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 criterion, from some set of available alternatives. It is generally divided into two subfi ...

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 imposes relations between the various partial derivatives of a multivariable function. The function is often thought of as an "unknown" to be solved for, similarly to ...

. Typical domains where these problems arise include

aerodynamics Aerodynamics, from grc, ἀήρ ''aero'' (air) + grc, δυναμική (dynamics), is the study of the motion of air, particularly when affected by a solid object, such as an airplane wing. It involves topics covered in the field of fluid dyn ...

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 flows. Computers are used to perform the calculations required to simulate t ...

image segmentation In digital image processing and computer vision, image segmentation is the process of partitioning a digital image into multiple image segments, also known as image regions or image objects ( sets of pixels). The goal of segmentation is to simpl ...

, and

inverse problems An inverse problem in science is the process of calculating from a set of observations the causal factors that produced them: for example, calculating an image in X-ray computed tomography, source reconstruction in acoustics, or calculating the ...

. 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 = u

where

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, that is, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidea ...

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 th ...

Applications

* Aerodynamic shape optimization *

Drug delivery Drug delivery refers to approaches, formulations, manufacturing techniques, storage systems, and technologies involved in transporting a pharmaceutical compound to its target site to achieve a desired therapeutic effect. Principles related to dr ...

Mathematical finance Mathematical finance, also known as quantitative finance and financial mathematics, is a field of applied mathematics, concerned with mathematical modeling of financial markets. In general, there exist two separate branches of finance that requir ...

Optimal control of bacterial chemotaxis system

The following example comes from p. 20-21 of Pearson. Chemotaxis 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 Chemotaxis (from '' chemo-'' + '' taxis'') is the movement of an organism or entity in response to a chemical stimulus. Somatic cells, bacteria, and other single-cell or multicellular organisms direct their movements according to certain chemi ...

, it is possible to formulate a boundary control problem:

+ \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:

&= 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 \end

where

\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 function on Euclidean space. It is usually denoted by the symbols \nabla\cdot\nabla, \nabla^2 (where \nabla is t ...

References

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

Applications

Optimal control of bacterial chemotaxis system

See also

References

Further reading

External links