【摘要】：本文主要對(duì)橢圓、拋物型方程的系數(shù)反演問(wèn)題進(jìn)行了研究,這兩類反問(wèn)題不論在金融、物理、醫(yī)學(xué)、地質(zhì)探測(cè)和無(wú)線電傳播領(lǐng)域還是在電磁金屬成型技術(shù)中都有著極其廣泛的應(yīng)用。文章第一部分的橢圓方程是基于弗雷歇偏導(dǎo),證明了所對(duì)應(yīng)泛函的凸性,由泛函的這個(gè)性質(zhì)可以得到解的唯一性;而文章中第二部分?jǐn)?shù)學(xué)模型是一個(gè)拋物型方程,利用最優(yōu)控制理論,探究了原問(wèn)題對(duì)應(yīng)的優(yōu)化問(wèn)題解的適定性。該問(wèn)題的難點(diǎn)在于:首先,二階拋物型方程中所需反演的是一個(gè)二階項(xiàng)系數(shù),這是一個(gè)強(qiáng)不適定且完全非線性的問(wèn)題;同時(shí),給出的附加條件并非通常意義下的終端觀測(cè)值,而是積分平均意義的觀測(cè)值,這種類型的附加條件會(huì)導(dǎo)致相應(yīng)的控制泛函的極小元所滿足的必要條件極為復(fù)雜。再者,由于所給的控制泛函沒(méi)有凸性,一般情況下很難得到最優(yōu)解的唯一性。通過(guò)仔細(xì)分析了極小元所滿足的必要條件,并結(jié)合正問(wèn)題的一些先驗(yàn)估計(jì)式,我們發(fā)現(xiàn),當(dāng)終端時(shí)刻T適當(dāng)小時(shí),可以證明極小元的局部唯一性和穩(wěn)定性,這也是本文的主要工作。文章主要包含以下四個(gè)部分:首先引言部分講述了反問(wèn)題的背景,國(guó)內(nèi)外的研究狀況,以及反問(wèn)題成長(zhǎng)的一個(gè)歷程。第一章從理論上重點(diǎn)分析了橢圓型方程的系數(shù)反演問(wèn)題,首先介紹了橢圓方程的數(shù)學(xué)模型,由于反問(wèn)題是不適定的,這時(shí)需要對(duì)橢圓方程的正問(wèn)題進(jìn)行能量估計(jì),運(yùn)用弗雷歇偏導(dǎo)理論對(duì)這個(gè)方程的能量估計(jì)式進(jìn)行變形,得到了橢圓方程對(duì)應(yīng)泛函的凸性,由泛函的這個(gè)性質(zhì)可以得到橢圓方程解的唯一性。第二章是對(duì)拋物型方程的擴(kuò)散系數(shù)反問(wèn)題的研究,由于原問(wèn)題的不適定性,將原問(wèn)題轉(zhuǎn)化為最優(yōu)控制問(wèn)題P,采用最優(yōu)化方法對(duì)擴(kuò)散系數(shù)反演問(wèn)題進(jìn)行了研究。再根據(jù)正問(wèn)題的能量估計(jì)和相應(yīng)的共軛方程的能量估計(jì),然后運(yùn)用能量估計(jì)式得到了最優(yōu)解所滿足的必要條件,最后在T比較小的情況下得到了最優(yōu)解的唯一性和穩(wěn)定性。第三章對(duì)拋物和橢圓型方程反問(wèn)題的后續(xù)工作進(jìn)行了總結(jié)與展望。
[Abstract]:In this paper, we mainly study the coefficient inversion of elliptical and parabola equations. these two kinds of inverse problems are in finance, physics and medicine. Geological exploration and radio propagation are also widely used in electromagnetic metal forming technology. In the first part of this paper, the elliptical equation is based on Frecher partial derivation, and the convexity of the corresponding functional is proved. The uniqueness of the solution can be obtained from this property of the functional. In the second part of the paper, the mathematical model is a parabola equation. By using the optimal control theory, the well-posedness of the solution of the optimization problem corresponding to the original problem is discussed. The difficulties of this problem are as follows: firstly, what needs to be inversed in the second-order parabola equation is a second-order term coefficient, which is a strongly ill-posed and completely nonlinear problem; At the same time, the additional conditions given are not terminal observations in the usual sense, but observations in the mean sense of integral. This type of additional conditions will lead to the very complex necessary conditions satisfied by the minimum elements of the corresponding control functional. Moreover, because the given control functional is not convex, it is difficult to obtain the uniqueness of the optimal solution in general. By carefully analyzing the necessary conditions satisfied by the minimum element and combining with some prior estimates of the positive problem, we find that when the terminal time T is properly small, the local uniqueness and stability of the minimum element can be proved. This is also the main work of this paper. The article mainly includes the following four parts: first of all, the introduction describes the background of the anti-problem, the research situation at home and abroad, and a course of the growth of the anti-problem. In the first chapter, the coefficient inversion problem of Elliptic equation is analyzed theoretically. Firstly, the mathematical model of Elliptic equation is introduced. Because the inverse problem is ill-posed, it is necessary to estimate the energy of the positive problem of Elliptic equation. The energy estimation formula of the equation is deformed by using Frecher partial derivation theory, and the convexity of the corresponding functional of the elliptical equation is obtained. from this property of the functional, the uniqueness of the solution of the elliptical equation can be obtained. In the second chapter, the inverse problem of diffusion coefficient of parabola equation is studied. because of the discomfort of the original problem, the original problem is transformed into the optimal control problem P, and the inverse problem of diffusion coefficient is studied by using the optimization method. Then according to the energy estimation of the positive problem and the energy estimation of the corresponding conjugated equation, then the necessary conditions for the optimal solution are obtained by using the energy estimation formula. Finally, the uniqueness and stability of the optimal solution are obtained when T is relatively small. In the third chapter, the follow-up work of inverse problems of parabola and Elliptic equations is summarized and prospected.
【學(xué)位授予單位】：蘭州交通大學(xué)
【學(xué)位級(jí)別】：碩士
【學(xué)位授予年份】：2017
【分類號(hào)】：O175

【參考文獻(xiàn)】

相關(guān)期刊論文 前1條

1 Zuicha DENG;Liu YANG;;An Inverse Problem of Identifying the Radiative Coefficient in a Degenerate Parabolic Equation[J];Chinese Annals of Mathematics(Series B);2014年03期

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本文編號(hào)：2481809