liu.seSearch for publications in DiVA
Change search

Cite
Citation style
• apa
• harvard1
• ieee
• modern-language-association-8th-edition
• vancouver
• oxford
• Other style
More styles
Language
• de-DE
• en-GB
• en-US
• fi-FI
• nn-NO
• nn-NB
• sv-SE
• Other locale
More languages
Output format
• html
• text
• asciidoc
• rtf
Optimization methods for postprocessing finite element solutions
Linköping University, The Institute of Technology. Linköping University, Department of Mathematics, Optimization .ORCID iD: 0000-0003-1836-4200
Institute of Numerical Mathematics Russian Academy of Sciences, Moscow, Russia.
Institute of Numerical Mathematics Russian Academy of Sciences, Moscow, Russia.
2007 (English)In: International Conference on Numerical Optimization and Numerical Linear Algebra,2007, 2007Conference paper, Published paper (Other academic)
##### Abstract [en]

Simulation of transport phenomena based on advection-diffusion equation is very popular in many engineering applications. Non-monotonicity of the numerical solution is the typical drawback of the conventional methods of approximation, such as finite elements (FE), finite volumes, and mixed finite elements. The problem of monotonicity is particularly important in cases of highly anisotropic diffusion tensors or distorted unstructured meshes. For instance, in the nuclear waste transport simulation, the non-monotonicity results in the presence of negative concentrations which may lead to unacceptable concentration and chemistry calculations failure. Another drawback of the conventional methods is a possible violation of the discrete maximum principle, which establishes lower and upper bounds for solution. We suggest here a least-change correction to the available FE solution $\bar{x} \in R^n$. This postprocessing procedure is aimed on recovering the monotonicity and some other important properties that may not be exhibited by $\bar{x}$. The mathematical formulation of the postprocessing problem is reduced to the following convex quadratic programming problem %Given the FE solution $\bar{x} \in R^n$, find $x_* \in R^n$ which %solves the list-distance problem. $$\label{ls2} \begin{array}{cl} \mbox{min} & \|x-\bar{x}\|^2 \\ \mbox{s.t.} & Mx \ge 0, \\ & l \le x \le u, \\ & e^Tx = m. \end{array}$$ The set of constraints $Mx \ge 0$ represents here the monotonicity requirements. It establishes relations between some of the adjacent mesh cells in the form $x_i \le x_j$, which relates cells $i$ and $j$. The corresponding row of the matrix $M$ is composed mainly of zeros, but its $i$th and $j$th elements, which are equal to $-1$ and $+1$, respectively. The set of constraints $l \le x \le u$ originates from the discrete maximum principle. In the last constraint, $e=(1,1, \ldots ,1)^T \in R^n$. It formulates the conservativity requirement. The postprocessing based on (\ref{ls2}) is typically a large scale problem. We introduce here algorithms for solving this problem. They are based on the observation that, in the presence of the monotonicity constraints only, problem (\ref{ls2}) is the classical monotonic regression problem, which can be solved efficiently by some of the available monotonic regression algorithms. This solution is used then for producing the optimal solution to problem (\ref{ls2}) in the presence of all the constraints. We present results of numerical experiments to illustrate the efficiency of our algorithms.

2007.
##### Keyword [en]
quadratic programming, large scale optimization, least distance problem, isotonic regression, finite elements.
Mathematics
##### Identifiers
Local ID: 53035OAI: oai:DiVA.org:liu-40344DiVA: diva2:261193
##### Note
Invited plenary talkAvailable from: 2009-10-10 Created: 2009-10-10 Last updated: 2015-06-02

No full text

Burdakov, Oleg

#### Search in DiVA

Burdakov, Oleg
##### By organisation
The Institute of TechnologyOptimization
Mathematics

urn-nbn

#### Altmetric score

urn-nbn
Total: 359 hits

Cite
Citation style
• apa
• harvard1
• ieee
• modern-language-association-8th-edition
• vancouver
• oxford
• Other style
More styles
Language
• de-DE
• en-GB
• en-US
• fi-FI
• nn-NO
• nn-NB
• sv-SE
• Other locale
More languages
Output format
• html
• text
• asciidoc
• rtf