| 1. |
- Burdakov, Oleg, 1953-, et al.
(författare)
-
Monotonic data fitting and interpolation with application to postprocessing of FE solutions
- 2007
-
Ingår i: CERFACS 20th Anniversary Conference on High-performance Computing,2007. ; , s. 11-12
-
Konferensbidrag (övrigt vetenskapligt/konstnärligt)abstract
- In this talk we consider the isotonic regression (IR) problem which can be formulated as follows. Given a vector $\bar{x} \in R^n$, find $x_* \in R^n$ which solves the problem: \begin{equation}\label{ir2} \begin{array}{cl} \mbox{min} & \|x-\bar{x}\|^2 \\ \mbox{s.t.} & Mx \ge 0. \end{array} \end{equation} The set of constraints $Mx \ge 0$ represents here the monotonicity relations of the form $x_i \le x_j$ for a given set of pairs of the components of $x$. 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 most challenging applications of (\ref{ir2}) are characterized by very large values of $n$. We introduce new IR algorithms. Our numerical experiments demonstrate the high efficiency of our algorithms, especially for very large-scale problems, and their robustness. They are able to solve some problems which all existing IR algorithms fail to solve. We outline also our new algorithms for monotonicity-preserving interpolation of scattered multivariate data. In this talk we focus on application of our IR algorithms in postprocessing of FE solutions. Non-monotonicity of the numerical solution is a 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 the 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 \begin{equation}\label{ls2} \begin{array}{cl} \mbox{min} & \|x-\bar{x}\|^2 \\ \mbox{s.t.} & Mx \ge 0, \quad l \le x \le u, \quad e^Tx = m, \end{array} \end{equation} where$e=(1,1, \ldots ,1)^T \in R^n$. The set of constraints $Mx \ge 0$ represents here the monotonicity relations between some of the adjacent mesh cells. The constraints $l \le x \le u$ originate from the discrete maximum principle. The last constraint 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.
|
|
| 2. |
- Burdakov, Oleg, 1953-, et al.
(författare)
-
Monotonicity recovering and accuracy preserving optimization methods for postprocessing finite element solutions
- 2011
-
Rapport (övrigt vetenskapligt/konstnärligt)abstract
- We suggest here a least-change correction to available finite element (FE) solution.This postprocessing procedure is aimed at recoveringthe monotonicity and some other important properties that may not beexhibited by the FE solution. It is based on solvinga monotonic regression problem with some extra constraints.One of them is a linear equality-type constraint which models the conservativityrequirement. The other ones are box-type constraints, andthey originate from the discrete maximum principle.The resulting postprocessing problem is a large scale quadratic optimization problem. It is proved that the postprocessedFE solution preserves the accuracy of the discrete FE approximation.We introduce an algorithm for solving the postprocessingproblem. It can be viewed as a dual ascent method basedon the Lagrangian relaxation of the equality constraint.We justify theoretically its correctness.Its efficiency is demonstrated by the presented results of numerical experiments.
|
|
| 3. |
- Burdakov, Oleg, et al.
(författare)
-
Monotonicity recovering and accuracy preserving optimization methods for postprocessing finite element solutions
- 2012
-
Ingår i: Journal of Computational Physics. - : Elsevier. - 0021-9991 .- 1090-2716. ; 231:8, s. 3126-3142
-
Tidskriftsartikel (refereegranskat)abstract
- We suggest here a least-change correction to available finite element (FE) solution. This postprocessing procedure is aimed at recovering the monotonicity and some other important properties that may not be exhibited by the FE solution. Although our approach is presented for FEs, it admits natural extension to other numerical schemes, such as finite differences and finite volumes. For the postprocessing, a priori information about the monotonicity is assumed to be available, either for the whole domain or for a subdomain where the lost monotonicity is to be recovered. The obvious requirement is that such information is to be obtained without involving the exact solution, e.g. from expected symmetries of this solution. less thanbrgreater than less thanbrgreater thanThe postprocessing is based on solving a monotonic regression problem with some extra constraints. One of them is a linear equality-type constraint that models the conservativity requirement. The other ones are box-type constraints, and they originate from the discrete maximum principle. The resulting postprocessing problem is a large scale quadratic optimization problem. It is proved that the postprocessed FE solution preserves the accuracy of the discrete FE approximation. less thanbrgreater than less thanbrgreater thanWe introduce an algorithm for solving the postprocessing problem. It can be viewed as a dual ascent method based on the Lagrangian relaxation of the equality constraint. We justify theoretically its correctness. Its efficiency is demonstrated by the presented results of numerical experiments.
|
|
| 4. |
- Burdakov, Oleg, 1953-, et al.
(författare)
-
Optimization methods for postprocessing finite element solutions
- 2007
-
Ingår i: International Conference on Numerical Optimization and Numerical Linear Algebra,2007.
-
Konferensbidrag (övrigt vetenskapligt/konstnärligt)abstract
- 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. \begin{equation}\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} \end{equation} 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.
|
|
