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Research Interests
Computers were invented to automate tedious and error-prone tasks, like the vast hoards of arithemtic operations required to perform advanced numerical simulations of science and engineering problems. However, programming computers is itself a tedious and error-prone task. So, why not get a computer to do it?
At the intersection of mathematics and computer science, one finds “metanumerical computing” – the use of mathematical structure to generate, manipulate, and optimize numerical software. I have contributed to software projects such as the FEniCS project, and Firedrake. The goal is to fuse together aspects of domain-specific languages with structural and algorithmic aspects of finite elements to produce easy-to-use yet highly efficient code systems that provide efficient implementations of state-of-the-art numerical methods. Or, you can call it “numerical methods with a universal quantifier”.
These are under active development and yet widely used in applications. I am also interested in pressing forward basic research in algorithms and finite element analysis. One ongoing project is to develop low-complexity simplicial finite element methods based on Bernstein polynomials. These allow calculations on unstructured meshes with the same complexity as tensor-product elements, and also admit a wide range of different approximating spaces. How we do present such structured calculations to high-level finite element code? This problem is tackled in FInAT, which, unlike FIAT, is not a tabulator.
Also, given the ability to solve one problem well, how do we solve two problems glued together? Multiphysics problems, both from the standpoint of theoretical analysis and the development of efficient preconditioners, are one way I utilize the ability of codes such as FEniCS and Firedrake.
Given the ability to efficiently produce efficient simulation codes, how can we turn back toward a deeper understanding of numerical methods and their application? On this front, I have an ongoing interest in wave equations such as shallow water. In addition to traditional stability and error estimates, how can we understand energy conservation or dissipation in the presence of damping? How closely do numerical methods track physical expectations? How can we use the kinds of properties to get more refined estimates?
Computers were invented to automate tedious and error-prone tasks, like the vast hoards of arithemtic operations required to perform advanced numerical simulations of science and engineering problems. However, programming computers is itself a tedious and error-prone task. So, why not get a computer to do it?
At the intersection of mathematics and computer science, one finds “metanumerical computing” – the use of mathematical structure to generate, manipulate, and optimize numerical software. I have contributed to software projects such as the FEniCS project, and Firedrake. The goal is to fuse together aspects of domain-specific languages with structural and algorithmic aspects of finite elements to produce easy-to-use yet highly efficient code systems that provide efficient implementations of state-of-the-art numerical methods. Or, you can call it “numerical methods with a universal quantifier”.
These are under active development and yet widely used in applications. I am also interested in pressing forward basic research in algorithms and finite element analysis. One ongoing project is to develop low-complexity simplicial finite element methods based on Bernstein polynomials. These allow calculations on unstructured meshes with the same complexity as tensor-product elements, and also admit a wide range of different approximating spaces. How we do present such structured calculations to high-level finite element code? This problem is tackled in FInAT, which, unlike FIAT, is not a tabulator.
Also, given the ability to solve one problem well, how do we solve two problems glued together? Multiphysics problems, both from the standpoint of theoretical analysis and the development of efficient preconditioners, are one way I utilize the ability of codes such as FEniCS and Firedrake.
Given the ability to efficiently produce efficient simulation codes, how can we turn back toward a deeper understanding of numerical methods and their application? On this front, I have an ongoing interest in wave equations such as shallow water. In addition to traditional stability and error estimates, how can we understand energy conservation or dissipation in the presence of damping? How closely do numerical methods track physical expectations? How can we use the kinds of properties to get more refined estimates?
研究兴趣
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Computers & Mathematics with Applications (2024): 22-32
arxiv(2023)
arxiv(2023)
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SIAM REVIEWno. 1 (2023): 319-+
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CoRR (2023)
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SIAM Journal on Scientific Computing (2023)
SIAM REVIEWno. 1 (2023): 317-+
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