First Graduate Fellowship in CMSE Department Established by MSU Alumnus

MSU Today  

2017-02-27

Andrew Christlieb, CMSE chair, said that the endowment is incredibly important to the department because it provides a mechanism for recruiting the best students. “To build a world-leading department, we need talented graduate students who are enthusiastic about being here, and enthusiastic about what we’re doing,” Christlieb said. “This endowment gives us the ability to recruit that level of talent. It means a lot to us.”...

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Prestigious Research Fellowships Awarded to MSU College of Natural Science Faculty

MSU Today  

2016-02-24

Andrew Christlieb, mathematics professor and CMSE chair, was thrilled about Hirn’s selection as a Sloan fellow and the additional support it will lend to his innovative research. “Matt’s work is unique in that he is developing next generation machine learning algorithms in order to address some of the most pressing and challenging problems arising in large scale computational modeling,” Christlieb said. “This aspect of Matt’s work makes him a critical link in CMSE, and really points to an untapped potential at the interface of computing and data science. We are truly honored that the Sloan Foundation has taken this opportunity to recognize one of our energetic, young scientists.”...

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MSU Recruiting 100 Scientists to Meet Global Challenges

WKAR  

2015-10-20

Michigan State University will hire 100 new faculty members to embark on a long-term effort to expand its scientific research profile. We talk with MSU’s Dr. Steven Hsu, Vice-President for Research and Graduate Studies, and Dr. Andrew Christlieb, chair of the new Department of Computational Mathematics, Science and Engineering...

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Adaptive mesh refinement based on high order finite difference WENO scheme for multi-scale simulations

2011 In this paper, we propose a finite difference AMR-WENO method for hyperbolic conservation laws. The proposed method combines the adaptive mesh refinement (AMR) framework and with the high order finite difference weighted essentially non-oscillatory (WENO) method in space and the total variation diminishing (TVD) Runge–Kutta (RK) method in time (WENO-RK) and by a high order coupling. Our goal is to realize mesh adaptivity in the AMR framework, while maintaining very high (higher than second) order accuracy of the WENO-RK method in the finite difference setting. The high order coupling of AMR and WENO-RK is accomplished by high order prolongation in both space (WENO interpolation) and time (Hermite interpolation) from coarse to fine grid solutions, and at ghost points. The resulting AMR-WENO method is accurate, robust and efficient, due to the mesh adaptivity and very high order spatial and temporal accuracy. We have experimented with both the third and the fifth order AMR-WENO schemes. We demonstrate the accuracy of the proposed scheme using smooth test problems, and their quality and efficiency using several 1D and 2D nonlinear hyperbolic problems with very challenging initial conditions. The AMR solutions are observed to perform as well as, and in some cases even better than, the corresponding uniform fine grid solutions. We conclude that there is significant improvement of the fifth order AMR-WENO over the third order one, not only in accuracy for smooth problems, but also in its ability in resolving complicated solution structures, due to the very low numerical diffusion of high order schemes. In our work, we found that it is difficult to design a robust AMR-WENO scheme that is both conservative and high order (higher than second order), due to the mass inconsistency of coarse and fine grid solutions at the initial stage in a finite difference scheme. Resolving these issues as well as conducting comprehensive evaluation of computational efficiency constitute our future work.

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Parallel high-order integrators

2010 In this work we discuss a class of defect correction methods which is easily adapted to create parallel time integrators for multicore architectures and is ideally suited for developing methods which can be order adaptive in time. The method is based on integral deferred correction (IDC), which was itself motivated by spectral deferred correction by Dutt, Greengard, and Rokhlin [BIT, 40 (2000), pp. 241–266]. The method presented here is a revised formulation of explicit IDC, dubbed revisionist IDC (RIDC), which can achieve pth-order accuracy in “wall-clock time” comparable to a single forward Euler simulation on problems where the time to evaluate the right-hand side of a system of differential equations is greater than latency costs of interprocessor communication, such as in the case of the N-body problem. The key idea is to rewrite the defect correction framework so that, after initial start-up costs, each correction loop can be lagged behind the previous correction loop in a manner that facilitates running the predictor and $M=p-1$ correctors in parallel on an interval which has K steps, where $K\gg p$. We prove that given an rth-order Runge–Kutta method in both the prediction and M correction loops of RIDC, then the method is order $r\times(M+1)$. The parallelization in RIDC uses a small number of cores (the number of processors used is limited by the order one wants to achieve). Using a four-core CPU, it is natural to think about fourth-order RIDC built with forward Euler, or eighth-order RIDC constructed with second-order Runge–Kutta. Numerical tests on an N-body simulation show that RIDC methods can be significantly faster than popular Runge–Kutta methods such as the classical fourth-order Runge–Kutta scheme. In a PDE setting, one can imagine coupling RIDC time integrators with parallel spatial evaluators, thereby increasing the parallelization. The ideas behind RIDC extend to implicit and semi-implicit IDC and have high potential in this area.

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A conservative high order semi-Lagrangian WENO method for the Vlasov equation

2010 In this paper, we propose a novel Vlasov solver based on a semi-Lagrangian method which combines Strang splitting in time with high order WENO (weighted essentially non-oscillatory) reconstruction in space. A key insight in this work is that the spatial interpolation matrices, used in the reconstruction process of a semi-Lagrangian approach to linear hyperbolic equations, can be factored into right and left flux matrices. It is the factoring of the interpolation matrices which makes it possible to apply the WENO methodology in the reconstruction used in the semi-Lagrangian update. The spatial WENO reconstruction developed for this method is conservative and updates point values of the solution. While the third, fifth, seventh and ninth order reconstructions are presented in this paper, the scheme can be extended to arbitrarily high order. WENO reconstruction is able to achieve high order accuracy in smooth parts of the solution while being able to capture sharp interfaces without introducing oscillations. Moreover, the CFL time step restriction of a regular finite difference or finite volume WENO scheme is removed in a semi-Lagrangian framework, allowing for a cheaper and more flexible numerical realization. The quality of the proposed method is demonstrated by applying the approach to basic test problems, such as linear advection and rigid body rotation, and to classical plasma problems, such as Landau damping and the two-stream instability. Even though the method is only second order accurate in time, our numerical results suggest the use of high order reconstruction is advantageous when considering the Vlasov–Poisson system.

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Integral deferred correction methods constructed with high order Runge-Kutta integrators

2010 Spectral deferred correction (SDC) methods for solving ordinary differential equations (ODEs) were introduced by Dutt, Greengard and Rokhlin (2000). It was shown in that paper that SDC methods can achieve arbitrary high order accuracy and possess nice stability properties. Their SDC methods are constructed with low order integrators, such as forward Euler or backward Euler, and are able to handle stiff and non-stiff terms in the ODEs. In this paper, we use high order Runge-Kutta (RK) integrators to construct a family of related methods, which we refer to as integral deferred correction (IDC) methods. The distribution of quadrature nodes is assumed to be uniform, and the corresponding local error analysis is given. The smoothness of the error vector associated with an IDC method, measured by the discrete Sobolev norm, is a crucial tool in our analysis. The expected order of accuracy is demonstrated through several numerical examples. Superior numerical stability and accuracy regions are observed when high order RK integrators are used to construct IDC methods.

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Grid-free plasma simulation techniques

2006 A common approach to modeling kinetic problems in plasma physics is to represent the plasma as a set of Lagrangian macro-particles which interact through long-range forces. In the well-known particle-in-cell (PIC) method, the particle charges are interpolated to a mesh and the fields are obtained using a fast Poisson solver. The advantage of this approach is that the electrostatic forces can be evaluated in time O(NlogN), where N is the number of macro-particles, but the scheme has difficulty resolving steep gradients and handling nonconforming domains unless a sufficiently fine mesh is used. The current work describes a grid-free alternative, the boundary integral/treecode (BIT) method. Using Green's theorem, we express the solution to Poisson's equation as the sum of a volume integral and a boundary integral which are computed using particle discretizations. The treecode replaces particle-particle interactions by particle-cluster interactions which are evaluated by Taylor expansions. In addition, the Green's function is regularized and adaptive particle insertion is implemented to maintain resolution. Like PIC, the operation count is O(NlogN), but BIT avoids using a regular grid, so it can potentially resolve steep gradients and handle complex domains more efficiently. We applied BIT to several bounded plasma problems including a one-dimensional (1-D) sheath in direct current (dc) discharges, 1-D virtual cathode, cold two-stream instability, two-dimensional (2-D) planar and cylindrical ion optics, and particle dynamics in a Penning-Malmberg trap. Some comparisons of BIT and PIC were performed. These results and ongoing work will be reviewed.

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