A Subgradient Method for Free Material Design

Kód výsledku v IS VaVaI

<a href="https://www.isvavai.cz/riv?ss=detail&h=RIV%2F67985556%3A_____%2F16%3A00507124" target="_blank" >RIV/67985556:_____/16:00507124 - isvavai.cz</a>

Výsledek na webu

<a href="https://epubs.siam.org/doi/10.1137/15M1019660" target="_blank" >https://epubs.siam.org/doi/10.1137/15M1019660</a>

DOI - Digital Object Identifier

<a href="http://dx.doi.org/10.1137/15M1019660" target="_blank" >10.1137/15M1019660</a>

Jazyk výsledku

angličtina

Název v původním jazyce

A Subgradient Method for Free Material Design

Popis výsledku v původním jazyce

A small improvement in the structure of the material could save the manufactory a lot of money. The free material design can be formulated as an optimization problem. However, due to its large scale, second order methods cannot solve the free material design problem in reasonable size. We formulate the free material optimization (FMO) problem into a saddle-point form in which the inverse of the stiffness matrix A(E) in the constraint is eliminated. The size of A(E) is generally large, denoted as N × N. We apply the primal-dual subgradient method to solve the restricted saddle-point formula. This is the first gradient-type method for FMO. Each iteration of our algorithm takes a total of O(N^2) floating-point operations and an auxiliary vector storage of size O(N), compared with formulations having the inverse of A(E) which requires O(N^3) arithmetic operations and an auxiliary vector storage of size O(N^2). To solve the problem, we developed a closed-form solution to a semidefinite least squares problem and an efficient parameter update scheme for the gradient method, which are included in the appendix. We also approximate a solution to the bounded Lagrangian dual problem. The problem is decomposed into small problems each only having an unknown of k × k (k = 3 or 6) matrix, and can be solved in parallel. The iteration bound of our algorithm is optimal for general subgradient scheme. Finally we present promising numerical results.n

Název v anglickém jazyce

A Subgradient Method for Free Material Design

Popis výsledku anglicky

A small improvement in the structure of the material could save the manufactory a lot of money. The free material design can be formulated as an optimization problem. However, due to its large scale, second order methods cannot solve the free material design problem in reasonable size. We formulate the free material optimization (FMO) problem into a saddle-point form in which the inverse of the stiffness matrix A(E) in the constraint is eliminated. The size of A(E) is generally large, denoted as N × N. We apply the primal-dual subgradient method to solve the restricted saddle-point formula. This is the first gradient-type method for FMO. Each iteration of our algorithm takes a total of O(N^2) floating-point operations and an auxiliary vector storage of size O(N), compared with formulations having the inverse of A(E) which requires O(N^3) arithmetic operations and an auxiliary vector storage of size O(N^2). To solve the problem, we developed a closed-form solution to a semidefinite least squares problem and an efficient parameter update scheme for the gradient method, which are included in the appendix. We also approximate a solution to the bounded Lagrangian dual problem. The problem is decomposed into small problems each only having an unknown of k × k (k = 3 or 6) matrix, and can be solved in parallel. The iteration bound of our algorithm is optimal for general subgradient scheme. Finally we present promising numerical results.n

Druh

J_imp - Článek v periodiku v databázi Web of Science

CEP obor

—

OECD FORD obor

10101 - Pure mathematics

Projekt

—

Návaznosti

I - Institucionalni podpora na dlouhodoby koncepcni rozvoj vyzkumne organizace

Rok uplatnění

2016

Kód důvěrnosti údajů

S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů

Název periodika

SIAM Journal on Optimization

ISSN

1052-6234

e-ISSN

—

Svazek periodika

26

Číslo periodika v rámci svazku

4

Stát vydavatele periodika

US - Spojené státy americké

Počet stran výsledku

41

Strana od-do

2314-2354

Kód UT WoS článku

000391853600014

EID výsledku v databázi Scopus

2-s2.0-85007240765