## Seminar information archive

Seminar information archive ～02/25｜Today's seminar 02/26 | Future seminars 02/27～

### 2016/06/13

#### Numerical Analysis Seminar

16:30-18:00 Room #056 (Graduate School of Math. Sci. Bldg.)

Dissection : A direct solver with kernel detection for finite element matrices

(日本語)

**Atsushi Suzuki**(Osaka University)Dissection : A direct solver with kernel detection for finite element matrices

(日本語)

[ Abstract ]

Large-scale sparse matrices are solved in finite element analyses of elasticity and/or flow problems. In some cases, the matrix may be singular, e.g. due to pressure ambiguity of the Navier-Stokes equations, or due to rigid body movements of sub-domain elasticity problems by a domain decomposition method. Therefore, it is better the linear solver understands rank-deficiency of the matrix.

By assuming the matrix is factorized into LDU form with a symmetric partial permutation, and by introducing a threshold to postpone factorization for pseudo null pivots, solvability of the last Schur complement matrix will be examined. Usual procedure for rank-deficiency problem is based on computation of eigenvalues or singular values and an introduced threshold determines the null space. However, developed new algorithm in DOI:10.1002/nme.4729 is based on computation of residuals combined with orthogonal projections onto supposed image spaces and there is no necessary to introduce a threshold for understanding zero value in floating point. The algorithm uses higher precision arithmetic, e.g. quadruple precision, to distinguish numerical round-off errors that occurred during factorization of the whole sparse matrix from ones during the kernel detection procedure itself.

This is joint work with François-Xavier Roux (LJLL, UPMC/ONERA).

Large-scale sparse matrices are solved in finite element analyses of elasticity and/or flow problems. In some cases, the matrix may be singular, e.g. due to pressure ambiguity of the Navier-Stokes equations, or due to rigid body movements of sub-domain elasticity problems by a domain decomposition method. Therefore, it is better the linear solver understands rank-deficiency of the matrix.

By assuming the matrix is factorized into LDU form with a symmetric partial permutation, and by introducing a threshold to postpone factorization for pseudo null pivots, solvability of the last Schur complement matrix will be examined. Usual procedure for rank-deficiency problem is based on computation of eigenvalues or singular values and an introduced threshold determines the null space. However, developed new algorithm in DOI:10.1002/nme.4729 is based on computation of residuals combined with orthogonal projections onto supposed image spaces and there is no necessary to introduce a threshold for understanding zero value in floating point. The algorithm uses higher precision arithmetic, e.g. quadruple precision, to distinguish numerical round-off errors that occurred during factorization of the whole sparse matrix from ones during the kernel detection procedure itself.

This is joint work with François-Xavier Roux (LJLL, UPMC/ONERA).

#### Tokyo Probability Seminar

16:50-18:20 Room #128 (Graduate School of Math. Sci. Bldg.)

Jump processes on boudaries of random trees

**Yuki Tokushige**(Research Institute for Mathematical Sciences, Kyoto University)Jump processes on boudaries of random trees

#### Seminar on Geometric Complex Analysis

10:30-12:00 Room #128 (Graduate School of Math. Sci. Bldg.)

(JAPANESE)

**Masanori Adachi**(Tokyo University of Science)(JAPANESE)

#### Operator Algebra Seminars

16:45-18:15 Room #118 (Graduate School of Math. Sci. Bldg.)

Lattices models for topological orders and boundary-bulk duality

**Liang Kong**(Univ. New Hampshire/Harvard Univ.)Lattices models for topological orders and boundary-bulk duality

### 2016/06/08

#### Number Theory Seminar

16:00-18:30 Room #16:00-17:00は002, 17:30-18:30は056 (Graduate School of Math. Sci. Bldg.)

Torsion order of smooth projective surfaces (English)

Local and global geometric structures of perfectoid Shimura varieties (English)

**Bruno Kahn**(Institut de mathématiques de Jussieu-Paris Rive Gauche) 16:00-17:00Torsion order of smooth projective surfaces (English)

[ Abstract ]

To a smooth projective variety $X$ whose Chow group of $0$-cycles is $\mathbb{Q}$-universally trivial one can associate its torsion order ${\mathrm{Tor}}(X)$, the smallest multiple of the diagonal appearing in a cycle-theoretic decomposition à la Bloch-Srinivas. We show that ${\mathrm{Tor}}(X)$ is the exponent of the torsion in the Néron-Severi-group of $X$ when $X$ is a surface over an algebraically closed field $k$, up to a power of the exponential characteristic of $k$.

To a smooth projective variety $X$ whose Chow group of $0$-cycles is $\mathbb{Q}$-universally trivial one can associate its torsion order ${\mathrm{Tor}}(X)$, the smallest multiple of the diagonal appearing in a cycle-theoretic decomposition à la Bloch-Srinivas. We show that ${\mathrm{Tor}}(X)$ is the exponent of the torsion in the Néron-Severi-group of $X$ when $X$ is a surface over an algebraically closed field $k$, up to a power of the exponential characteristic of $k$.

**Xu Shen**(Morningside Center of Mathematics) 17:30-18:30Local and global geometric structures of perfectoid Shimura varieties (English)

[ Abstract ]

In this talk, we will investigate some geometric structural properties of perfectoid Shimura varieties of abelian type. In the global part, we will construct some minimal and toroidal type compactifications for these spaces, and we will describe explicitly the degeneration of Hodge-Tate period map at the boundaries. In the local part, we will show that each Newton stratum of these perfectoid Shimura varieties can be described by the related (generalized) Rapoport-Zink space and Igusa variety. As a consequence of our local and global constructions, we can compute the stalks of the relative cohomology under the Hodge-Tate period map of the intersection complex (on the minimal compactification), in terms of cohomology of Igusa varieties at the boundary with truncated coefficients.

In this talk, we will investigate some geometric structural properties of perfectoid Shimura varieties of abelian type. In the global part, we will construct some minimal and toroidal type compactifications for these spaces, and we will describe explicitly the degeneration of Hodge-Tate period map at the boundaries. In the local part, we will show that each Newton stratum of these perfectoid Shimura varieties can be described by the related (generalized) Rapoport-Zink space and Igusa variety. As a consequence of our local and global constructions, we can compute the stalks of the relative cohomology under the Hodge-Tate period map of the intersection complex (on the minimal compactification), in terms of cohomology of Igusa varieties at the boundary with truncated coefficients.

### 2016/06/07

#### Tuesday Seminar on Topology

17:00-18:30 Room #056 (Graduate School of Math. Sci. Bldg.)

Topology of holomorphic Lefschetz pencils on the four-torus (JAPANESE)

**Kenta Hayano**(Keio University)Topology of holomorphic Lefschetz pencils on the four-torus (JAPANESE)

[ Abstract ]

In this talk, we will show that two holomorphic Lefschetz pencils on the four-torus are (smoothly) isomorphic if they have the same genus and divisibility. The proof relies on the theory of moduli spaces of polarized abelian surfaces. We will also give vanishing cycles of some holomorphic pencils of the four-torus explicitly. This is joint work with Noriyuki Hamada (The University of Tokyo).

In this talk, we will show that two holomorphic Lefschetz pencils on the four-torus are (smoothly) isomorphic if they have the same genus and divisibility. The proof relies on the theory of moduli spaces of polarized abelian surfaces. We will also give vanishing cycles of some holomorphic pencils of the four-torus explicitly. This is joint work with Noriyuki Hamada (The University of Tokyo).

### 2016/06/06

#### Seminar on Geometric Complex Analysis

10:30-12:00 Room #128 (Graduate School of Math. Sci. Bldg.)

(JAPANESE)

**Shin Kikuta**(Kogakuin University)(JAPANESE)

#### Operator Algebra Seminars

16:45-18:15 Room #118 (Graduate School of Math. Sci. Bldg.)

Minimal ambient nuclear $C^*$-algebras

**Yuhei Suzuki**(Chiba Univ.)Minimal ambient nuclear $C^*$-algebras

### 2016/06/03

#### Geometry Colloquium

13:00-14:30 Room #118 (Graduate School of Math. Sci. Bldg.)

On a construction of holomorphic disks (Japanese)

**Takeo Nishinou**(Rikkyo University)On a construction of holomorphic disks (Japanese)

[ Abstract ]

Recent study of algebraic and symplectic geometry revealed that holomorphic disks play an important role in several situations, deforming the classical geometry in some sense. In this talk we give a construction of holomorphic disks based on deformation theory, mainly on certain algebraic surfaces.

Recent study of algebraic and symplectic geometry revealed that holomorphic disks play an important role in several situations, deforming the classical geometry in some sense. In this talk we give a construction of holomorphic disks based on deformation theory, mainly on certain algebraic surfaces.

#### Geometry Colloquium

15:00-16:30 Room #118 (Graduate School of Math. Sci. Bldg.)

Caldero's toric degenerations and mirror symmetry (Japanese)

**Makoto Miura**(KIAS)Caldero's toric degenerations and mirror symmetry (Japanese)

[ Abstract ]

In this talk, we explain some basic facts on toric degenerations of Fano varieties. In particular, we focus on the toric degenerations of Schubert varieties proposed by Caldero, where we use the string parametrizations of Lusztig--Kashiwara's dual canonical basis. As an application, we introduce a conjectural mirror construction of a linear section Calabi--Yau 3-fold in an orthogonal Grassmannian OG(2,7). This talk is based on joint works with Daisuke Inoue and Atsushi Ito.

In this talk, we explain some basic facts on toric degenerations of Fano varieties. In particular, we focus on the toric degenerations of Schubert varieties proposed by Caldero, where we use the string parametrizations of Lusztig--Kashiwara's dual canonical basis. As an application, we introduce a conjectural mirror construction of a linear section Calabi--Yau 3-fold in an orthogonal Grassmannian OG(2,7). This talk is based on joint works with Daisuke Inoue and Atsushi Ito.

### 2016/06/01

#### Mathematical Biology Seminar

16:30-17:30 Room #128演習室 (Graduate School of Math. Sci. Bldg.)

A variational problem associated with the minimal speed of traveling waves for the spatially

periodic KPP equation (ENGLISH)

**Xiao Dongyuan**(Graduate School of Mathematical Sciences, The University of Tokyo)A variational problem associated with the minimal speed of traveling waves for the spatially

periodic KPP equation (ENGLISH)

[ Abstract ]

We consider a spatially periodic KPP equation of the form

$$u_t=u_{xx}+b(x)u(1-u).$$

This equation is motivated by a model in mathematical ecology describing the invasion of an alien species into spatially periodic habitat. We deal with the following variational problem:

$$\underset{b\in A_i}{\mbox{Maximize}}\ \ c^*(b),\ i=1,2,$$

where $c^*(b)$ denotes the minimal speed of the traveling wave of the above equation, and sets $A_1$, $A_2$ are defined by

$$A_1:=\{b\ |\ \int_0^Lb=\alpha L,||b||_{\infty}\le h \},$$

$$A_2:=\{b\ |\ \int_0^Lb^2=\beta L\},$$

with $h>\alpha>0$ and $\beta>0$ being arbitrarily given constants. It is known that $c^*(b)$ is given by the principal eigenvalue $k(\lambda,b)$ associated with the one-dimensional elliptic operator under the periodic boundary condition:

$$-L_{\lambda,b}\psi=-\frac{d^2}{dx^2}\psi-2\lambda\frac{d}{dx}\psi-(b(x)

+\lambda^2)\psi\ \ (x\in\mathbb{R}/L\mathbb{Z}).$$

It is important to note that, in one-dimensional reaction-diffusion equations, the minimal speed $c^*(b)$ coincides with the so-called spreading speed. The notion of spreading speed was introduced in mathematical ecology to describe how fast the invading species expands its territory. In other words, our goal is to find an optimal coefficient $b(x)$ that gives the fastest spreading speed under certain given constraints and to study the properties of such $b(x)$.

We consider a spatially periodic KPP equation of the form

$$u_t=u_{xx}+b(x)u(1-u).$$

This equation is motivated by a model in mathematical ecology describing the invasion of an alien species into spatially periodic habitat. We deal with the following variational problem:

$$\underset{b\in A_i}{\mbox{Maximize}}\ \ c^*(b),\ i=1,2,$$

where $c^*(b)$ denotes the minimal speed of the traveling wave of the above equation, and sets $A_1$, $A_2$ are defined by

$$A_1:=\{b\ |\ \int_0^Lb=\alpha L,||b||_{\infty}\le h \},$$

$$A_2:=\{b\ |\ \int_0^Lb^2=\beta L\},$$

with $h>\alpha>0$ and $\beta>0$ being arbitrarily given constants. It is known that $c^*(b)$ is given by the principal eigenvalue $k(\lambda,b)$ associated with the one-dimensional elliptic operator under the periodic boundary condition:

$$-L_{\lambda,b}\psi=-\frac{d^2}{dx^2}\psi-2\lambda\frac{d}{dx}\psi-(b(x)

+\lambda^2)\psi\ \ (x\in\mathbb{R}/L\mathbb{Z}).$$

It is important to note that, in one-dimensional reaction-diffusion equations, the minimal speed $c^*(b)$ coincides with the so-called spreading speed. The notion of spreading speed was introduced in mathematical ecology to describe how fast the invading species expands its territory. In other words, our goal is to find an optimal coefficient $b(x)$ that gives the fastest spreading speed under certain given constraints and to study the properties of such $b(x)$.

### 2016/05/31

#### Tuesday Seminar on Topology

17:00-18:30 Room #056 (Graduate School of Math. Sci. Bldg.)

A linking invariant for algebraic curves (ENGLISH)

**Benoît Guerville-Ballé**(Tokyo Gakugei University)A linking invariant for algebraic curves (ENGLISH)

[ Abstract ]

We construct a topological invariant of algebraic plane curves, which is in some sense an adaptation of the linking number of knot theory. As an application, we show that this invariant distinguishes a new Zariski pair of curves (ie a pair of curves having same combinatorics, yet different topology).

We construct a topological invariant of algebraic plane curves, which is in some sense an adaptation of the linking number of knot theory. As an application, we show that this invariant distinguishes a new Zariski pair of curves (ie a pair of curves having same combinatorics, yet different topology).

#### Algebraic Geometry Seminar

15:30-17:00 Room #122 (Graduate School of Math. Sci. Bldg.)

A Characterization of Symplectic Grassmannians (JAPANESE)

**Kiwamu Watanabe**(Saitama University)A Characterization of Symplectic Grassmannians (JAPANESE)

[ Abstract ]

In the series of their works, J. M. Hwang and N. Mok have been developing the theory of Varieties of Minimal Rational Tangents (VMRT for short). In this direction, the results of Mok and J. Hong-Hwang allow us to recognize a homogeneous Fano manifold X of Picard number one by looking at its VMRT at a general point. This characterization works for all rational homogeneous manifolds of Picard number one whenever the VMRT is rational homogeneous, which is always the case except for the short root cases; namely for symplectic Grassmannians, and for two varieties of type F*4*.

In this talk we show that, if we impose that the VMRT is the expected one at every point of the variety, we may still characterize symplectic Grassmannians. This is a joint work with G. Occhetta and L. E. Sola Conde (arXiv:1604.06867).

In the series of their works, J. M. Hwang and N. Mok have been developing the theory of Varieties of Minimal Rational Tangents (VMRT for short). In this direction, the results of Mok and J. Hong-Hwang allow us to recognize a homogeneous Fano manifold X of Picard number one by looking at its VMRT at a general point. This characterization works for all rational homogeneous manifolds of Picard number one whenever the VMRT is rational homogeneous, which is always the case except for the short root cases; namely for symplectic Grassmannians, and for two varieties of type F*4*.

In this talk we show that, if we impose that the VMRT is the expected one at every point of the variety, we may still characterize symplectic Grassmannians. This is a joint work with G. Occhetta and L. E. Sola Conde (arXiv:1604.06867).

### 2016/05/30

#### Seminar on Geometric Complex Analysis

10:30-12:00 Room #128 (Graduate School of Math. Sci. Bldg.)

(JAPANESE)

**Takeo Ohsawa**(Nagoya University)(JAPANESE)

#### Tokyo Probability Seminar

16:50-18:20 Room #128 (Graduate School of Math. Sci. Bldg.)

**Takafumi Otsuka**(Graduate school of science and engineering, Tokyo metropolitan university)#### Operator Algebra Seminars

16:45-18:15 Room #118 (Graduate School of Math. Sci. Bldg.)

TBA

**Yosuke Kubota**(Univ. Tokyo)TBA

#### Seminar on Probability and Statistics

13:00-14:10 Room #052 (Graduate School of Math. Sci. Bldg.)

Statistical genetics contributes to elucidation of disease biology and genomic drug discovery

**OKADA, Yukinori**(Osaka University)Statistical genetics contributes to elucidation of disease biology and genomic drug discovery

### 2016/05/27

#### Colloquium

15:30-16:30 Room #123 (Graduate School of Math. Sci. Bldg.)

Moduli spaces of linear representations and splittings of 3-manifolds

**Takahiro Kitayama**(Graduate School of Mathematical Sciences, University of Tokyo)Moduli spaces of linear representations and splittings of 3-manifolds

#### Geometry Colloquium

10:00-11:30 Room #118 (Graduate School of Math. Sci. Bldg.)

Compact Special Lagrangian T^2-conifolds (Japanese)

**Yohsuke Imagi**(Kavli IPMU)Compact Special Lagrangian T^2-conifolds (Japanese)

[ Abstract ]

Special Lagrangian submanifolds may be defined as volume-minimizing Lagrangian submanifolds of Calabi--Yau manifolds. Some interesting but difficult topics are (1) the SYZ conjecture, (2) counting compact special Lagrangian homology spheres, and (3) relation to Fukaya categories---all concerned with singularity of special Lagrangian submanifolds. I first recall some basic facts about these things and then talk about a simple class of singularity modelled on a certain T^2-cone.

Special Lagrangian submanifolds may be defined as volume-minimizing Lagrangian submanifolds of Calabi--Yau manifolds. Some interesting but difficult topics are (1) the SYZ conjecture, (2) counting compact special Lagrangian homology spheres, and (3) relation to Fukaya categories---all concerned with singularity of special Lagrangian submanifolds. I first recall some basic facts about these things and then talk about a simple class of singularity modelled on a certain T^2-cone.

#### Geometry Colloquium

13:00-14:30 Room #118 (Graduate School of Math. Sci. Bldg.)

Deformation of Einstein metrics and $L^2$ cohomology on strictly pseudoconvex domains (Japanese)

**Yoshihiko Matsumoto**(Osaka University)Deformation of Einstein metrics and $L^2$ cohomology on strictly pseudoconvex domains (Japanese)

[ Abstract ]

Any bounded strictly pseudoconvex domain of a Stein manifold carries a complete Kähler-Einstein metric of negative scalar curvature, which is unique up to homothety, as shown by S. Y. Cheng and S. T. Yau. I will discuss the fact that this Cheng-Yau metric deforms into a family of Einstein metrics parametrized by partially integrable CR structures on the boundary under the assumption that the dimension is at least three. The necessary analysis on the linearized Einstein operator can be reduced to a vanishing result of the $L^2$ Dolbeault cohomology with values in the holomorphic tangent bundle.

Any bounded strictly pseudoconvex domain of a Stein manifold carries a complete Kähler-Einstein metric of negative scalar curvature, which is unique up to homothety, as shown by S. Y. Cheng and S. T. Yau. I will discuss the fact that this Cheng-Yau metric deforms into a family of Einstein metrics parametrized by partially integrable CR structures on the boundary under the assumption that the dimension is at least three. The necessary analysis on the linearized Einstein operator can be reduced to a vanishing result of the $L^2$ Dolbeault cohomology with values in the holomorphic tangent bundle.

### 2016/05/24

#### Algebraic Geometry Seminar

15:30-17:00 Room #122 (Graduate School of Math. Sci. Bldg.)

ON PSEUDO KOBAYASHI HYPERBOLICITY OF SUBVARIETIES OF ABELIAN VARIETIES

(tba)

**Katsutoshi Yamanoi**(Osaka University)ON PSEUDO KOBAYASHI HYPERBOLICITY OF SUBVARIETIES OF ABELIAN VARIETIES

(tba)

[ Abstract ]

We prove that the Kobayashi pseudo distance of a closed subvariety X of an abelian variety A is a true distance outside the special set Sp(X) of X, where Sp(X) is the union of all positive dimensional translated abelian subvarieties of A which are contained in X. More strongly, we prove that a closed subvariety X of an abelian variety is taut modulo Sp(X); Every sequence fn : D → X of holomorphic mappings from the unit disc D admits a subsequence which converges locally uniformly, unless the image fn(K) of a fixed compact set K of D eventually gets arbitrarily close to Sp(X) as n gets larger. These generalize a classical theorem on algebraic degeneracy of entire holomorphic curves in irregular varieties.

We prove that the Kobayashi pseudo distance of a closed subvariety X of an abelian variety A is a true distance outside the special set Sp(X) of X, where Sp(X) is the union of all positive dimensional translated abelian subvarieties of A which are contained in X. More strongly, we prove that a closed subvariety X of an abelian variety is taut modulo Sp(X); Every sequence fn : D → X of holomorphic mappings from the unit disc D admits a subsequence which converges locally uniformly, unless the image fn(K) of a fixed compact set K of D eventually gets arbitrarily close to Sp(X) as n gets larger. These generalize a classical theorem on algebraic degeneracy of entire holomorphic curves in irregular varieties.

#### Tuesday Seminar on Topology

17:00-18:30 Room #056 (Graduate School of Math. Sci. Bldg.)

Independence of Roseman moves for surface-knot diagrams (JAPANESE)

**Kokoro Tanaka**(Tokyo Gakugei University)Independence of Roseman moves for surface-knot diagrams (JAPANESE)

[ Abstract ]

Roseman moves are seven types of local modifications for surface-knot diagrams in 3-space which generate ambient isotopies of surface-knots in 4-space. In this talk, I will discuss independence among the seven Roseman moves. In particular, I will focus on Roseman moves involving triple points and on those involving branch points. The former is joint work with Kanako Oshiro (Sophia University) and Kengo Kawamura (Osaka City University), and the latter is joint work with Masamichi Takase (Seikei University).

Roseman moves are seven types of local modifications for surface-knot diagrams in 3-space which generate ambient isotopies of surface-knots in 4-space. In this talk, I will discuss independence among the seven Roseman moves. In particular, I will focus on Roseman moves involving triple points and on those involving branch points. The former is joint work with Kanako Oshiro (Sophia University) and Kengo Kawamura (Osaka City University), and the latter is joint work with Masamichi Takase (Seikei University).

### 2016/05/23

#### Seminar on Geometric Complex Analysis

10:30-12:00 Room #128 (Graduate School of Math. Sci. Bldg.)

A computation method for algebraic local cohomology and its applications (JAPANESE)

**Katsusuke Nabeshima**(The University of Tokushima)A computation method for algebraic local cohomology and its applications (JAPANESE)

[ Abstract ]

Local cohomology was introduced by A. Grothendieck. Subsequent development to a great extent has been motivated by Grothendieck's ideas. Nowadays, local cohomology is a key ingredient in algebraic geometry, commutative algebra, topology and D-modules, and is a fundamental tool for applications in several fields.

In this talk, an algorithmic method to compute algebraic local cohomology classes (with parameters), supported at a point, associated with a given zero-dimensional ideal, is considered in the context of symbolic computation. There are several applications of the method. For example, the method can be used to analyze properties of singularities and deformations of Artin algebra. As the applications, methods for computing standard bases of zero-dimensional ideals and solving ideal membership problems, are also introduced.

Local cohomology was introduced by A. Grothendieck. Subsequent development to a great extent has been motivated by Grothendieck's ideas. Nowadays, local cohomology is a key ingredient in algebraic geometry, commutative algebra, topology and D-modules, and is a fundamental tool for applications in several fields.

In this talk, an algorithmic method to compute algebraic local cohomology classes (with parameters), supported at a point, associated with a given zero-dimensional ideal, is considered in the context of symbolic computation. There are several applications of the method. For example, the method can be used to analyze properties of singularities and deformations of Artin algebra. As the applications, methods for computing standard bases of zero-dimensional ideals and solving ideal membership problems, are also introduced.

#### Tokyo Probability Seminar

16:50-18:20 Room #128 (Graduate School of Math. Sci. Bldg.)

Sub-Riemannian diffusions on foliated manifolds

**Fabrice Baudoin**(Department of mathematics, Purdue university)Sub-Riemannian diffusions on foliated manifolds

[ Abstract ]

We study the horizontal diffusion of a totally geodesic Riemannian foliation. We particularly focus on integration by parts formulas on the path space of the diffusion and present several heat semigroup gradient bounds as a consequence. Connections with a generalized sub-Riemannian curvature dimension inequality are made.

We study the horizontal diffusion of a totally geodesic Riemannian foliation. We particularly focus on integration by parts formulas on the path space of the diffusion and present several heat semigroup gradient bounds as a consequence. Connections with a generalized sub-Riemannian curvature dimension inequality are made.

#### Numerical Analysis Seminar

16:30-18:00 Room #056 (Graduate School of Math. Sci. Bldg.)

Inner-iteration preconditioning for least squares problems and its applications (日本語)

**Keiichi Morikuni**(University of Tsukuba)Inner-iteration preconditioning for least squares problems and its applications (日本語)

[ Abstract ]

We discuss inner-iteration preconditioning for Krylov subspace methods for solving large-scale linear least squares problems. The preconditioning uses several steps of stationary iterative methods, and is efficient when the successive overrelaxation (SOR) method for the normal equations is employed. The SOR inner-iteration left/right-preconditioned generalized minimal residual (BA/AB-GMRES) methods determine a least squares solution/the minimum-norm solution of linear systems of equations without breakdown even in the rank-deficient case. The inner-iteration preconditioning requires less memory than incomplete matrix factorization-type one, and is effective for ill-conditioned and/or rank-deficient least squares problems.

We present applications of inner-iteration preconditioning to solutions of (1) general least squares problems, which is to find a least squares solution whose Euclidean norm is minimum; (2) linear systems of equations which arise in an interior-point method for solving linear programming problems. In (1), we focus on a two-step procedure for the solution of general least squares problems; the first step is to determine a least squares solution and the second step is to determine the minimum-norm solution to a linear system of equation. The solution of each step can be done by using the inner-iteration preconditioned GMRES methods. Numerical experiments show that the SOR inner-iteration preconditioned GMRES methods are more efficient than previous methods for some problems. In (2), the linear systems of equations at each interior-point step become ill-conditioned in the late phase of the interior-point iterations. To solve the linear systems of equation robustly, the inner-iteration preconditioning applies. We present efficient techniques to apply the inner-iteration preconditioning to the linear systems of equations. Numerical experiments on benchmark problems show that the inner-iteration preconditioning is robust compared to previous methods. (2) is joint work with Yiran Cui (University College London), Takashi Tsuchiya (National Graduate Institute for Policy Studies) , and Ken Hayami (National Institute of Informatics and SOKENDAI).

We discuss inner-iteration preconditioning for Krylov subspace methods for solving large-scale linear least squares problems. The preconditioning uses several steps of stationary iterative methods, and is efficient when the successive overrelaxation (SOR) method for the normal equations is employed. The SOR inner-iteration left/right-preconditioned generalized minimal residual (BA/AB-GMRES) methods determine a least squares solution/the minimum-norm solution of linear systems of equations without breakdown even in the rank-deficient case. The inner-iteration preconditioning requires less memory than incomplete matrix factorization-type one, and is effective for ill-conditioned and/or rank-deficient least squares problems.

We present applications of inner-iteration preconditioning to solutions of (1) general least squares problems, which is to find a least squares solution whose Euclidean norm is minimum; (2) linear systems of equations which arise in an interior-point method for solving linear programming problems. In (1), we focus on a two-step procedure for the solution of general least squares problems; the first step is to determine a least squares solution and the second step is to determine the minimum-norm solution to a linear system of equation. The solution of each step can be done by using the inner-iteration preconditioned GMRES methods. Numerical experiments show that the SOR inner-iteration preconditioned GMRES methods are more efficient than previous methods for some problems. In (2), the linear systems of equations at each interior-point step become ill-conditioned in the late phase of the interior-point iterations. To solve the linear systems of equation robustly, the inner-iteration preconditioning applies. We present efficient techniques to apply the inner-iteration preconditioning to the linear systems of equations. Numerical experiments on benchmark problems show that the inner-iteration preconditioning is robust compared to previous methods. (2) is joint work with Yiran Cui (University College London), Takashi Tsuchiya (National Graduate Institute for Policy Studies) , and Ken Hayami (National Institute of Informatics and SOKENDAI).

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