Frank Giraldo

News Articles​

1. A Stanford University study ranked the worlds's top 2% of scientist through 2023 [click here]. A total of 6 faculty members (including me) from the Department of Applied Mathematics at the Naval Postgraduate School made the list

2. HPC Wire article highlighting NUMA and xNUMA work on hurricanes and multi-scale modeling framework (2023) [click here]

3. Lawrence Livermore National Lab (LLNL) article on the Center for Efficient Exascale Discretization (CEED) annual meeting which mentions our work on leveraging GPUs for multi scale modeling framework (2023) [click here]

4. NPS article on NUMA project related to large-eddy simulations to better understand  hurricane dynamics using:  high-resolution modeling, super-parameterization, scientific machine learning, and GPU computing (2022) [click here]

5. Nature article that discusses the Fortran programming language which is used in NUMA/NUMO (2021) [click here]

6. Nature article on the Julia programming language (2019) [click here]

7. Science magazine article about the CLIMA project (2018) [click here]

8. Nature article that mentions the NUMO model (2015) [click here]

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Presentations On-line​

1. May 16 2023: LLNL MFEM group on element-based Galerkin method for hurricane simulations  [YouTube]

2. June 20, 2023: SIAM Geosciences plenary talk on element-based Galerkin Methods for weather, climate, and ocean [PDF]

3. March 20, 2024: FSU Department of Scientific Computing on HEVI time-integration for weather and climate models [YouTube]

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Research Contributions

(In what follows, all the publications mentioned below can be found in the Publications tab)

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• I am first and foremost an applied mathematician that prefers paper and pencil analysis. In particular, my main strength is the derivation of new forms of the governing equations of motion. E.g., NUMA contains 5 different forms of the governing equations (NUMA solves the compressible and incompressible Navier-Stokes equations, as well as the shallow water equations on the plane and on the sphere), each having a number of variants in order to improve conservation and stability.  This focus then allows me to construct numerical algorithms that are streamlined for accuracy and efficiency.

• I am a reluctant computational mathematician. I have had no choice but to fully embrace scientific computing and to this end I have sought to improve my skills in this area (e.g., scaling our models using both distributed computing and accelerators such as GPUs).

• In the period 1997-2006, I wrote a series of papers on semi-Lagrangian (I called them Lagrange-Galerkin) methods for both finite elements and spectral element methods (see Giraldo JCP 1997, JCP 1998, MWR 1999, IJNMF 2000, JCP 2000, CMA 2006, and Giraldo et al. JCP 2003); in JCP 1998, I performed the stability and accuracy analysis of the semi-Lagrangian method using high-order interpolation.  I enjoyed performing this beautiful analysis but moved away from these methods because I was worried about the computational cost they would incur.

• In 2001, I wrote a paper on applying spectral element (SE) methods on spherical domains for geophysical fluid dynamics applications using Cartesian coordinates. This paper (Giraldo IJNMF 2001) was the idea that spawned both the NSEAM and NUMA models.

• In 2002, my colleagues and I wrote a very nice paper on constructing discontinuous Galerkin (DG) methods on spherical domains for geophysical fluid dynamics applications (the first paper on this topic, see Giraldo et al. JCP 2002). This work has made its way into the NUMA model.

• In the period 2004-2005, I developed a spectral element hydrostatic atmospheric model (NSEAM). In Giraldo-Rosmond MWR 2004, we showed how to construct a 3D model on the sphere using Cartesian coordinates and ran classical test cases such as the baroclinic instability and Held-Suarez test cases. In Giraldo QJRMS 2005, I extended this model to implicit-explicit time-integration (I called it semi-implicit) and showed the amazing scalability of this method.

• In 2008, we introduced the spectral element and discontinuous Galerkin methods for nonhydrostatic atmospheric modeling (see Giraldo-Restelli JCP 2008); this was the first paper on SE and DG methods for these equations (compressible Navier-Stokes with buoyancy).  In this paper, we derived SE and DG discretizations for various forms of the compressible Navier-Stokes equations and presented a benchmark suite along with metrics for testing nonhydrostatic atmospheric models. 

• In 2009, Marco Restelli and I showed how to construct IMEX methods for DG; we were able to derive a Schur complement but only under special conditions (see Restelli-Giraldo SISC 2009).  Recently, we have been able to generalize this idea (see Reddy et al JCP 2023). 

• From 2010 through the present day, I have been working to make SE and DG methods more stable and efficient.  To this end, we have developed linear implicit-explicit (IMEX) time-integrators (see Giraldo et al. SISC 2010, Giraldo et al. SISC 2013, Abdi et al. IJHPCA 2019, Abdi et al. IJHPCA 2019, and Mueller et al. IHPCA 2019). Along these lines, we also explored hybridizable discontinuous Galerkin (HDG) methods (see Kang et al. JCP 2020).

• In 2014-2015, we published two papers on non-conforming adaptive mesh refinement (AMR) using both SE and DG with implicit-explicit methods for the compressible Navier-Stokes equations with buoyancy (see Kopera-Giraldo JCP 2014 and Kopera-Giraldo JCP 2015). In Kopera-Giraldo JCP 2015, we show and rigorously prove that SE with non-conforming AMR conserves mass (an important result).

• In 2016, we published a paper (see Abdi-Giraldo JCP 2016) that describes in detail how to write continuous Galerkin (i.e., spectral elements) and discontinuous Galerkin (i.e., discontinuous spectral element) within the same code-base. The importance of this paper is that we are able to show how to store three different methods within the same code. We called these methods Discontinuous Galerkin (DG), Continuous Galerkin with continuous storage (CGc) and Continuous Galerkin with discontinuous storage (CGd). CGc uses the classical finite element grid point storage (no duplicate degrees of freedom) whereas CGd uses a storage similar to DG.

• In 2020, I published a book on SE and DG methods that I worked on for over ten years. I learned many hard lessons during those years (such as that tensor-product bases are much easier to implement and the resulting models are much more efficient than other high-order methods) and my hope was to share them with young researchers (for the Springer link to the book click here).

• All the hard work that we put into spectral element methods for the compressible Euler equations has paid off and made its way into the U.S. Navy's NEPTUNE weather system (for details click here).  We continue to work with NRL to improve this weather prediction system. We have worked closely with NRL to extend NEPTUNE (and also NUMA) to space weather applications (for details click here). We have also improved the boundary conditions for high-altitude simulations in Kelly et al. JCP 2023.

• In 2022, we published a paper on entropy-stable DG methods for global atmospheric modeling. The value of the  paper Waruszewski et al. JCP 2022 is that we can maintain unconditional stability due to the spatial discretization indefinitely. To prove this point, we run the baroclinic instability for hundreds of days using NO dissipation mechanism - the stability is mathematically guaranteed by the entropy-stable fluxes. We have also applied a similar technique to the Held-Suarez test case and ran it for over a thousand days with NO dissipation (see Souza et al.  JAMES 2022).

• In 2023, we developed nonlinear and linear horizontally explicit vertically implicit IMEX methods which has allowed us to develop unconditionally stable time-integration methods along the vertical direction for nonhydrostatic atmospheric models (see Giraldo et al. JCP 2023). 

• Work on time-integrators continue with papers that will appear in 2024 on multirate IMEX methods (look for Mugg et al. 2024) and preconditioned IMEX methods (look for Reddy et al. 2024)

• ​Although not our immediate mandate, in collaboration with my colleague, Prof. Michal Kopera (Boise State University), we have developed a variant of NUMA called NUMO for incompressible nonhydrostatic ocean modeling.  This work can be found in Kopera et al.  International Journal on Geomathematics 2023.

• Current Focus: We continue to improve the mathematics and numerics of NUMA; recently, we have resurrected work on GPU computing, and began work on multi-scale modeling framework (see my LLNL presentation and look for Kang et al. 2024), and scientific machine learning. In addition, we are working with our collaborators to better understand hurricanes (see Hasan et al. JAMES 2022) and extending cloud simulations to unstructured grids (see Tissaoui et al. JAMES 2022); a paper on dynamically adaptive grids is coming soon.

• GPU work: NUMA is OpenACC capable and is included in the Nvidia Testing Analysis suite that is run nightly across various hardware architectures (including Grace-Hopper). The latest results (for explicit time-stepping) show that the GPU results are 5x-8x faster than the CPU per Watt (energy consumption); the range in speedup is dependent on the GPU hardware considered (with the H100 being the fastest). We are also working with the Hartree center of the United Kingdom Science and Technology Facilities Council on using the API/DSL PSyclone to port NUMA to GPUs. Look for an upcoming white paper (in 2024) by the ICAMS HPC working group that will include these results.