### Schedule

#### 2016-08-22

TimeSpeaker/ Title/ Abstract/ VODLocation
08:45 - 09:00 Registration
CAMP
09:00 - 09:10 Opening
CAMP
09:10 - 09:55 The Hybrid Spectral Difference Methods for the Elliptic and Navier-Stokes Equations / Youngmok Jeon (Ajou University)
In this presentation we introduce the hybrid spectral differences(HSD) for the elliptic and Navier-Stokes equations. The HSD is a finite difference version of the hybridized discontinuous Galerkin method. The HSD is comparable with the finite difference methods and spectral difference methods. The main difference between the FDM and HDM is that the FD formula of a single type is deployed for all interior nodes in the FDM, while the cell finite difference and the interface difference are combined in the HDM. The finite difference method is simple to implement and it can solve many physical problems efficiently. The HDM is as easy to implement as the FDM, and it apparently seems to possess several advantages over the FDM. Those advantages are listed below. (1) The method can be applied to nonuniform grids, retaining the optimal order of convergence. The FD formulas in a reference cell can be applied to cells of any dimension, multiplied by scaling factors. (2) Numerical methods with an arbitrarily high order convergence can be obtained by simply locating more cell points. (3) Problems on a complicated geometry can be treated reasonably well, and the boundary condition can be imposed exactly on the exact boundary (no variational crime). (4) Stability problems when solving the Stokes/Navier-Stokes problem can be resolved without introducing a staggered grid or a stream-vorticity formulation. Therefore, the HDM requires less programming efforts compared to the staggered grid method, and the method can be applied without dimensionality restriction. (5) In the HDMthe grids for pressure are sub-located on the grids for velocity fields for flow problems. (6) Numerical analysis is based on a discrete divergence theory on each cell. (7) The flux conservation property holds in each cell and flux continuation holds across intercell boundaries. (8) The embedded static condensation property of the HDM reduces degrees of freedom a lot.
CAMP
09:55 - 10:40 On the Refinement of Constructive a Priori Error Estimates of the Finite Element Methods with Applications to verified Computation for PDEs / Mitsuhiro Nakao (Kyushu University)
On the Refinement of Constructive a Priori Error Estimates of the Finite Element Methods with Applications to verified Computation for PDEs We consider the constructive a priori error estimates of the higher order finite element methods for the Poisson equation. This can be applied to some improvements of computational efficiency in the numerical verification method of solutions for nonlinear elliptic boundary value problems. We also mention some refinements in the optimal order error estimates for a finite element approximation of parabolic initial boundary value problems, which has a possibility to lead the efficient verification of solutions for nonlinear parabolic problems as well.
CAMP
10:40 - 11:00 Coffee Break
CAMP
11:00 - 11:45 Staggered Discontinuous Galerkin Methods and their Applications / Hyea Hyun Kim (Kyung Hee University)
Staggered discontinuous Galerkin methods are presented for second order elliptic problems. Advantages of the proposed methods over the stangerd discontinous Galerkin methods are first discussed. The methods have been successfully applied to wave equations, convection-diffusion equations, elasticity problems, and fluid problems. Among them, applications to fluid problems are presented for incompressible Stokes equations and incompressible Navier-Stokes equations. The accuracy of the approximation is analyzed and numerical results are presented.
CAMP
11:45 - 12:30 Modulus-Based Iteration Methods for Nonnegative and Box Constrained Least Square Problems / Ken Hayami (National Institute of Informatics / SOKENDAI)
For the solution of large sparse nonnegative constrained least squares (NNLS) problems, an iterative method is proposed[1] which uses the CGLS method for the inner iterations and the modulus iterative method[2] for the outer iterations to solve the linear complementarity problem resulting from the Karush-Kuhn-Tucker conditions of the NNLS problem. Theoretical convergence analysis including the optimal choice of the parameter matrix is presented for the proposed method. In addition, the method can be further enhanced by incorporating the active set strategy, which contains two stages where the first stage consists of modulus iterations to identify the active set, while the second stage solves the reduced unconstrained least squares problems only on the inactive variables, and projects the solution into the nonnegative region. Numerical experiments show the efficiency of the proposed methods compared to projection gradient-type methods with less iteration steps and CPU time. We will also apply the method to image restoration. We will also present extension of the method to box constrained least squares problems. This is the work of my Ph.D. student Ning Zheng (SOKENDAI), and is also coauthored by Prof. Jun-Feng Yin (Tongji University). Reference: [1] Zheng, N., Hayami, K., and Yin, J.-F., Modulus-type inner outer iteration methods for nonnegative constrained least squares problems, (accepted for publication in SIAM Journal on Matrix Analysis and Applications). http://www.nii.ac.jp/TechReports/public_html/16-001E.pdf [2] Bai, Z.-Z., Modulus-based matrix splitting iteration methods for linear complementarity problems, Numer. Linear Algebra Appl., 6 (2010), pp. 917--933.
CAMP
12:30 - 14:00 Lunch
CAMP
14:00 - 14:45 Nonconforming Approach to Generalized Multiscale Methods / Dongwoo Sheen (Seoul National University)
A framework is introduced for nonconforming multiscale approach based on GMsFEM (Generalized Multiscale Finite Element Methods) . Snapshot spaces are constructed for each {macro--scale} block. The snapshot spaces can be based on either conforming or nonconforming elements. With suitable dimension reduction, offline spaces are constructed. Moment function spaces are then introduced to impose continuity among the local offline spaces, which results in nonconforming GMsFE spaces. Oversampling and randomized boundary condition strategies are considered. Steps for the nonconforming GMsFEM are given explicitly. Error estimates are derived. Numerical results are presented to support the efficiency of the proposed approach.
CAMP
14:45 - 15:30 Staggered Finite Difference Methods for some Flow Problems / Hongxing Rui (Shandong University)
In this talk we will present some finite difference methods based on nonuniform staggered grids for some flow problems including the Non-Darcy flow in porous media and fluid flow.
CAMP
15:30 - 15:50 Coffee Break
CAMP
15:50 - 16:35 Multiscale Methods for Seismic Wave Simulations / Eric Chung (The Chinese University of Hong Kong)
Numerical simulations of wave propagation in heterogeneous media are important in many applications such as seismic propagation and seismic inversion. In this talk, we will present a new multiscale approach for seismic wave propagation. The method is able to compute the solution with much fewer degrees of freedom compared with fine mesh simulation. The idea is to capture the multiscale features of the solutions by carefully designed multiscale basis functions. We will also present applications to inverse problems. The research is partially supported by Hong Kong RGC General Research Fund (Project 14301314).
CAMP
16:35 - 17:05 Finite Difference Approximation for Nonlinear Schrödinger Equations with Application to Blow-Up Computation / Takiko Sasaki (Waseda University)
Finite difference schemes for computing blow-up solutions of one dimensional nonlinear Schrodinger equations are presented. By applying time increments control technique, we can introduce a numerical blow-up time which is an approximation of the exact blow-up time of nonlinear Schrodinger equation. After having verified the convergence of our proposed schemes, we proved that the solution of a finite-difference scheme actually blows up in the numerical blow-up time. Then, we proved that the numerical blow-up time converges to the exact blow-up time as the discretization parameters tend to zero.
CAMP

#### 2016-08-23

TimeSpeaker/ Title/ Abstract/ VODLocation
09:10 - 09:55 On Solving Ill Conditioned Linear Systems / Craig C. Douglas (University of Wyoming)
This paper presents the first results to combine theoretically sound methods (spectral projection, deflation methods, and multigrid methods together to attack ill conditioned linear systems. Our preliminary results show that the proposed algorithm applied to a Krylov subspace method takes much fewer iterations for solving an ill conditioned problem downloaded from a popular online sparse matrix collection. To our best knowledge, the constructions of most, if not all, deflation subspace matrices in the literature are problem dependent. Further, some of them are ad-hoc. The method proposed here is problem independent.
CAMP
09:55 - 10:40 On the Computation of the Inverse Sturm-Liouville Problem in Impedance Form / Zhengda Huang (Zhejiang University)
This is a report of our group's work on the computations of a inverse Sturm-Liouville problem in impedance form with Dirichlet boundary conditions. Two cases, without and with the symmetric assumption for the impedance, are considered. For the case without the symmetric assumption, the problem is computed by solving an optimal problem based on the descent flow method, the modified descent flow method and a ULM-like descent flow method, separately. For the case of the symmetric impedance, the problem is computed by solving a nonlinear equation with the simple Newton's method. Numerical examples for smooth, non-smooth and discontinuous impedance functions are performed to show the efficiency of these methods.
CAMP
10:40 - 11:00 Coffee Break
CAMP
11:00 - 11:45 Explicit Stabilized Runge-Kutta Methods for Stiff Stochastic Differential Equations with a Semilinear Drift Term / Yoshio Komori (Kyushu Institute of Technology)
We are concerned with numerical methods which can effectively solve stiff Itô stochastic differential equations (SDEs). In general, implicit methods can be expected to solve such SDEs with relatively large step size compared with explicit methods, but they may suffer from computational cost if the dimension of SDEs is large. In order to deal with high-dimensional stiff SDEs, one of promising methods is explicit and stabilized methods. In fact, the class of stochastic orthogonal Runge-Kutta Chebyshev (SROCK) methods has been recently proposed for some types of stiff SDEs. As a different approach from the SROCK methods, in this talk we will propose explicit exponential Runge-Kutta methods for stiff SDEs with a semilinear drift term. We will carry out numerical experiments to compare our methods with the SROCK methods. (This is a joint work with D. Cohen and K. Burrage.)
CAMP
11:45 - 12:30 Primal Domain Decomposition Methods for the Total Variation Minimization, based on Dual Decomposition / Chang-Ock Lee (KAIST)
We propose nonoverlapping domain decomposition methods for solving the total variation minimization problem. We decompose the domain of the dual problem into nonoverlapping rectangular subdomains, where local total variation problems are solved. We convert the local dual problems into the equivalent primal forms which reproduce the original problem at smaller dimension. Sequential and parallel algorithms are presented. The convergence of both algorithms is analyzed and numerical results are presented.
CAMP
12:30 - 14:00 Lunch
CAMP
14:00 - 14:45 T-splines and Generalized T-splines / Durkbin Cho (Dongguk University)
T-splines are a generalization of the classical tensor-product B-splines based on meshes (called T-meshes) which allow T-junctions, that is vertices which are endpoints of less than 4 edges, unlike in the tensor-product case. The use of T-splines gives some considerable improvements on the classical tensor-product splines and NURBS, such as the possibility to apply local refinements, a heavy reduction of the number of control points needed, and the ability to easily avoid gaps when joining several surfaces. In this talk, we will present several results about T-splines, in particular concerning their linear independence and their generalization to a noteworthy non-polynomial case (Generalized T-splines), and then construct a dual basis for a noteworthy class of Generalized T-splines.
CAMP
14:45 - 15:30 A Fast and Asymmetric Structure-Preserving Numerical Method for Partial Differential Equations / Daisuke Furihata (Osaka University)
We have developed some asymmetric numerical schemes based on the discrete variational derivative method for partial differential equations. Those are structure-preserving, and we can expect that they are faster in computations than the normal structure-preserving nonlinear schemes in general. In this talk, we introduce the asymmetric structure-preserving methods in detail and show some properties of them with some numerical computation results.
CAMP
15:30 - 15:50 Coffee Break
CAMP
15:50 - 16:35 Numerical Integration Based on the Hyperfunction Theory / Hidenori Ogata (The University of Electro-Communications)
We present a numerical integration method based on the hyperfunction theory, which is a generalized function theory proposed by M. Sato and based on complex analysis. In the presented method, a desired integral is transformed into a complex integral by the definition of hyperfunction integrals, and it is evaluated by the trapezoidal rule. The approximation by the presented method is shown to converge geometrically from theoretical error estimate, and it is efficient especially for integrals with end-point singularities from some numerical examples.
CAMP
16:35 - 17:05 The Finite Element Heterogeneous Multiscale Method based on the DSSY Nonconforming Element / Jaeryun Yim (Seoul National University)
The homogenization approach is one of the successful strategies to solve multiscale problems approximately. The Finite Element Heterogeneous Multiscale Method (FEHMM) which is based on the nite element method estimates the eective homogenized coecients associated with basis functions on each sampling domain to solve homogenized problems numerically. In this talk, we propose a FEHMM scheme for multiscale elliptic problems based on the DSSY (Douglas-Santos-Sheen-Ye) nonconforming nite element. A priori error estimates for the homogenized solution are derived by properties of the FEHMM and the nonconforming element. We provide several numerical results that conrm our analysis. Some technical issues about implementations are mentioned.
CAMP
18:00 - 20:00 Banquet
TBA

#### 2016-08-24

TimeSpeaker/ Title/ Abstract/ VODLocation
09:10 - 09:55 Discrete Maximal Regularity and the Finite Element Method for Parabolic Problems / Norikazu Saito (The University of Tokyo)
Maximal regularity is a fundamental concept in the theory of partial differential equations. In this talk, I report some results of our recent study on a fully discrete version of maximal regularity for parabolic equations. First, we derive various stability results in $L^p(0,T;L^q(\Omega))$ norm, $p,q\in (1,\infty)$ for the finite element approximation with the mass-lumping to the linear heat equation. Our method of analysis is an operator theoretical one using pure imaginary powers of operators and might be a discrete version of Dore and Venni (1987). As an application, optimal order error estimates in that norm are proved. Furthermore, we study the finite element approximation for semilinear heat equations with locally Lipschitz continuous nonlinearity and offer a new method for deriving optimal order error estimates. This is a joint work with Tomoya Kemmochi.
CAMP
09:55 - 10:40 Hybrid Discontinuous Galerkin Methods / Eun-Jae Park (Yonsei University)
In this talk we present a priori and posteriori error estimators for hybrid discontinuous Galerkin (HDG) methods for elliptic equations. First, we present arbitrary-order HDG methods to solve the Poisson problem and propose residual type error estimators. Next, we present guaranteed type error estimators by postprocessing scalar and flux unknowns. Then, we consider diffusion problems with discontinuous coefficients. Some numerical examples are presented to show the performance of the methods.
CAMP
10:40 - 11:00 Coffee Break
CAMP
11:00 - 11:45 Dissection, a Parallel Direct Solver for Finite Element Matrices and its Usage with FreeFem++ / Atsushi Suzuki (Osaka University)
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 underdeterminant of the Navier-Stokes equations, or to rigid body movements of subdomain-wise elasticity problem by a domain decomposition method. Therefore it is better that the linear solver understands rank-deficiency of the matrix. By assuming the matrix is factorized into LDU form with a symmetric partial permutation, which is satisfied by finite element matrices, efficient factorization procedure is obtained. However, block strategy for parallelization limits the search range of pivots and may request usage of combination of 1x1 and 2x2 pivot entries. Postponing strategy for pseudo null pivots with a given threshold recomputes a Schur complement, which will be examined to understand the whole matrix is invertible or the dimension of the null space by a new kernel detection algorithm. Usages of Dissection solver in solving Navier-Stokes equations without introducing pressure perturbation and in implementing FETI algorithm for an elasticity problem are shown with FreeFem++, finite element software.
CAMP
11:45 - 12:30 Coffee break
CAMP
12:30 - 14:00 Lunch
CAMP
14:00 - 18:00 Excursion
TBA

#### 2016-08-25

TimeSpeaker/ Title/ Abstract/ VODLocation
09:10 - 09:55 Stability of $Q_1-P_0$ for Stokes Equations / Chunjae Park (Konkuk University)
It is well known that $Q_1-P_0$ finite element spaces over rectangular meshes do not satisfy the inf-sup condition for Stokes equations. Nevertheless, $Q_1-P_0$ shows an optimal error convergence of $O(h)$. In this talk, we will prove that $Q_1-P_0$ is stable for Stokes equations.
CAMP
09:55 - 10:40 Multilevel Monte Carlo Immersed FEM for Elliptic Problems with Random Coefficients / Imbo Sim (National Institute for Mathematical Sciences)
In this talk, we propose a multilevel Monte Carlo-immersed finite element method (MLMC-IFEM) for elliptic problems with discontinuous random coefficients. The multilevel Monte Carlo method is an effective sampling method for uncertainty quantification. Exploiting features of immersed finite element methods, we provide the multilevel method on the hierarchical meshes which are independent of discontinuity of the coefficients. The error estimation and computational complexity of the MLMC-IFEM are derived by properties of the MLMC method and the IFEM. A series of numerical experiments demonstrate accuracy and efficiency of the proposed method.
CAMP
10:40 - 11:00 Coffee Break
CAMP
11:00 - 11:30 Nonconforming Finite Element Method Applied to the Driven Cavity Problem / Roktaek Lim (Nanyang Technological University)
A cheapest stable nonconforming finite element method is presented for solving the incompressible flow in a square cavity without smoothing the corner singularities. The stable cheapest nonconforming finite element pair based on $P_1\times P_0$ on rectangular meshes [1] is employed with a minimal modification of the discontinuous Dirichlet data on the top boundary, where $\widetilde{\mathscr{P}_0^h}$ is the finite element space of piecewise constant pressures with the globally one-dimensional checker-board pattern subspace eliminated. The proposed Stokes elements have the least number of degrees of freedom compared to those of known stable Stokes elements. Three accuracy indications for our elements are analyzed and numerically verified. Also, various numerous computational results obtained by using our proposed element show excellent accuracy. References [1] S. Kim, J. Yim, and D. Sheen. Stable cheapest nonconforming finite elements for the Stokes equations. J. Comput. Appl. Math., 299:2-14, 2016.
CAMP
11:30 - 12:00 Laplace Transform Method with the Magnus Expansion for Parabolic Problems / Kanghun Cho (Seoul National University)
The Laplace transform method is efficient for solving parabolic problems whose coefficients are time independent. However the method is inappropriate to deal with problems whose coefficients are time dependent. We can solve the difficulty with the aid of the Magnus expansion. In this talk, we propose a Laplace transform method with Magnus expansion for solving general parabolic problems. Some numerical examples are presented to show the efficiency of the method.
CAMP
12:00 - 14:00 Lunch
CAMP
14:00 - 14:30 Three Type of Condition Numbers of Quadratic Matrix Equation / Syed Muhammad Raza Shah Naqvi (Pusan National University)
This paper is devoted to normwise, mixed and componentwise condition numbers of Quadratic matrix equation. The results are explained by the example.
TBA
14:30 - 15:00 A Modified Newton's Method to Solve Matrix Equations / Geonwoo Kim (Pusan National University)
There are several types of matrix equations. In particular, Newton's Method is the most popular and effective way to find a numerical solution of the equations. In this talk, we will discuss modified Newton's Method under special matrix equations with some conditions. Moreover, we will consider some examples which give the advantage of the modified iteration.
CAMP
15:45 - 16:05 Closing
CAMP

#### 2016-08-26

TimeSpeaker/ Title/ Abstract/ VODLocation
09:00 - 10:40 Organizer Discussion
CAMP
10:40 - 11:00 Closing
CAMP