PDE-constrained Optimization
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PDE-constrained optimization is a subset of
mathematical optimization Mathematical optimization (alternatively spelled ''optimisation'') or mathematical programming is the selection of a best element, with regard to some criteria, from some set of available alternatives. It is generally divided into two subfiel ...
where at least one of the constraints may be expressed as a
partial differential equation In mathematics, a partial differential equation (PDE) is an equation which involves a multivariable function and one or more of its partial derivatives. The function is often thought of as an "unknown" that solves the equation, similar to ho ...
. Typical domains where these problems arise include
aerodynamics Aerodynamics () is the study of the motion of atmosphere of Earth, air, particularly when affected by a solid object, such as an airplane wing. It involves topics covered in the field of fluid dynamics and its subfield of gas dynamics, and is an ...
,
computational fluid dynamics Computational fluid dynamics (CFD) is a branch of fluid mechanics that uses numerical analysis and data structures to analyze and solve problems that involve fluid dynamics, fluid flows. Computers are used to perform the calculations required ...
, image segmentation, and
inverse problems ''Inverse Problems'' is a peer-reviewed, broad-based interdisciplinary journal for pure and applied mathematicians and physicists produced by IOP Publishing. It combines theoretical, experimental and mathematical papers on inverse problems wit ...
. A standard formulation of PDE-constrained optimization encountered in a number of disciplines is given by:\min_ \; \frac 1 2 \, y-\widehat\, _^2 + \frac\beta2 \, u\, _^2, \quad \text \; \mathcaly = uwhere u is the control variable and \, \cdot\, _^ is the squared
Euclidean norm Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are ''Euclidean spaces'' ...
and is not a norm itself. Closed-form solutions are generally unavailable for PDE-constrained optimization problems, necessitating the development of
numerical methods Numerical analysis is the study of algorithms that use numerical approximation (as opposed to symbolic manipulations) for the problems of mathematical analysis (as distinguished from discrete mathematics). It is the study of numerical methods t ...
.


Applications

* Aerodynamic shape optimization *
Drug delivery Drug delivery involves various methods and technologies designed to transport pharmaceutical compounds to their target sites helping therapeutic effect. It involves principles related to drug preparation, route of administration, site-specif ...
*
Mathematical finance Mathematical finance, also known as quantitative finance and financial mathematics, is a field of applied mathematics, concerned with mathematical modeling in the financial field. In general, there exist two separate branches of finance that req ...
*
Epidemiology Epidemiology is the study and analysis of the distribution (who, when, and where), patterns and Risk factor (epidemiology), determinants of health and disease conditions in a defined population, and application of this knowledge to prevent dise ...


Optimal control of bacterial chemotaxis system

The following example comes from p. 20-21 of Pearson.
Chemotaxis Chemotaxis (from ''chemical substance, chemo-'' + ''taxis'') is the movement of an organism or entity in response to a chemical stimulus. Somatic cells, bacteria, and other single-cell organism, single-cell or multicellular organisms direct thei ...
is the movement of an organism in response to an external chemical stimulus. One problem of particular interest is in managing the spatial dynamics of bacteria that are subject to chemotaxis to achieve some desired result. For a cell density z(t,) and concentration density c(t,) of a chemoattractant, it is possible to formulate a boundary control problem:\min_ \; \int_\left (T,)-\widehat \right + \int_\left (T,)-\widehat \right + \int_^\int_u^where \widehat is the ideal cell density, \widehat is the ideal concentration density, and u is the control variable. This objective function is subject to the dynamics:\begin - D_\Delta z - \alpha \nabla \cdot \left z \right&= 0 \quad \text \quad \Omega \\ - \Delta c + \rho c - w &= 0 \quad \text \quad \Omega \\ &= 0 \quad \text \quad \partial\Omega \\ + \zeta (c-u) &= 0 \quad \text \quad \partial\Omega \endwhere \Delta is the
Laplace operator In mathematics, the Laplace operator or Laplacian is a differential operator given by the divergence of the gradient of a Scalar field, scalar function on Euclidean space. It is usually denoted by the symbols \nabla\cdot\nabla, \nabla^2 (where \ ...
.


See also

* Multiphysics * Shape optimization * SU2 code


References


Further reading

* Antil, Harbir; Kouri, Drew. P; Lacasse, Martin-D.; Ridzal, Denis (2018).
Frontiers in PDE-Constrained Optimization
'. The IMA Volumes in Mathematics and its Applications, Springer. . * Tröltzsch, Fredi (2010).
Optimal Control of Partial Differential Equations: Theory, Methods, and Applications
'. Graduate Studies in Mathematics, American Mathematical Society. {{ISBN, 978-0-8218-4904-0}.


External links


A Brief Introduction to PDE Constrained Optimization

PDE Constrained Optimization

Optimal solvers for PDE-Constrained Optimization

Model Problems in PDE-Constrained Optimization
Mathematical optimization Optimal control Partial differential equations