Krylov methods for low‐rank commuting generalized Sylvester equations

Jarlebring Elias; Mele Giampaolo; Palitta Davide; Ringh Emil

摘要

Summary >We consider generalizations of the Sylvester matrix equation, consisting of the sum of a Sylvester operator and a linear operator Π with a particular structure. More precisely, the commutators of the matrix coefficients of the operator Π and the Sylvester operator coefficients are assumed to be matrices with low rank. We show (under certain additional conditions) low‐rank approximability of this problem, that is, the solution to this matrix equation can be approximated with a low‐rank matrix. Projection methods have successfully been used to solve other matrix equations with low‐rank approximability. We propose a new projection method for this class of matrix equations. The choice of the subspace is a crucial ingredient for any projection method for matrix equations. Our method is based on an adaption and extension of the extended Krylov subspace method for Sylvester equations. A constructive choice of the starting vector/block is derived from the low‐rank commutators. We illustrate the effectiveness of our method by solving large‐scale matrix equations arising from applications in control theory and the discretization of PDEs. The advantages of our approach in comparison to other methods are also illustrated. </abstract> </span> <span class="z_kbtn z_kbtnclass hoverxs" style="display: none;">展开▼</span> </div> <div class="translation abstracttxt"> <span class="zhankaihshouqi fivelineshidden" id="abstract"> <span>机译：</span><Abstract Type =“Main”XML：Lang =“en”XML：ID =“NLA2176-ABS-ABS-ABS-0001”> <标题类型=“main”>摘要</ title> >我们考虑Sylvester矩阵方程的概括，由具有特定结构的Sylvester操作员和线性操作员的总和组成。更确切地说，假设操作员π和Sylvester运算符系数的矩阵系数的换向器是具有低等级的矩阵。我们展示（在某些额外条件下）该问题的低秩近似性，即，该矩阵方程的解决方案可以用低秩矩阵近似。投影方法已成功地用于解决具有低秩近似性的其他矩阵方程。我们为这类矩阵方程提出了一种新的投影方法。子空间的选择是用于矩阵方程的任何投影方法的重要成分。我们的方法基于SylVester方程的扩展Krylov子空间方法的适应和扩展。起始载体/块的建设性选择是从低级换向器衍生的。我们通过求解来自控制理论中的应用和PDE的离散化的大规模矩阵方程来说明我们的方法的有效性。还示出了与其他方法相比的方法的优点。</ p> </ abstract> </span> <span class="z_kbtn z_kbtnclass hoverxs" style="display: none;">展开▼</span> </div> </div> <div class="record"> <h2 class="all_title" id="enpatent33" > 著录项</h2> <ul> <li> <span class="lefttit">来源</span> <div style="width: 86%;vertical-align: text-top;display: inline-block;"> <a href='/journal-foreign-26104/'>《Numerical linear algebra with applications 》</a> <b style="margin: 0 2px;">|</b><span>2018年第6期</span><b style="margin: 0 2px;">|</b> <span>共17页</span> </div> </li> <li> <div class="author"> <span class="lefttit">作者</span> <p id="fAuthorthree" class="threelineshidden zhankaihshouqi"> <a href="/search.html?doctypes=4_5_6_1-0_4-0_1_2_3_7_9&sertext=Jarlebring Elias&option=202" target="_blank" rel="nofollow">Jarlebring Elias;</a> <a href="/search.html?doctypes=4_5_6_1-0_4-0_1_2_3_7_9&sertext=Mele Giampaolo&option=202" target="_blank" rel="nofollow">Mele Giampaolo;</a> <a href="/search.html?doctypes=4_5_6_1-0_4-0_1_2_3_7_9&sertext=Palitta Davide&option=202" target="_blank" rel="nofollow">Palitta Davide;</a> <a href="/search.html?doctypes=4_5_6_1-0_4-0_1_2_3_7_9&sertext=Ringh Emil&option=202" target="_blank" rel="nofollow">Ringh Emil;</a> </p> <span class="z_kbtnclass z_kbtnclassall hoverxs" id="zkzz" style="display: none;">展开▼</span> </div> </li> <li> <div style="display: flex;"> <span class="lefttit">作者单位</span> <div style="position: relative;margin-left: 3px;max-width: 639px;"> <div class="threelineshidden zhankaihshouqi" id="fOrgthree"> <p>Department of MathematicsKTH Royal Institute of Technology Swedish e‐science research center (SeRC)Lindstedtsv?gen 25 SE‐100 44 Stockholm Sweden;</p> <p>Department of MathematicsKTH Royal Institute of Technology Swedish e‐science research center (SeRC)Lindstedtsv?gen 25 SE‐100 44 Stockholm Sweden;</p> <p>Dipartimento di MatematicaUniversità di BolognaPiazza di Porta S. Donato 5 I‐40127 Bologna Italy;</p> <p>Department of MathematicsKTH Royal Institute of Technology Swedish e‐science research center (SeRC)Lindstedtsv?gen 25 SE‐100 44 Stockholm Sweden;</p> </div> <span class="z_kbtnclass z_kbtnclassall hoverxs" id="zhdw" style="display: none;">展开▼</span> </div> </div> </li> <li> <span class="lefttit">收录信息</span> <span style="width: 86%;vertical-align: text-top;display: inline-block;"></span> </li> <li> <span class="lefttit">原文格式</span> <span>PDF</span> </li> <li> <span class="lefttit">正文语种</span> <span>eng</span> </li> <li> <span class="lefttit">中图分类</span> <span> <a href="https://www.zhangqiaokeyan.com/clc/4602.html" title="代数方程论、线性代数">代数方程论、线性代数 ;</a></span> </li> <li class="antistop"> <span class="lefttit">关键词</span> <p style="width: 86%;vertical-align: text-top;"> <a style="color: #3E7FEB;" href="/search.html?doctypes=4_5_6_1-0_4-0_1_2_3_7_9&sertext=generalized Sylvester equation&option=203" rel="nofollow">generalized Sylvester equation;</a> <a style="color: #3E7FEB;" href="/search.html?doctypes=4_5_6_1-0_4-0_1_2_3_7_9&sertext=iterative solvers&option=203" rel="nofollow">iterative solvers;</a> <a style="color: #3E7FEB;" href="/search.html?doctypes=4_5_6_1-0_4-0_1_2_3_7_9&sertext=Krylov subspace&option=203" rel="nofollow">Krylov subspace;</a> <a style="color: #3E7FEB;" href="/search.html?doctypes=4_5_6_1-0_4-0_1_2_3_7_9&sertext=low‐rank commutation&option=203" rel="nofollow">low‐rank commutation;</a> <a style="color: #3E7FEB;" href="/search.html?doctypes=4_5_6_1-0_4-0_1_2_3_7_9&sertext=matrix equation&option=203" rel="nofollow">matrix equation;</a> <a style="color: #3E7FEB;" href="/search.html?doctypes=4_5_6_1-0_4-0_1_2_3_7_9&sertext=projection methods&option=203" rel="nofollow">projection methods;</a> </p> <div class="translation"> 机译：广义Sylvester方程;迭代求解器;Krylov子空间;低秩换向;矩阵方程;投影方法; 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href="/patent-detail/06130419345857.html">METHOD FOR SEEKING SEMI-ANALYTICAL SOLUTIONS TO MULTI-SPECIES CONTAMINANT TRANSPORT EQUATIONS COUPLED WITH SEQUENTIAL FIRST-ORDER REACTIONS THROUGH SIMILARITY TRANSFORMATION OF CLEMENT AND A GENERALIZED INTEGRAL TRANSFORM TECHNIQUE</a> <b>[P]</b> . <span> 外国专利：  KR101262260B1 </span> <span> . 2013-05-08</span> </div> <p class="zwjiyix translation" style="max-width: initial;height: auto;word-break: break-all;white-space: initial;text-overflow: initial;overflow: initial;"> <span>机译：通过相似相似转换和广义积分变换技术求与顺序一阶反应耦合的多污染物传输方程的半解析解的方法 </span> </p> </li> </ul> </div> </div> </div> <div class="theme cardcommon" style="overflow: auto;display:none"> <h3 class="all_title" id="enpatent55">相关主题</h3> <ul id="subject"> </ul> </div> </div> </div> </div> <div class="right rightcon"> <div class="details_img cardcommon clearfix" style="margin-bottom: 10px;display:none;" > </div> </div> </div> <div id="thesis_get_original1" class="downloadBth" style="bottom: 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