discuss properties of differential Operators associated with Bethe vectors in the Gaudin model and, in particular, prove a conditional statement: we deduce the transversality of certain Schubert cycles in a Grassmannian from the simplicity of the spectrum of the Gaudin Hamiltonians.
Bernhard Maschke - One of the best experts on this subject based on the ideXlab platform.
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Dirac structures and Boundary Control Systems associated with Skew-Symmetric Differential Operators
Siam Journal on Control and Optimization, 2005Co-Authors: Y. Le Gorrec, Hans Zwart, Bernhard MaschkeAbstract:Associated with a skew-Symmetric Linear Operator on the spatial domain $[a,b]$ we define a Dirac structure which includes the port variables on the boundary of this spatial domain. This Dirac structure is a subspace of a Hilbert space. Naturally, associated with this Dirac structure is an infinite-dimensional system. We parameterize the boundary port variables for which the \( C_{0} \)-semigroup associated with this system is contractive or unitary. Furthermore, this parameterization is used to split the boundary port variables into inputs and outputs. Similarly, we define a Linear port controlled Hamiltonian system associated with the previously defined Dirac structure and a Symmetric positive Operator defining the energy of the system. We illustrate this theory on the example of the Timoshenko beam.
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a semigroup approach to port hamiltonian systems associated with Linear skew Symmetric Operator
16th International Symposium on Mathematical Theory of Networks and Systems MTNS 2004, 2004Co-Authors: Le Y Gorrec, Heiko J Zwart, Bernhard MaschkeAbstract:In this paper we first define a Dirac structure on a Hilbert spaces associated with a skew-Symmetric Linear Operator including port variables on the boundary of its domain. Secondly, we associate $C_0$-semigroup with some parameterization of the boundary port variables and define a family of boundary control systems. Thirdly we define a Linear port controlled Hamiltonian system associated with the previously defined Dirac structure and generated by a Symmetric positive Operator defining the energy of the system.
