Solution of an inverse problem for ＂fixed-fixed＂ and ＂fixed-free＂ spring-mass systems
线性方程组的几种求解方法
甘肃政法学院本科学年论文（设计）题目浅议线性方程组的几种求解方法学号：姓名：指导教师：成绩：__________________完成时间： 2012 年 11 月目录第一章引言 (1)第二章线性方程组的几种解法 (1)2.1 斯消元法 (1)2.1.1 消元过程 (1)2.1.2 回代过程 (2)2.1.3 解的判断 (2)2.2 克莱姆法则 (3)2.3 LU分解法 (4)2.4 追赶法 (6)第三章结束语 (8)致谢 (8)参考文献 (9)摘要:线性方程组是线性代数的核心内容之一，其解法研究是代数学中经典且重要的研究课题.下面将综述几种不同类型的线性方程组的解法，如消元法、克莱姆法则、直接三角形法、、追赶法，并以具体例子介绍不同解法的应用技巧. 在这些解法中，高斯消元法方法，具有表达式清晰，使用范围广的特点.另外，这些方法有利于快速有效地解决线性方程组的求解问题，为解线性方程组提供一个简易平台，促进了理论与实际的结合。
关键词:线性方程组;解法;应用Several methods of solving linear equation groupAbstract: The system of linear equations is one of linear algebra core contents, its solution research is in the algebra the classics also the important research topic. This article summarized several kind of different type system of linear equations solution, like the elimination, the Cramer principle, the generalized inverse matrix law, the direct triangle law, the square root method, pursue the law, and by concrete example introduction different solution application skill. In these solutions, the generalized inverse matrix method, has the expression to be clear, use scope broad characteristic. Moreover, these methods favor effectively solve the system of linear equations solution problem fast, provides a simple platform for the solution system of linear equations, promoted the theory and the actual union.Key word: Linear equations; Solution ; Example第一章 引言线性方程组理论是高等数学中十分重要的内容，而线性方程组的解法是利用线性方程组理论解决问题的关键.下面将介绍线性方程组的消元法、追赶法、直接三角形法等求解方法，为求解线性方程组提供一个平台。
Inverse Problems for Partial Differential Equations
Mathematics Subject Classification (1991): 35R30, 86A22, 80A23, 65M30, 35JXX, 35L With four illustrations.
Library of Congress Cataloging-in-Publication Data Isakov, Victor, 1947— Inverse problems for partial differential equations / Victor Isakov. p. cm. — (Applied mathematical sciences ; 127) Includes bibliographical references (p. - ) and index. ISBN 978-1-4899-0032-6 1. Inverse problems (Differential equations) 2. Differential equations, Partial. I. Title. II. Series: Applied mathematical sciences (Springer-Verlag New York Inc.) ; v. 127. QA1.A647 vol.127 [QA374] 510s-^dc21 97-16663 [515'.353] Printed on acid-free paper. © 1998 Springer Science+Business Media New York Originally published by Springer-Verlag New York, Inc. in 1998 Softcover reprint of the hardcover 1st edition 1998 All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher, except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, similar or dissimilar methodology now known or hereafter developed is forbidden. The use of general descriptive names, trade names, trademarks, etc., in this publication, even if the former are not especially identified, is not to be taken as a sign that such names, as understood by the Trade Marks and Merchandise Marks Act, may accordingly be used freely by anyone. Production managed by Timothy Taylor; manufacturing supervised by Joe Quatela. Typeset by the Bartlett Press, Inc., Marietta, GA. SPIN 10628737 9 8 7 6 5 4 3 2 1 ISBN 978-1-4899-0032-6 ISBN 978-1-4899-0030-2 (eBook) DOI 10.1007/978-1-4899-0030-2
试图找到解决办法英语作文
试图找到解决办法英语作文Searching for Solutions。
In today’s complex and ever-changing world, problems are inevitable. However, what sets successful individuals and organizations apart is their ability to find solutions to these problems. Whether it’s a personal or professional issue, there are several steps that can be taken toidentify and resolve the problem at hand.The first step in finding a solution is to clearly define the problem. This involves asking questions, gathering information, and analyzing the situation. It’s important to understand the root cause of the problem and the factors that contribute to it. Once the problem is clearly defined, it’s easier to identify potential solutions.The next step is to brainstorm potential solutions. This involves generating as many ideas as possible, withoutjudging or dismissing any of them. It’s important to involve others in the brainstorming process, as different perspectives can lead to more creative and effective solutions. Once a list of potential solutions is generated, it’s important t o evaluate each one based on its feasibility, effectiveness, and potential consequences.After evaluating potential solutions, it’s time to select the best one. This involves weighing the pros and cons of each solution and determining which one is most likely to solve the problem and have the least negative consequences. It’s important to consider the resources and time required to implement the solution, as well as the potential impact on stakeholders.Once a solution is selected, it’s important to develop an action plan. This involves outlining the steps required to implement the solution, assigning responsibilities, and setting deadlines. It’s important to communicate the action plan to all relevant stakeholders and to monitor progress regularly.Finally, it’s important to evaluate the effectiveness of the solution. This involves measuring the results and determining whether the problem has been fully resolved. If the solution was not effective, it’s important to go back to the previous steps and identify a new solution.In conclusion, finding solutions to problems requires a systematic and thoughtful approach. By clearly defining the problem, brainstorming potential solutions, evaluating and selecting the best one, developing an action plan, and evaluating the effectiveness of the solution, individuals and organizations can overcome even the most complex challenges.。
Inverse Problems and Regularizatio An Introduction
Introduction to Regularization
Example: Landweber iteration
Landweber iteration
[Landweber ’51]
Solve normal equation by Richardson iteration Landweber iteration xk +1 = xk − F ∗ (F (xk ) − y ) k = 0, . . .
xα = (F ∗ F + αI )−1 F ∗ y
Introduction to Regularization
Example: Tikhonov Regularization
Error estimates (under some conditions) δ2 + α (Stability)
xα − x †
What are Inverse Problems ?
One possible definition [Engl, Hanke, Neubauer ’96]: Inverse problems are concerned with determining causes for a desired or an observed effect.
(Parameter, Unknown,
Solution of Inv. Prob, . . .)
Cause
Direct Problem =⇒ Inverse Problem ⇐=
Effect (Data, Observation, . . .)
Introduction to Regularization
Theorem: An linear operator with finite dimensional range is always well-posed (in Nashed’s sense).
如何处理问题面对挑战的技巧的英语作文
如何处理问题面对挑战的技巧的英语作文Facing challenges and solving problems are inevitable in our lives. Whether it's a difficult project at work, a conflict in relationships, or a personal crisis, learning how to effectively handle problems and overcome challenges is a crucial skill that can greatly impact our success and well-being. In this article, I will discuss some key techniques and strategies for dealing with problems and facing challenges head-on.First and foremost, it is important to stay calm and composed when facing a problem. Emotions such as anger, fear, and frustration can cloud our judgment and hinder our ability to think clearly and rationally. Taking a deep breath, stepping back from the situation, and giving yourself some time to cool off can help you approach the problem with a clearer mind and make better decisions.Next, it is essential to define the problem accurately and thoroughly. Take the time to identify the root cause of the issue and determine the scope and impact of the problem. By understanding the problem thoroughly, you can develop a more effective and targeted solution.Once you have a clear understanding of the problem, it is important to brainstorm possible solutions. Consider all possible options and weigh the pros and cons of each. It can also be helpful to seek input and advice from others who may have different perspectives or experiences that can help you generate new ideas and solutions.After identifying potential solutions, it is important to develop a plan of action. Break down the problem into smaller, more manageable tasks and create a timeline for completing each step. Setting specific goals and milestones can help you stay motivated and focused on making progress towards solving the problem.While working towards a solution, it is important to stay flexible and open-minded. Not every solution will work, and it may be necessary to adjust your approach as you learn more about the problem and its complexities. Be willing to adapt and try new strategies if needed to find a successful resolution.Finally, it is important to stay positive and maintain a growth mindset. View problems and challenges as opportunities for learning and growth rather than insurmountable obstacles. By approaching problems with a positive attitude and a willingness to learn from your experiences, you can develop resilience, buildconfidence, and become better equipped to handle future challenges.In conclusion, dealing with problems and facing challenges requires a combination of effective problem-solving skills, emotional intelligence, and a positive mindset. By staying calm, defining the problem, brainstorming solutions, developing a plan of action, staying flexible, and maintaining a positive attitude, you can overcome obstacles and emerge stronger and more resilient. Remember that problems are a natural part of life, and by developing the skills to handle them effectively, you can achieve greater success and fulfillment in all areas of your life.。
数学物理方程讲义姜礼尚答案
数学物理方程讲义姜礼尚答案11许绍浦《数学分析教程》南京大学出版社这些书应该够了，其他书不一一列举。
从中选择一本当作课本就可以了。
外国数学分析教材：11《微积分学教程》菲赫金格尔茨著数学分析第一名著，不要被它的大部头吓到。
我大四上半年开始看，发现写的非常清楚，看起来很快的。
强烈推荐大家看一下，哪怕买了收藏。
买书不建议看价格，而要看书好不好。
一本好的教科书能打下坚实的基础，影响今后的学习。
12《数学分析原理》菲赫金格尔茨著上本书的简写，不提倡看，要看就看上本。
13《数学分析》卓立奇观点很新，最近几年很流行，不过似乎没有必要。
14《数学分析简明教程》辛钦课后没有习题，但是推荐了《吉米多维奇数学分析习题集》里的相应习题。
但是随着习题集的更新，题已经对不上号了，不过辛钦的文笔还是不错的。
15《数学分析讲义》阿黑波夫等著莫斯科大学的讲义，不过是一本讲义，看着极为吃力，不过用来过知识点不错。
16《数学分析八讲》辛钦大师就是大师，强烈推荐。
17《数学分析原理》rudin中国的数学是从前苏联学来的，和俄罗斯教材比较像，看俄罗斯的书不会很吃力。
不过这本美国的书还是值得一看的。
写的简单明了，可以自己试着把上面的定理推导一遍。
18《微积分与分析引论》库朗又一本美国的经典数学分析书。
有人认为观点已经不流行了，但是数学分析是一门基础课目的是打下一个好的基础。
19《流形上的微积分》斯皮瓦克分析的进一步。
中国的数学分析一般不讲流形上的微积分，不过流形上的微积分是一种潮流，还是看一看的好。
20《在南开大学的演讲》陈省身从中会有一些领悟，不过可惜好像网络上流传的版本少了一些内容。
21华罗庚《高等数学引论》科学出版社数学分析习题集不做题就如同没有学过一样。
希望将课本后的习题一道道自己做完，不要看答案。
买习题集也要买习题集，不买习题集的答案。
1《吉米多维奇数学分析习题集》最近几年人们人云亦云的说这本书多么不好，批评计算题数目过多，不适合数学系等等。
9SiCr合金刃具钢在不同介质淬火后性能比较
9SiCr合金刃具钢在不同介质淬火后性能比较程赫明,谢建斌,李建云(昆明理工大学工程力学系,云南昆明650093)摘要:通过对9SiCr合金刃具钢在清水、锭子油和高压气体等淬火介质中淬火对比实验,研究了9SiCr合金刃具钢在不同介质淬火工艺处理后的性能。
研究结果表明,水淬火时试件表面与中心的温差较大,锭子油次之,高压气体较小;试件高压气体淬火时,温度梯度小,整个断面冷却比较均匀,可以预计,相应的热应力和热变形也比较小;应用适当压力的氮气能够实现淬透性比较好的9SiCr合金刃具钢的淬火处理。
关键词:9SiCr合金刃具钢;气体淬火;温度;相变中图分类号:T G156.31 文献标识码:A 文章编号:1005 5053(2004)04 0014 04近二十多年以来,随着现代材料技术的不断发展,为了找到对高速钢、模具钢、合金钢等材料进行淬火处理的最佳途径,国外于20世纪70年代初期开始研制高压气体淬火设备进行高压气体淬火技术的研究,在油淬火和压力低于105Pa气体淬火取得一定经验的基础上,引进了压力大于105Pa的气体淬火技术。
该技术很快受到世界范围热处理界的关注。
目前国外用于高压气体淬火时的气体压力已高达2 106Pa以上。
气体淬火是一种现代的有效材料加工工艺。
金属及合金高压气体淬火技术具有高冷却速度,生产效率高、成本低(比盐浴炉低50%),无环境污染,改善淬火工作环境,易于控制淬火工艺参数;金属及合金工件经高压气体淬火技术淬火处理后,表面不氧化,不增碳,淬火均匀性好,工件变形小,工件内外温差小,工件内外热应力小等优点[1]。
通过对9SiCr合金刃具钢在清水、锭子油和高压气体等淬火介质中淬火对比实验,研究了9SiCr 合金刃具钢在不同介质淬火工艺处理后的性能。
研究结果表明,水淬火时试件表面与中心的冷却曲线温差较大,锭子油次之,高压气体较小。
试件高压气体淬火时,内部温差小,内部冷却比较均匀,可以预计,相应的热应力和热变形也比较小;应用较小压力的氮气能够实现淬透性比较好的9SiCr合金钢的高压氮气淬火处理。
不连续介质反演的原对偶牛顿法和全变差正则化
不连续介质反演的原对偶牛顿法和全变差正则化冯立新;李媛;张磊【摘要】研究利用散射场测量数据反演非均匀介质的逆散射问题,特别是平面波在非均匀介质中传播时所产生逆散射问题的数值计算.为克服非均匀介质不连续变化和反演具有不适定性的困难,提出基于全变差正则化的原对偶牛顿方法,避免了一般正则化方法对不连续介质交界处反演的过光滑性作用.数值试验显示,本算法可以在观测数据带有一定噪声的境况下有效地重构不连续介质系数.%An inverse scattering problem of inhomogeneous mediums from the measurements of the scattered field is considered.In particular,it focuses on the numerical computation of the inverse scattering generated by the interaction of a plane wave and an inhomogeneous medium.To solve the ill-posedness as well as the difficulties caused by inhomogeneous media,a primal-dual Newton method based on the total variation regularization is constructed.The overly smooth effect of the usual regularization method for the inversion is overcome.Numerical experiments show that this method can recover discontinuous coefficients under moderate amount of noise in the observation data.【期刊名称】《黑龙江大学自然科学学报》【年(卷),期】2018(035)001【总页数】9页(P1-9)【关键词】全变差正则化;原对偶牛顿法;逆介质散射【作者】冯立新;李媛;张磊【作者单位】黑龙江大学数学科学学院,哈尔滨150080;黑龙江大学数学科学学院,哈尔滨150080;黑龙江大学数学科学学院,哈尔滨150080【正文语种】中文【中图分类】O1750 IntroductionIn this paper, we focus on the inverse medium scattering problem of determining electromagnetic properties of unknown inhomogeneous objects embedded in a homogeneous background from noisy measurements of the scattered field corresponding to one incident wave impinged on the objects. We describe the scattering model mathematically as follows:Δu(x)+k2η(x)u(x)=0, x∈R2,(1)where u is the total field, k is the wavenumber, η(x)>0 for all x, andm(x)=1-η(x) is the scatterer with a compact support. We assume that B containing the compact support of the scatterer m(x) be a bounded domain in R2. Let ui which is assumed to satisfyΔui+k2ui=0, x∈R2,(2)denote the incident on the inhomogeneity. Assume that the incident fieldui is a plane wave, i.e., ui(x)=exp(ikd·x), where d∈R2 is the propagation direction (a unit vector). The total field u consists of the incident field ui and the scattered field us, that is,u=ui+us.It follows immediately from Eqs. (1) and (2) that the scattered field satisfies Δus+k2us=k2m(x)ui, x∈R2,(3)and the scattered field is required to satisfy the following Sommerfeld radiation condition(4)uniformly in all directions x/|x|.The inverse medium scattering problems arise naturally in many applications such as geophysical exploration, medical imaging, and radar detection [1-5]. For the practical significance of the inverse problem, some inverse scattering methods have been developed in the literature, which may be divided into two categories: the direct methods and the indirect methods. The direct methods aims at detecting the scatterer support and shape, and includes linear sampling method (LSM)[6-8], factorization method (paper [9], Chapter 5, paper [10]), and multiple signal classification (MUSIC)[11-13]. In contrast, the indirect methods provides a distributed estimate of the refractive index by applying regularization techniques. We just mention recursive linearization [14-18] , Gauss-Newton method [19-21], and level set method [22] for an incomplete list. Generally, theestimates by an indirect method can provide more details of inclusions, but at the expense of increased computational efforts. There has been considerable interest in considering efficient and stable inversion techniques. However, due to the strong nonlinearity of the map from the refractive index to the scattered field, severe ill-posedness of the inverse problem and the limited availability of the scattered data, it is still a very challenging problem.In this work, we assume that the scattered data with fixed wavenumber are known in the domain B, i.e., the scattered field is measured forxj∈B,j=1,…,J for a given incident field. We develop an iterated method for accurately detecting the scatterer support. In the case, from the Lippmann-Schwinger integral equation, the inverse problem can be seen as the operator equation of the first kind with unknown m(x). An efficient numerical method is presented for solving the inverse medium scattering problem which is to reconstruct the inhomogeneous medium from inner measurements of the scattered field. We construct the total variation (TV) regularization approximation of the integral equation. The main reason of choosing TV regularization is that TV can penalize highly oscillatory solutions while allowing jumps in the regularized solution. Consequently, a primal-dual Newton method is used to minimize the TV functional [23-25]. This is an iterated method, which need solve direct scattering problem for each step of the process. The scattering data is generated by numerical solution of the direct scattering problem, which is implemented by using the efficient fast algorithm [26]. In this work, we develop a TVregularization and primal-dual method for solving the inverse medium problem with discontinuous coefficients. The remainder of the paper is organized as follows. Section 1 gives the integral formulation of inverse medium problem (1)～(4) and describes the TV regularization approximation. In Section 2, we employ a primal-dual Newton method for solving the corresponding inverse problem. Section 3 presents the numerical results for the inverse medium problem. Finally, we give the relevant conclusions in Section 4.1 Formulation of the integral equation and TV approximationThe integral formulation of problem (1)～(4) is the main ingredient in the proposed method. Let be the Hankel function of first kind and order 0, see paper [1] for details. We have the following Lippmann-Schwinger integral equation for u:(5)K(x,y)m(y)dy.So, we have following operator equation(Km)(x)=g.Define the Tikhonov-TV functional(6)where TV(m) is the TV of a function m defined on the B. If m is smooth, one can obtain the representation|m|dx.However, the representation is not suitable for the implementation of the numerical procedure, due to the nondifferentiability of the Euclidean norm at the origin. To overcome this difficulty, one can take an approximation to the Euclidean norm |x| like where β is a small positive parameter. This yields the following approximation to TV(m):Instead of the Tikhonov-TV functional (6), we will consider minimization of the functional(7)Suppose m=mi,j is defined on an equispaced grid in two space dimensions, {(x1i,x2j)|x1i=iΔx1,x2j=jΔx2,i=0,…,n1,j=0,…,n2}. We defi ne the discrete penalty functional Jβ(m):R(n1+1)×(n2+1)→R by(8)where To simplify notation, we drop a factor of Δx1Δx2 from the right-hand side of (8). This factor can be absorbed in the regularization α in (7). In the following section, we consider minimization of the discretized functional:(9)where the matrix K is a discretization of the operator K, the vector g and J denote discrete data, and discretization of TV approximation Jβ,respectively.2 A primal-dual Newton methodTo minimize the functional (9), the primal-dual Newton method is employed. Consider the convex functional defined on C=R2. One can show that the conjugate set C* is the unit ball in R2 and corresponding conjugate functional to φβ is(10)We can obtain the dual representation by (10):(11)The following theorem relates the gradient of a convex functional φβ to the gradient of its conjugate see paper [25], p.138 for a proof.Theorem 1 Suppose that φβ is differentiable in a neighborhood ofx0∈C⊂Rd, and the mapping F=gradφβ:Rd→R d is invertible in that neighborhood. Then is Frechét differentiable in a neighborhood ofy0=φβ(x0) withEmploying the dual representation (11), we obtain(12)We stack the array components mi,j,ui,j and vi,j into column vector m,u, and v, let Dx1 and Dx2 be matrix representation for the grid operators and and let <·,·> denote the Euclidean inner product. Then (12) can be rewritten asMinimization of the least squares functional (10) is equivalent to computing the saddle point(u*,v*,m*(13)where We refer to m as the primal variable, and to u and v as the dual variables.Since (13) is unconstrained with respect to m, a first order necessary condition for a saddle point is(14)An additional necessary condition is that the duality gap in (11) must vanish, i.e., for each grid index i, j,(15)Finally, the dual variables must lie in the conjugate set, i.e.,(ui,j,vi,j)∈C*.(16)Suppose (15) holds for a point (ui,j,vi,j) in the interior of C*. ThenUsing Theorem 1, above equations is equivalent toDx1m=B(m)u, Dx2m=B(m)v,(17)where B(m)=diag(1/ψ′(m)).We can reformulate the first order necessary conditions (14) and (17) as a nonlinear system:(18)The derivative of G can be expressed asHere B′(m)u has matrix representationB′(m)u-Dx1=-E11Dx1-E12Dx2(19)withwhere the products and quotients are computed pointwise. Similarly,B′(m)v-Dx2=-E21Dx1-E22Dx2(20)withNewton’s method for the system (18) requires solutions of systems of the formG′(u,v,m)(Δu,Δv,Δm)=-G(u,v,m).Substituting (19)～(20) and app lying block row reduction to convert G′ to block upper triangular form, consequently we obtainwhereandWe employ backtracking to the boundary to maintain the constraint (16). In other words, we computeall i,j}.We then updateAlgorithm Primal-dual Newton’s method for TV functional.ν:=0;m0:=initial guess for primal variable;u0,v0:=initial guess for dual variables;begin primal-dual Newton iterations;Δm:=[KTK+αLν]-1rν;mν+1:=mν+Δm;τν:=max{0≤τ≤1|(uν+τΔu,vν+τΔv)∈C*};uν+1:=uν+τνΔu;vν+1:=vν+τνΔv;ν:=ν+1;end primal-dual Newton iteration.3 Numerical experimentsSome numerical examples are presented to illustrate the performance of the proposed method. Here, the scattering data are generated by numerical solution of the direct scattering problem, which is implemented by using the efficient fast algorithm (see Appendix).In experiments we consider the following inhomogeneities,★ the scattered fields are measured on the domain B=[-2,2]×[-2,2];★ the incident direction d=(cos(π/3),sin(π/3)), and wa ve number k=1;★ {Qi,j⊂R2,i=1,2,…,n1,j=1,2,…,n2} is a partition of B;★ the parameters n1=20(40),n2=20(40), α=0.001, β=0.05.Fig.1 True scatterer (n1)Fig.2 Reconstruction n1 using primal-dual Newton’s methodFig.3 True scatterer (n2)Fig.4 Reconstruction n2 using primal-dual Newton’s methodTo test the stability, some relative random noise is added to the data, i.e., the measurement data takes the formu|B:=(1+δrand)u|B,where ‘rand’ gives the Gaussian white noise and δ is a noise level parameter taken to be 5% in our numerical experiments.We have set the parameter α in the TV formulation. Then, we use the above primal-dual Newton’s algorithm to solve the inverse problem. The stopping criterion for the iteration is a relative decrease of the nonlinear residual by a factor of 10-4 or through 50 times iterations. Here, the initial guess for m can be chosen for different constant. The reconstruction got faster when the the initial guess for m is better. Fig.2 and Fig.4 present reconstruction results of the refractive index from the measurement data. We can see that the method has the ability to resolve the inverse medium problem with discontinuous coefficients efficiently.4 ConclusionsWe have proposed and discussed a reconstruction technique of the inverse medium problem with discontinuous coefficients based on the measurements of the scattering field. Our aim to overcome the difficulties caused by the the ill-posedness as well as the difficulties caused by inhomogeneous media has been realized. To make the discontinuous coefficients of the medium can be reliably reconstructed, we convert the problem to an equivalent optimization problem, and then introduce the regularization term. At the same time, we use primal dual Newton method to solve the above problem. Finally, an inversion algorithm, based on TV regularization, for the inverse medium problem from the measurement data of the scattered field is shown. Numerical experiments show that the method is effective.AppendixHere, we give a fast algorithm to solve forward scattering problem (1)～(4).The method was proposed by Nadaniela Egidi etc, see paper [26] for details. The problem (1)～(4) can be reformulated as a Fredholm integral equation (5) of the second kind:Let {Ql,j⊂R2,l=1,2,…,n1,j=1,2,…,n2} be a p artition of B, the integral equation (5) is discretized as follows:(21)where ξl,j∈R2 is the center of ml,j=m(ξl,j), and ul,j is an approximation of u(ξl,j). Linear system (21) can be rewritten as following formAu=b,where b∈Cn1n2 contains u∈Cn1n2 contains ul, j, and the entry of matrix A, at row corresponding to indices l,j, and at column corresponding to i1, j1, containswhere δ is the Kronecker function.In following we give an approximation of the coefficient matrix A by using the properties of Hankel functions We havewhere J0 is the Bessel function of order 0, and Y0 is the Newmann function of order 0. From the power series expansions of these functions we have tlρ2l, ρ>0,where γ≈0.577 215 7 is the Euler constant, andhere B>0 is a suitable constant that depends on the size of D, and L is a truncation parameter.Substituting above express into (21), we obtain the following linear system:(22)where is the approximation of ul,j .We restrict our attention to coordinate partitions of B, whereQl,j,l=1,…,n1,j=1,…,n2 are given by rectangles having edges parallel to the coordinate axes. Moreover, we use the following notations :Ql,j=[al,bl]×[aj,bj],ξl,j=(ξ1l,ξ2j)∈R2,l=1,…,n1,j=1,…,n2. So, calculating the integral in (22), and rearranging the resulting terms, we obtain-×(-ξ1l)2q+1-l(-ξ2j)2p-2q+1-s)-,l=1,…,n1,j=1,…,n2.(23)Linear system (23) can be solved efficiently due to the special structure of its coefficient matrix, which is given by the identity matrix plus 2(L+1)2 rank-one matrices. We conclude that (23) can be rewritten as follows :(I+UVT)u=b,where U,V are following complex matrices having n1n2 rows and 2(L+1)2 columns.here×(-ξ1,i1)2q+1-l(-ξ2,i2)2p-2q+1-s,i1=1,2,...,n1,i2=1,2,...,n2，M=2L+1,N(l)=2L-2⎣l/2」+1.References[1] COLTON D, KRESS R. Inverse acoustic and electromagnetic scattering theory [M]. Berlin: Springer-Verlag, 1998.[2] CUI T J, CHEW W C, AYDINER A. Inverse scattering of two-dimensional dielectric objects buried in a lossy earth using the distorted Born iterative method [J]. IEEE Transactions on Geoscience and Remote Sensing, 2001, 39(2): 339-346.[3] BAO G, LI P. Numerical solution of an inverse medium scattering problem for Maxwell’s equations at fixed frequency [J]. Journal of Computational Physics, 2009, 228(12): 4638-4648.[4] BAO G, CHOW S N, LI P, et al. Numerical solution of an inverse medium scattering problem with a stochastic source [J]. Inverse Problems, 2010, 26 (7): 74014.[5] BAO G, LIN J, MEFIRE S M. Numerical reconstruction of electromagnetic inclusions in three dimensions [J]. SIAM Journal on Imaging Sciences, 2014, 7(1): 558-577.[6] CAKONI F, COLTON D, MONK P. The linear sampling method in inverse electromagnetic scattering [M]. Philadelphia: SIAM, 2010.[7] ITO K, JIN B, ZOU J. A direct sampling method to an inverse medium scattering problem [J]. Inverse Problems, 2012, 28(2): 25003.[8] ITO K, JIN B, ZOU J. A direct sampling method for inverse electromagnetic medium scattering [J]. Inverse Problems, 2013, 29(9): 95018.[9] KIRSCH A, GRINBERG N. The factorization method for inverse problems [M]. Oxford: Oxford University Press, 2008.[10] KIRSCH A. The MUSIC-algorithm and the factorization method in inverse scattering theory for inhomogeneous media [J]. Inverse Problems, 2002, 18(4): 1025-1040.[11] AMMARI H, IAKOVLEVA E, LESSELIER D, et al. MUSIC-type electromagnetic imaging of a collection of small three-dimensional inclusions [J]. SIAM Journal on Scientific Computing, 2007, 29(2): 674-709.[12] CHEN X D, ZHONG Y. MUSIC electromagnetic imaging with enhanced resolution for small inclusions [J]. Inverse Problems, 2009, 25(1): 15008. [13] BAO G, HOU S, LI P. Inverse scattering by a continuation method with initial guesses from a direct imaging algorithm [J]. Journal of Computational Physics, 2007, 227(1): 755-762.[14] BAO G, LI P. Inverse medium scattering for the Helmholtz equation at fixed frequency [J]. Inverse Problems, 2005, 21(5): 1621-1641.[15] BAO G, LI P. Numerical solution of inverse scattering for near-field optics [J]. Optics Letters, 2007, 32(11): 1465-1467.[16] BAO G, LIU J. Numerical solution of inverse scattering problems with multi-experimental limited aperture data [J]. SIAM Journal on ScientificComputing, 2003, 25(3): 1102-1117.[17] BAO G, TRIKI F. Error estimates for the recursive linearization of inverse medium problems [J]. Journal of Computational Mathematics, 2010, 28(6): 725-744.[18] BAO G, LI P, LIN J, et al. Inverse scattering problems with multi-frequencies [J]. Inverse Problems, 2015, 31 (9): 93001.[19] ZAEYTIJD D J, FRANCHOIS A, EYRAUD C, et al. Full-wave three-dimensional microwave imaging with a regularized Gauss-Newton method-theory and experiment [J]. IEEE Transactions on Antennas and Propagation, 2007, 55(11): 3279-3292.[20] HOHAGE T, LANGER S. Acceleration techniques for regularized Newton methods applied to electromagnetic inverse medium scattering problems [J]. Inverse Problems, 2010, 26(7): 74011.[21] KRESS R, RUNDELL W. Inverse scattering for shape and impedance [J]. Inverse Problems, 2001, 17(4): 1075-1085.[22] DORN O, LESSELIER D. Level set methods for inverse scattering: some recent developments [J]. Inverse Problems, 2009, 25(12): 125001.[23] BAO G, HUANG K, LI P, et al. A direct imaging method for inverse scattering using the generalized Foldy-Lax formulation [J]. Contemporary Mathematics, 2014, 615: 49-70.[24] CHAN T F, GOLUB G H, MULET P. A nonlinear primal-dual method for total varlation-based 冯立新image restoration [J]. SIAM Journal on Scientific Computing, 1999, 20(6): 1964-1977.[25] VOGEL C R. Computational methods for inverse problems [J].Philadelphia: SIAM, 2002.[26] EGIDI N, GOBBI R, MAPONI P. The efficient solution of electromagnetic scattering for inhomogeneous media [J]. Journal of Computational and Applied Mathematics, 2007, 210(1-2): 175-182.。
给数学遇到难题的人提意见的英语作文
给数学遇到难题的人提意见的英语作文全文共6篇示例，供读者参考篇1Math Can Be Fun If You Don't Give Up!Hi there! My name is Bobby and I'm 10 years old. I love math and want to share some tips that have helped me when I get stuck on a really hard problem. Math can be super frustrating sometimes, but if you don't quit, you can get through it!First off, take a deep breath and don't get too worked up. It's just math - it's not the end of the world if you can't figure it out right away. Getting all anxious and upset will just make it harder to think clearly. Stay calm and don't be mean to yourself. We all get stuck sometimes!Next, read through the problem again really slowly and carefully. I know that might sound obvious, but you'd be surprised how many times I've missed an important detail or instruction just by skimming too fast. Sometimes I'll even read the problem out loud to myself to make sure I'm not missing anything.After you understand exactly what the problem is asking, take a look at the information they give you. Underline or circle the key numbers and facts. That will help keep you focused on just the stuff you need to solve it.Now it's time to start working! Don't be afraid to draw pictures or diagrams - that really helps me visualize what's going on, especially with geometry problems or word problems about objects moving. Label everything clearly so you don't get mixed up.If the problem seems crazy complicated, try breaking it down into smaller steps. Solve one part at a time and don't move on until you've got that piece right. Rushing ahead and trying to do everything at once just gets me confused.When you get stuck on a step, try looking at it a different way. Sometimes I pretend I'm explaining it to my little sister. Having to put it in simpler terms can give me a new perspective that clicks in my brain.You can also try guessing and checking by picking some numbers to plug in and see if your answer makes sense. This won't work for every problem, but it's a good way to spotted if you're way off track.If you've been working on it forever and are feeling really frustrated, take a short break and do something else for a bit. Go play outside, grab a snack, or just walk around. Getting your mind off the problem for a little while can give you fresh eyes when you come back to it.While you're taking that break, think about if there are any tools or resources that could help. Maybe you have formulas or tables in your math book you could use. Or maybe you need a calculator, ruler, or protractor. Organize anything that might be useful so it's right there when you get back to work.When you eventually get the answer, double check it by going through every step again slowly. I've had so many problems where I made a silly mistake at the very end and got the wrong final solution. It's so disappointing to work that hard and then miss it at the last second!Don't be afraid to ask a friend, parent or teacher if you're still really stumped after giving it your best effort. There's no shame in needing a hint or some guidance to point you in the right direction. Even my math teacher says she can't do every problem all by herself.Finally, and most importantly, don't ever give up! I know math problems can make you want to tear your hair outsometimes. But stubbornly sticking with it, even when it's super hard, is the only way to ever get better. It's okay to struggle - that's how you learn!Every time I crack a really tough nut, I feel so proud of myself for not quitting. And you know what? The next challenge doesn't seem quite as scary. You start to believe that you can conquer anything if you just keep chipping away at it patiently.I hope these tips help you tackle those mean math monsters! Just take a deep breath, read carefully, draw pictures, break it into chunks, stay positive, and most of all...never give up! You've got this!篇2Math is Super Hard Sometimes! But Don't Give Up!Hi there! My name is Timmy and I'm in 5th grade. I know math can be really, really hard sometimes. Like, crazy hard! I used to hate math and think I was just bad at it. But I've learned some tricks that have really helped me, and I want to share them with you.First of all, it's totally normal to find math difficult. I mean, we're basically learning a whole new language with all thosenumbers, symbols, and formulas. It's like trying to read ancient Egyptian hieroglyphics or something! No wonder it makes our brains hurt.But here's the thing - math is super important. It helps us understand the world around us and solve all kinds of problems, big and small. Without math, we couldn't have awesome things like video games, roller coasters, space travel, and so much more. Math is everywhere!So we can't just give up on it, no matter how hard it gets. We have to keep trying and working at it, even when it's the last thing we want to do. Math is kind of like a muscle - the more we exercise it, the stronger it gets. Pretty neat, right?Now, I know what you're thinking - "Easy for you to say, Timmy! You're probably some kind of math genius!" But I'm really not. I used to be terrible at math and get super frustrated all the time. I would cry, throw tantrums, you name it. My parents and teachers must have wanted to pull their hair out dealing with me!It was only when I started using some special tricks that things began to click for me. So I'm going to share those with you now. Maybe they'll help you out too!Trick #1: Don't be afraid to ask questions!This one is huge. If you don't understand something in math, you have to speak up and ask your teacher to explain it in a different way. There's no such thing as a dumb question. I asked my math teacher a billion questions, and you know what? It really helped me learn.Trick #2: Take breaks when you need them.Math can make your brain feel like an overloaded computer that's about to crash. When you start feeling frustrated or overwhelmed, walk away for a little bit. Go play outside, watch some TV, eat a snack. Giving your mind a rest can help everything make more sense when you get back to it.Trick #3: Make it visual.A lot of math stuff can be pretty abstract and hard to picture in your head. But using visuals and hands-on objects can make it easier to understand. Draw pictures, use counting blocks or coins, act it out with your buddies. Anything to make those numbers more concrete!Trick #4: Find a study buddy.Studying math with a friend can be a lifesaver. You can quiz each other, compare notes, and explain things in your own words.Sometimes having someone else explain a concept clicks better than how your teacher does it. My best friend Jake and I get together all the time to study math, and it really works for us!Trick #5: Be positive!I'll be honest, this one is hard for me sometimes. When I get really frustrated with a math problem, it can put me in a grumpy mood. But getting all negative doesn't help at all. It's important to stay positive and tell yourself "I can do this!" A little positive self-talk can go a long way. And celebrate your small victories - they'll keep you motivated!Trick #6: Practice, practice, practice.Repetition is so important for building those math skills. Do lots of practice problems and examples from your textbook or that your teacher gives you. The more practice you get, the better you'll understand all the methods and concepts. Math is like learning an instrument or sport - you have to keep doing it regularly to get better at it.So those are my top tips for anyone struggling with math. I really hope they help you out! Just remember to stick with it and never give up. Math is hard, no doubt about it, but it's notimpossible. You've just got to find what works best for your own learning style.And one last piece of advice: Don't be too hard on yourself. Everyone learns at their own pace, and that's totally okay. As long as you're putting in your best effort, that's what matters most. You'll get there!Okay, time for me to go play some video games (hey, math makes those possible)! Let me know if you have any other questions. Math can be our friend if we approach it with the right strategies. You've got this!Your pal,Timmy篇3Math Can Be Hard But Don't Give Up!Hi there! My name is Tommy and I'm 9 years old. I go to Oakwood Elementary School and math is one of my favorite subjects. But even for me, math can sometimes be really, really hard! I get stuck on problems and feel like giving up. But I've learned some tricks to help me keep going when math gets tough. Let me share them with you!First of all, it's totally normal to get frustrated with math sometimes. Word problems can be super confusing with all those words. Fractions are like a weird puzzle. Algebra looks like a secret code! Even grown-ups say math is one of the hardest subjects. So if you're having trouble, don't feel bad about yourself. It happens to everybody!When you get stuck on a math problem, my number one tip is: DON'T PANIC! Take some deep breaths and relax. Getting all worked up and upset will just make it harder to think clearly. Step away for a few minutes if you need to. Have a sip of water, stretch, or take a short walk around the room. Letting your brain chill out for a bit can help you see the problem fresh when you come back to it.After you've calmed down, try looking at the problem again from the very beginning. Read through it slowly and really think about what it's asking you to do. Sometimes I miss little details or make mistakes because I'm going too fast. It's like those reading comprehension questions we get - you have to truly understand what the problem wants before you can solve it.If you're still stumped, my next suggestion is to draw a picture or make a diagram. For some reason, visuals really help my brain get a grip on what's going on, especially with wordproblems or geometry stuff. Just doodle something on a piece of scrap paper to show what the problem describes. It's like a puzzle - drawing it out helps me see where all the pieces fit!When I'm totally lost on how to start working on a problem, I ask myself: "What do I already know?" Maybe I don't understand the whole problem, but I probably know some of the information or concepts it uses. I can pull together those pieces I do understand as a starting point to figure out the rest. It's like a video game - you use the skills and power-ups you've already got to beat the hard levels!Another tip is to try solving the problem a different way if my first approach isn't working. Maybe I need to plug in numbers or use a formula instead of drawing a picture. Or maybe I should work backwards from the answer instead of forwards. There's often multiple paths that can get me to the solution.One of the biggest things that helps me is talking through the problem out loud, either to myself or to a friend or adult nearby. When I say each step I'm thinking out loud, I can hear if it makes sense or not. And sometimes my friends will be like "Hey Tommy, you missed this part!" Or they'll ask me a question that gives me a new way to look at it.If all else fails, there's no shame in asking for help! My teachers are happy when I raise my hand because it shows I'm trying and want to understand. They'll re-explain it a different way or give me a hint to get me unstuck without just giving the answer. Friends and family members can also talk through the problem with me.The most important thing is to never give up! Math can be super hard and frustrating, but that makes it even more rewarding when you finally "get" it and solve a tough problem. It feels like a huge victory! Stick with it, use strategies that work for your brain, and ask for help when you need it. Before you know it, math will start to make a lot more sense. I believe in you!Those are my tips for not letting math get you down. What works best for you might be different than what works for me, and that's okay! The key is finding strategies that fit your learning style to boost your confidence. With practice and perseverance, I know you can conquer any math concept. Just don't give up - you've got this!篇4Math Can Be Hard, But Don't Give Up!Hi there! My name is Timmy and I'm 10 years old. I love math and want to share some tips for when you get really stuck on a tough math problem. I've been there before and it can be super frustrating! But don't worry, there are ways to get "unstuck"!First of all, take a deep breath and don't get discouraged. Math is hard for everyone sometimes. Even super smartgrown-ups and mathematicians hit roadblocks. Getting frustrated won't help solve the problem any faster. It's better to stay calm and positive.Next, go back to the beginning and read the problem again carefully. Sometimes we miss an important detail or misunderstand part of the question. Reading slowly can help make sure you know exactly what the problem is asking you to do. Maybe draw a picture or diagram if that helps visualize it better.If you're still stuck after that, try talking it through with someone else. Explaining the problem and your thinking out loud can jog your memory or give you a new perspective. Ask a parent, teacher, or friend to listen and see if they have any suggestions.Taking a short break can work wonders too! Go outside for some fresh air or do an activity you enjoy for 15-20 minutes.Giving your brain a little rest allows you to approach the problem with a fresh mind when you return to it. I find math is easier after recharging my brain batteries.Another idea is to try a different strategy or method for solving it. If you've been using one approach that isn't working, switch it up! There are often multiple ways to solve a problem. Drawing pictures, making a list, looking for patterns, or breaking it into smaller steps could unstick you.Here's an important tip - don't be afraid to start over from the beginning! Rereading the full problem and redoing the first few steps can sometimes jog your memory or show you where you made a mistake. A fresh start beats banging your head against the wall.If you've tried everything and still can't crack it, that's okay! Ask your teacher for help or look for resources like a math buddy, videos, or websites that explain similar problems. Getting support is not cheating, it's being resourceful.Lastly, and most importantly, don't give up! Persisting and developing a growth mindset are vital for learning math. The struggle and hard work make you smarter in the long run. Easy problems don't help your brain grow.I'll leave you with an inspiring quote: "It's not that I'm so smart, it's just that I stay with problems longer." That's from Albert Einstein, one of the greatest mathematicians ever! If he could persist through difficulty, so can you.Math is awesome and rewarding, even when it's hard. Stick with it, use strategies to get unstuck, ask for support, and have faith in your ability to grow. You've got this! Just take it one step at a time.篇5How to Deal With Really Hard Math ProblemsHi there! My name is Tommy and I'm a 5th grader. Math can be really tough sometimes, I know. Those word problems and equations with all the x's and y's get super confusing. But don't worry, I've got some good tips for you on how to deal when a math problem has you totally stumped!First of all, take a deep breath and don't freak out. I used to get really frustrated when I couldn't figure out a problem right away. My face would get all red and I'd start breathing really fast. But that never helped me solve the problem. It just made me more mixed up. So the first step is to stay calm. The problemmight seem impossible right now, but with some tricks it can become a lot simpler.Next, read through the problem again really carefully. Sometimes I'd miss an important word or number because I was trying to go too fast. Maybe the question is actually asking you something different than what you thought at first. Or there's a hint hidden in there that you didn't notice before. Goingstep-by-step is way better than confusing yourself by skipping ahead.If you're still feeling stuck after reading it again, ask your teacher to explain it in a different way. Teachers are really good at saying the same thing using different words that might make more sense to you. They can break it down into smaller steps too. Don't be embarrassed or afraid to ask for help - that's what teachers are there for!Another tip is to draw a picture or make a diagram. I'm a visual learner, so seeing things helps my brain a lot. If it's a word problem about Amanda buying apples and giving some to her friends, I'll draw sticks for the people and circles for the apples. Then I can easily see what's happening and what numbers I need to use. For algebra problems, making a quick diagram of the x's and y's going up and down can be really useful too.You can also try using objects like counters or blocks to model the problem physically. Sometimes actually moving stuff around with your hands just clicks better for people instead of doing it all in your head. It's how we first learned about addition and subtraction when we were little kids! I still use my fingers for tricky multiplication problems sometimes.Another strategy is to look for patterns. Maybe the problem involves a list of numbers that follows some kind of rule, even if it's not obvious at first. Or there are repeated steps that keep happening over and over. Recognizing patterns is a huge part of math, so looking out for them can give you a big hint on where to start.Speaking of where to start, my teacher always tells us to try breaking the problem into smaller steps or pieces too. If you keep looking at the entire problem, it can seem way too big and complicated. But if you take it one step at a time, it becomes way more manageable. Do what you can, even if it's just the very first step, then move on to the next part. Boy, have I solved a lot of problems that way that initially looked impossible!Another good tip is to think of similar problems you've done before. Maybe you can spot something familiar that will help jog your memory on how to tackle this new one. Our brains loveusing prior knowledge and finding connections between things. It's a lot easier than trying to figure out each problem like it's something completely brand new every time.Don't be afraid to try out different approaches or strategies too. If one way isn't working, move on to plan B, or C, or D. There's usually multiple paths that can get you to the right solution. Be flexible and keep an open mind instead of getting stuck in one rigid way of thinking about it.If you're still completely stumped even after trying all those tips, take a break! I know it sounds weird, but sometimes walking away from a really hard problem for a little while can help so much. Go play outside, read a fiction book, hang out with friends - anything to get your mind off the problem. Then when you come back to it later, you'll be refreshed and able to look at it from a new perspective. I've had the solution just pop into my head so many times after taking a break.The most important thing is to never give up! Math can be brutally hard sometimes, no doubt about it. You're going to feel frustrated and want to quit. But keep persevering and chipping away at it, even if it's making super slow progress. Ask for help, try new strategies, and most importantly, believe in yourself! You're smarter and more capable than you think. With patienceand determination, even the knottiest, most twisty math problem can be solved.Trust me, the amazing feeling you get when you finally breakthrough and find the solution makes all the struggle 100% worth it. You'll feel like you can conquer anything! So keep trying your best and don't let those tough math problems win. You've got this!篇6Math Can Be Hard But Don't Give Up!Hi there! My name is Tommy and I'm 8 years old. I love math and want to share some advice if you're finding it really difficult. Math can definitely be super hard sometimes, but don't feel bad - everyone struggles now and then. The important thing is to never give up!I remember when I first started learning fractions. It was so confusing with all the funky numbers and lines. I just couldn't wrap my little brain around why 1/4 was smaller than 1/2. My teacher Mrs. Johnson tried explaining it a bunch of different ways, but I still didn't get it. I felt really dumb and wanted to cry. But my mom told me "Tommy, math is like riding a bike - it'swobbly at first until you get the hang of it. Don't get discouraged!"So I kept trying and trying. I asked Mrs. Johnson a million questions after class. I watched videos about。
反演基础inversion theory课件 Cambridge1
f(m) g(o) o = o(m) L(m) = g(o)/µ(o) o=o(m) h(m) = k f(m) L(m)
Jorge Landa (Chevron-Texaco)
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h(m) = k f(m) L(m)
Region of interest for 1994 Northridge earthquake simulation
SCEC geological model provides 3D soil properties in Greater LA Basin
Adaptive grid resolves up to 1Hz freq. w/100 million grid pts; uniform grid would require 2000x more points
Assume that we have a black box that generates samples m1, m2... of the a priori probability density f(m) . In the Metropolis method, some samples of the prior are accepted, others are rejected, so that the accepted points are samples of the posterior probability density h(m) .
Snapshot of simulated ground motion (simulation requires 3hr on 6Tflops PSC machine, running at >80% parallel eff)