Automatic Type Tracking: Operator Result Type Priority and Applications in Algorithmic Differentiation (with Examples from Mathematical Finance using Stochastic Automatic Differentiation)

A Multiphase CMAQ Version 5.0 Adjoint

10.5194/gmd-2019-287 ◽

2019 ◽

Author(s):

Shunliu Zhao ◽

Matthew G. Russell ◽

Amir Hakami ◽

Shannon L. Capps ◽

Matthew D. Turner ◽

...

Keyword(s):

Gas Phase ◽

Automatic Differentiation ◽

Transport Model ◽

Source Attribution ◽

Chemical Transport Model ◽

Adjoint Model ◽

Algorithmic Differentiation ◽

Time Specific ◽

Continuous Adjoint ◽

Discrete Adjoints

Abstract. We present the development of a multiphase adjoint for the Community Multiscale Air Quality (CMAQ) model, a widely used chemical transport model. The adjoint provides location- and time-specific gradients that can be used in various applications such as backward sensitivity analysis, source attribution, optimal pollution control, data assimilation and inverse modeling. The science processes of the CMAQ model include gas-phase chemistry, aerosols, cloud chemistry and dynamics, diffusion and advection. Discrete adjoints are implemented for all the science processes, with an additional continuous adjoint for advection. The development of discrete adjoints is assisted with Algorithmic Differentiation (AD) tools. Particularly, the Kinetic PreProcessor (KPP) is implemented for gas-phase and aqueous chemistry, and two different automatic differentiation tools are used for other processes such as clouds, aerosols, diffusion, and advection. The continuous adjoint of advection is developed manually. For adjoint validation, the brute-force or Finite Difference Method (FDM) is implemented process by process with box- or column-model simulations. Due to the inherent limitations of the FDM caused by numerical round-off errors, the Complex Variable Method (CVM) is adopted where necessary. The adjoint model often shows better agreement with the CVM than with the FDM. The adjoints of all science processes compare favorably with the FDM/CVM. In an example application of the full, multiphase adjoint model, we provide the first estimates of how emissions of PM2.5 affect public health across the US.

Download Full-text

A multiphase CMAQ version 5.0 adjoint

Geoscientific Model Development ◽

10.5194/gmd-13-2925-2020 ◽

2020 ◽

Vol 13 (7) ◽

pp. 2925-2944 ◽

Cited By ~ 1

Author(s):

Shunliu Zhao ◽

Matthew G. Russell ◽

Amir Hakami ◽

Shannon L. Capps ◽

Matthew D. Turner ◽

...

Keyword(s):

Gas Phase ◽

Automatic Differentiation ◽

Transport Model ◽

Source Attribution ◽

Chemical Transport Model ◽

Adjoint Model ◽

Algorithmic Differentiation ◽

Time Specific ◽

Continuous Adjoint ◽

Discrete Adjoints

Abstract. We present the development of a multiphase adjoint for the Community Multiscale Air Quality (CMAQ) model, a widely used chemical transport model. The adjoint model provides location- and time-specific gradients that can be used in various applications such as backward sensitivity analysis, source attribution, optimal pollution control, data assimilation, and inverse modeling. The science processes of the CMAQ model include gas-phase chemistry, aerosol dynamics and thermodynamics, cloud chemistry and dynamics, diffusion, and advection. Discrete adjoints are implemented for all the science processes, with an additional continuous adjoint for advection. The development of discrete adjoints is assisted with algorithmic differentiation (AD) tools. Particularly, the Kinetic PreProcessor (KPP) is implemented for gas-phase and aqueous chemistry, and two different automatic differentiation tools are used for other processes such as clouds, aerosols, diffusion, and advection. The continuous adjoint of advection is developed manually. For adjoint validation, the brute-force or finite-difference method (FDM) is implemented process by process with box- or column-model simulations. Due to the inherent limitations of the FDM caused by numerical round-off errors, the complex variable method (CVM) is adopted where necessary. The adjoint model often shows better agreement with the CVM than with the FDM. The adjoints of all science processes compare favorably with the FDM and CVM. In an example application of the full multiphase adjoint model, we provide the first estimates of how emissions of particulate matter (PM2.5) affect public health across the US.

Download Full-text

Another Multibody Dynamics in Natural Coordinates through Automatic Differentiation and High-Index DAE Solving

Acta Cybernetica ◽

10.14232/actacyb.24.3.2020.4 ◽

2020 ◽

Vol 24 (3) ◽

pp. 315-341

Author(s):

John D Pryce ◽

Nedialko Nedialkov

Keyword(s):

Automatic Differentiation ◽

Mass Matrix ◽

Equations Of Motion ◽

3D Models ◽

High Index ◽

Natural Coordinates ◽

Differential Algebraic Equation ◽

Sparse Models ◽

Algorithmic Differentiation ◽

A Minor

The Natural Coordinates (NCs) method for Lagrangian modelling and simulation of multi-body systems is valued for giving simple, sparse models. We describe our version of it (NPNCs) and compare with the classical ap- proach of Jalón and Bayo (JBNCs). NPNCs use the high-index differential- algebraic equation solver DAETS. Algorithmic differentiation, not symbolic algebra, forms the equations of motion from the Lagrangian. NPNCs give significantly smaller equation systems than JBNCs, at the cost of a non- constant mass matrix for fully 3D models—a minor downside in the DAETS context. A 2D and a 3D example are presented, with numerical results.

Download Full-text

A mathematical view of automatic differentiation

Acta Numerica ◽

10.1017/s0962492902000132 ◽

2003 ◽

Vol 12 ◽

pp. 321-398 ◽

Cited By ~ 32

Author(s):

Andreas Griewank

Keyword(s):

Program Transformation ◽

Scientific Computing ◽

Automatic Differentiation ◽

Floating Point ◽

Complexity Bounds ◽

Algorithmic Differentiation ◽

Floating Point Arithmetic ◽

Efficient Calculation ◽

Point Arithmetic

Automatic, or algorithmic, differentiation addresses the need for the accurate and efficient calculation of derivative values in scientific computing. To this end procedural programs for the evaluation of problem-specific functions are transformed into programs that also compute the required derivative values at the same numerical arguments in floating point arithmetic. Disregarding many important implementation issues, we examine in this article complexity bounds and other more mathematical aspects of the program transformation task sketched above.

Download Full-text

Degenerate Diffusion Operators Arising in Population Biology (AM-185)

10.23943/princeton/9780691157122.001.0001 ◽

2017 ◽

Author(s):

Charles L. Epstein ◽

Rafe Mazzeo

Keyword(s):

Adjoint Operator ◽

Integral Kernel ◽

Mathematical Finance ◽

Elliptic Operators ◽

Complete Analysis ◽

Time Behavior ◽

Degenerate Elliptic Operators ◽

Regularity Properties ◽

Degenerate Elliptic ◽

Mathematical Foundations

This book provides the mathematical foundations for the analysis of a class of degenerate elliptic operators defined on manifolds with corners, which arise in a variety of applications such as population genetics, mathematical finance, and economics. The results discussed in this book prove the uniqueness of the solution to the martingale problem and therefore the existence of the associated Markov process. The book uses an “integral kernel method” to develop mathematical foundations for the study of such degenerate elliptic operators and the stochastic processes they define. The precise nature of the degeneracies of the principal symbol for these operators leads to solutions of the parabolic and elliptic problems that display novel regularity properties. Dually, the adjoint operator allows for rather dramatic singularities, such as measures supported on high codimensional strata of the boundary. The book establishes the uniqueness, existence, and sharp regularity properties for solutions to the homogeneous and inhomogeneous heat equations, as well as a complete analysis of the resolvent operator acting on Hölder spaces. It shows that the semigroups defined by these operators have holomorphic extensions to the right half plane. The book also demonstrates precise asymptotic results for the long-time behavior of solutions to both the forward and backward Kolmogorov equations.

Download Full-text