E?cient adaptive strategy for solving inverse
problems
M.Paszy′n ski(1),B.Barabasz(2),R.Schaefer(1)
(1)Department of Computer Science
(2)Department of Modeling and Information Technology
AGH University of Science and Technology,
Al.Mickiewicza30,30-059Cracow,Poland,
{paszynsk,schaefer}@https://www.sodocs.net/doc/096090875.html,.pl
barabasz@https://www.sodocs.net/doc/096090875.html,.pl
https://www.sodocs.net/doc/096090875.html,.pl/~paszynsk
Abstract.The paper describes the strategy for e?cient solving of di?-
cult inverse problems,utilizing Finite Element Method(FEM)as a direct
problem solver.The strategy consists of?nding an optimal balance be-
tween the accuracy of global optimization method and the accuracy of
an hp-adaptive FEM used for the multiple solving of the direct problem.
The crucial relation among errors was found for the objective function
being the energy of the system de?ning the direct problem.The strat-
egy was applied for searching the thermal expansion coe?cient(CTE)
parameter in the Step-and-?ash Imprint Lithography(SFIL)process.
Key words:Inverse problems,Finite Element Method,hp adaptivity,
Molecular Statics
1Introduction
Inverse parametric problems belong to the group of heaviest computational tasks. Their solution require a sequence of direct problem solutions,e.g.obtained by Finite Element Method(FEM),thus the accuracy of the inverse problem solution is limited by the accuracy of the direct problem solution.We utilize the fully automatic hp FEM codes[6,3]generating a sequence of computational meshes delivering exponential convergence of the numerical error with respect to the mesh size for solving direct https://www.sodocs.net/doc/096090875.html,ing the maximum accuracy for the direct problem solve by each iteration of inverse solver leads to needles computational costs(see e.g.[4]).A better strategy is to balance dynamically the accuracy of both iterations.
However,to be able to execute such strategy we need to relate the error of optimization method de?ned as the uncorrectness of objective function value with the FEM solution error.We propose such relation and the detailed error balance strategy for the objective being the energy of the system that is described by the simple problem.
2M.Paszy′n ski,B.Barabasz,R.Schaefer
The strategy is tested on the Step-and-Flash Impring Lithography(SFIL) simulations.The objective of the inverse analysis is to?nd value of the thermal expansion coe?cient enforcing shrinkage of the feature well comparable with experimental data.The energy used for the error estimation of the objective function was obtained from the experimental data and static molecular model calculations[5].
2The automatic hp adaptive Finite Element Method Sequential and parallel3D hp adaptive FEM codes[6],[3]generate in fully au-tomatic mode a sequence of hp FE meshes providing exponential convergence of the numerical error with respect to size of the mesh(number of degrees of freedom,CPU time).Given an initial mesh,called the coarse mesh,presented
Fig.1.The coarse mesh with p=2and?ne mesh with p=3on all elements edges, faces,and interiors.The optimal meshes after the?rst,second and third iterations. Various colors denote various polynomial orders of approximation.
on the?rst picture in Fig.1,with polynomial orders of approximations p=2 on elements edges,faces and interiors,we?rst perform global hp re?nement to produce the?ne mesh presented on the second picture in Fig.1,by breaking each element into8son elements,and increasing the polynomial order of ap-proximation by one.The direct problem is solved on the coarse and on the?ne mesh.The energy norm(see e.g.[1])di?erence between coarse and?ne mesh solutions is then utilized to estimate relative errors over coarse mesh elements. The optimal re?nements are then selected and performed for coarse mesh ele-ments with high relative errors.The coarse mesh elements can be broken into smaller son elements(this procedure is called h re?nement)or the polynomial order of approximation can be increased on element edges,faces or interiors(this procedure is called p re?nement),or both(this is called hp re?nement).For each ?nite element from the coarse mesh we consider locally several possible h,p or hp re?nements.For each?nite element the re?nement strategy providing maximum error decrease rate is selected.The error decrease rate
u h/2,p+1?u hp ? u h/2,p+1?w hp
rate=
E?cient solving of an inverse problem by relating inverse and direct errors.3 for the?ne mesh solution and w hp is the solution corresponding to proposed re?nement strategy,obtained by the projection based interpolation technique[2]. The optimal mesh generated in such a way becomes a coarse mesh for the next iteration,and the entire procedure is repeated as long as the global relative error estimation is larger then the required accuracy of the solution(see[6]for more details).The sequence of optimal meshes generated by automatic hp-adaptive code from the coarse mesh is presented on third,fourth and?fth pictures in Figure1.The relative error of the solution goes down from15%to5%.
3The relation between the objective function error and the Finite Element Method error
We assume the direct problem is modeled by the abstract variational equation
u∈u0+V
b(u,v)=l(v)?v∈V
(2)
where u0is the lift of the Dirichlet boundary conditions[2].Functionals b and l depend on the inverse problem parameters d.The variational problem(2)is equivalent with the minimization one(3),if b is symmetric and positive de?nite (see e.g.[2]) u∈u0+V
E(u)=1
2b(u,u)?l(u)is the functional of the total energy of the solution.
The Problem(2)may be approximated using the FEM on the?nite dimen-sional subspace V h,p?V
u h,p∈u0+V h,p
b(u h,p,v h,p)=l(v h,p)?v h,p∈V h,p
.(4)
For a sequence of meshes generated by the self-adaptive hp FEM code,for every coarse mesh,a coarse mesh space is a subset of the corresponding?ne mesh space,V h,p?V h/2,p+1?V.
An absolute relative FEM error utilized by the self-adaptive hp FEM code is de?ned as the energy norm di?erence between the coarse and?ne mesh solutions
err F EM= u h,p?u h/2,p+1 E.(5) The inverse problem can be formulated as
F ind?d:|J h,p ?d ?J(d?)|=lim h→0,p→∞min d k∈?|J h,p d k ?J(d?)|(6)
where d?denotes exact parameters of the inverse problem(exact solution of the variational formulation for these parameters is well comparable with exper-iment data),d k denotes approximated parameters of the inverse problem,?is a set of all admissible parameters d k,J(d?)=E(u(d?))is the energy of the
4M.Paszy′n ski,B.Barabasz,R.Schaefer
exact solution u(d?)of the variational problem(2)for exact parameters d?, J h,p d k =E u h,p d k is the energy of the solution u h,p d k of the approxi-mated problem(4)for approximated parameters d k.
Objective function error is de?ned as an energy di?erence between the solu-tion of the approximated problem(4)for approximated parameter d k and the exact solution of the problem(2)for exact parameter d?(assumed to be equal to the energy of the experiment)
e h,p d k =|J h,p d k ?J(d?)|.(7) In other words,the approximated parameter d k is placed into the approximated formulation(4),the solution o
f the problem u h,p d k (which depends on d k)is computed by FEM,and the energy of the solution E u h,p d k is computed. Lemma1.2 J h,p d k ?J h/2,p+1 d k = u h,p d k ?u h/2,p+1 d k 2E Proof:2 J h,p d k ?J h/2,p+1 d k =2 E u h,p d k ?E u h/2,p+1 d k = b u h,p d k ,u h,p d k ?2l u h,p d k ?b u h/2,p+1 d k ,u h/2,p+1 d k + 2l u h/2,p+1 d k =b u h,p d k ,u h,p d k ?b u h/2,p+1 d k ,u h/2,p+1 d k + 2l u h/2,p+1 d k ?u h,p d k =b u h,p d k ,u h,p d k ?
b u h/2,p+1 d k ,u h/2,p+1 d k +2b u h/2,p+1 d k ,u h/2,p+1 d k ?u h,p d k = b u h/2,p+1 d k ?u h,p d k ,u h/2,p+1 d k ?u h,p d k =
u h,p d k ?u h/2,p+1 d k 2E
where V h,p?V h/2,p+1?V stand for the coarse and?ne mesh subspaces. Lemma2.e h/2,p+1 d k ≤1
u h/2,p+1 d k ?u h,p d k 2E+|J h,p d k ?J(d?)|.
2
The objective function error over the?ne mesh is limited by the relative error of the coarse mesh with respect to the?ne mesh,plus the objective function error over the coarse mesh.
4Algorithm
Lemma2motivates the following algorithm relating the inverse error with the objective function error.We start with random initial values of the inverse prob-lem parameters
solve the problem on the coarse and fine FEM meshes
compute FEM error
inverse analysis loop
Propose new values for inverse problem parameters
E?cient solving of an inverse problem by relating inverse and direct errors.5 solve the problem on the coarse mesh
Compute objective function error
if(objective function error execute one step of the hp adaptivity, solve the problem on the new coarse and fine FEM meshes compute FEM error if(inverse error end Inverse error estimation proven in Lemma2allows us to perform hp adaptation in the right moment.If the objective function error is much smaller than the FEM error,the minimization of the objective function error does not make sense on current FE mesh,and the mesh quality improvement is needed. 5Step-and-?ash Imprint Lithography The above algorithm will be tested on the SFIL process simulation.The SFIL is a modern patterning process utilizing photopolymerization to replicate the to-pography of a template into a substrate.It can be summarized in the following steps,compare Fig.2:Dispense-the SFIL process employs a template/sub-strate alignment scheme to bring a rigid template and substrate into parallelism, trapping the etch barrier in the relief structure of the template;Imprint-the gap is closed until the force that ensures a thin base layer is reached;Exposure -the template is then illuminated through the backside to cure etch barrier; Separate-the template is withdrawn,leaving low-aspect ratio,high resolution features in the etch barrier;Breakthrough Etch-the residual etch barrier(base layer)is etched away with a short halogen plasma etch;Transfer Edge-the pattern is transferred into the transfer layer with an anisotropic oxygen reac-tive ion etch,creating high-aspect ratio,high resolution features in the organic transfer layer.The photopolymerization of the feature is often accompanied by the densi?cation,see Fig.2,which can be modeled by the linear elasticity with thermal expansion coe?cient(CTE)[5].We may de?ne the problem:Find u-displacement vector?eld,such that u∈V? H1(?) 3 .(8) b(u,v)=l(v)?v∈V where V= v∈ H1(?) 3:tr(v)=0onΓD ,??R3stands for the cubic-shape domain,ΓD is the bottom of the cube and H1(?)is the Sobolev space. b(u,v)= ?(E ijkl u k,l v i,j)dx;l(v)=α ?v i,i dx.(9) Here E ijkl=μ(δikδjl+δilδjk)+λδijδkl stands for the constitutive equation for the isotropic material,whereμandλare Lame coe?cients.The thermal expansion coe?cient(CTE)α=?V 6M.Paszy′n ski,B.Barabasz,R.Schaefer Fig.2.Modeling of the Step-and-Flash Imprint Lithography process. 6Numerical results The proposed algorithm was executed for the problem of?nding the proper value of the thermal expansion coe?cient enforcing shrinkage of the feature comparable with experiments.The algorithm performed43iterations on the ?rst optimal mesh(see the third picture in Fig.1)providing15%relative error of the direct problem solution.Then,the computational mesh was hp re?ned to increase the accuracy of the direct solver.The inverse algorithm continued by utilizing8%relative error mesh(see the fourth picture in Fig.1)for the direct problem.After39iterations the mesh was again hp re?ned(see the?fth picture in Fig.1)to provide5%relative error of the direct problem solution.After35 iterations of the inverse algorithm on the most accurate mesh the inverse problem was solved.The history of the(CTE)parameter convergence on the?rst,second and third optimal meshes is presented in Fig.3. Fig.3.History of convergence of CTE parameter on3meshes. E?cient solving of an inverse problem by relating inverse and direct errors.7 We compared the total execution time equal to0.1s+43×2×0.1s+1s+39×2×1s+10s+35×2×10s=8.7+79+710=797.7s with the classical algorithm, where the inverse problem was solved on the most accurate FEM mesh from the beginning.The classical algorithm required91iterations to obtain the same result.The execution time of the classical algorithm was10s+91×2×10s= 1830s. This di?erence will grow when the inverse algorithm will look for more inverse problem parameters at the same time,since number of direct problem solutions necessary to obtain the new propositions of the inverse parameters will grow. 7The molecular static model The energy of the experimental data J(d?)was estimated from the molecular static model,which provides realistic simulation results,well comparable with experiments[5].During the photopolymierization,the Van der Waals bound between particles forming polymer chain are converted into a stronger covalent bounds.The average distance between particles is decreasing and the volumetric contraction of the feature occurs.In the following,a general equations govern-ing the equilibrium con?gurations of the molecular lattice structure after the densi?cation and removing of the template are derived. Let us consider an arbitrary pair of bonded molecules with indicesαandβand given lattice position vector pα=(?xα,?yα,?zα).The unknown equilibrium position vector of particleα,under the action of all their intermolecular bonds, is denoted xα=(xα,yα,zα),the displacements from the initial position in the lattice to the equilibrium position is represented by the vector uα=xα?pα. Let · denote the vector norm or length in R3,let rαβ= xβ?xα be the distance between particlesαandβin initial con?guration.Then,the force Fαβ, along the vector xβ?xαis governed by the potential function V(rαβ), Fαβ=??V(rαβ) xβ?xα .(10) where?rst term represents the magnitude and second term is the direction.If the indices of bonded neighboring particles of particleαare collected in the set Nα,then we obtain its force equilibrium by applying the following sum: β∈NαFαβ=? β∈Nα?V(rαβ) xβ?xα =0.(11) The characteristics of the potential functions{V(rαβ)}β∈N αare provided by the Monte Carlo simulation[5].The covalent bounds are modeled by spring forces Fαβ=C1r+C2.(12) Spring like potential V(r)is quadratic.The Van der Waals bounds are model by non-linear forces and the Lennard-Jones potentials V(r)=Cαβ σαβr mαβ .(13) 8M.Paszy′n ski,B.Barabasz,R.Schaefer where r= xβ?xα . The equilibrium equations are non-linear and the Newton-Raphson lineariza-tion procedure is applied to solve the system.The resulting shrinkage of the feature is presented in Figure2. 8Conclusions and future work –The proper balance of errors of global optimization method and direct prob-lem solvers allows for e?cient speeding up the solution process of di?cult inverse problems.The analytic relation among both errors is necessary. –The relation between the objective function error and the relative error of the hp-adaptive FEM has been derived.The objective error was expressed as the energy di?erence between the numerical solution and experiment data.–The strategy relating the convergence ratios of the inverse and direct problem solution has been proposed and successfully tested for searching value of the CTE parameter in the SFIL process.We obtained about2.4speedup in comparison to the solution without error balancing for the simple test example.The higher speedup may be expected for problems with larger dimension. –The future work will include derivation of analytic relations between the hp-adaptive FEM error and objective function error de?ned in other ways.The possibilities of further speeding up of the solver will be tested by utilizing the parallel version of the hp-adaptive FEM codes[3]. Acknowledgments.The work reported in this paper was supported by Polish MNiSW grant no.3TO8B05529 References 1.Ciarlet P.,The Finite Element Method for Elliptic Problems,Society for Industrial &Applied,200 2. 2.Demkowicz L.,Computing with hp-Adaptive Finite Elements,Chapman&Hall/Crc Applied Mathematics&Nonlinear Science,2006 3.Paszy′n ski,M.,Demkowicz,L.,Parallel Fully Automatic hp-Adaptive3D Finite Element Package,Engineering with Computers,2006,in press. 4.Paszy′n ski,M.,Szeliga,D.,Barabasz,B.Macio l,P.,Inverse analysis with3D hp adaptive computations of the orthotropic heat transport and linear elasticity prob-lems,VII World Congress on Computational Mechanics,Los Angeles,July2006 5.Paszy′n ski,M.,Romkes,A.,Collister,E.,Meiring,J.,Demkowicz,L.,Willson,C. G.,On the Modeling of Step-and-Flash Imprint Lithography using Molecular Statics Models,ICES Report05-38,2005. 6.Rachowicz,W.,Pardo D.,Demkowicz,L.,Fully Automatic hp-Adaptivity in Three Dimensions,ICES Report04-22,2004. Efficient Market Hypothesis The Efficient Market Hypothesis (EMH) holds that in an open and efficient market, security prices fully reflect all available information and prices rapidly adjust to any new information. Fama first defined the term “efficient market” in financial literature in 1965. As a core theory to research efficient market, EMH still is a perfect assumption. There are two assumptions that must satisfy if EMH is established. The one is rational agent assumption. It says that all investors are the pursuit of individual utility maximization rational man, having the same understanding and the ability to analyze information. And the forecasts of stock prices are the same. Another assumption is that Random Walk Theory. In an efficient market, it is only when new information becomes available that significant changes in security prices will occur. Means stock price movements do not follow any pattern or trend. 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The?rst e?cient and secure method for Identity-Based Encryption was put forth by Boneh and Franklin[4].They proposed a solution using e?ciently computable bilinear maps that was shown to be secure in the random oracle model.Since then,there have been schemes shown to be secure without random oracles,but in a weaker model of security know as the Selective-ID model[9,1]. Most recently,Boneh and Boyen[2]described a scheme that was proved to be fully secure without random oracles;the possibility of such a scheme was to that point an open problem.However,their scheme is too ine?cient to be of practical use. We present the?rst e?cient Identity-Based Encryption scheme that is fully secure without random oracles.The proof of our scheme makes use of an algebraic method?rst used by Boneh and Boyen[1]and the security of our scheme reduces to the decisional Bilinear Di?e-Hellman (BDH)assumption. We additionally show that our IBE scheme implies a secure signature scheme under the compu-tational Di?e-Hellman assumption without random oracles.Previous practical signature schemes that were secure in the standard model relied on the Strong-RSA assumption[12,11]or the Strong-BDH assumption[3]. 1.1Related Work Shamir[16]?rst presented the idea of Identity-Based Encryption as a challenge to the research community.However,the?rst secure and e?cient scheme of Boneh and Franklin[4]did not appear until much later.The authors took a novel approach in using e?ciently computable bilinear maps in order to achieve their result. Canetti et.al.[9]describe a weaker model of security for Identity-Based Encryption that they term the Selective-ID model.In the Selective-ID model the adversary must?rst declare which 1 Chapter 6 Are Financial Markets Efficient? Multiple Choice Questions 1. How expectations are formed is important because expectations influence (a) the demand for assets. (b) bond prices. (c) the risk structure of interest rates. (d) the term structure of interest rates. (e) all of the above. Answer: E 2. According to the efficient market hypothesis, the current price of a financial security (a) is the discounted net present value of future interest payments. (b) is determined by the highest successful bidder. (c) fully reflects all available relevant information. (d) is a result of none of the above. Answer: C 3. The efficient market hypothesis (a) is based on the assumption that prices of securities fully reflect all available information. (b) holds that the expected return on a security equals the equilibrium return. (c) both (a) and (b). (d) neither (a) nor (b). Answer: C 4. If the optimal forecast of the return on a security exceeds the equilibrium return, then (a) the market is inefficient. (b) an unexploited profit opportunity exists. (c) the market is in equilibrium. (d) only (a) and (b) of the above are true. (e) only (b) and (c) of the above are true. Answer: D Classical Works in Financial Economics Eugene F. Fama 效率资本市场:理论与实证研究评述 效率资本市场:理论和实证研究评述Eugene F. Fama 金融经济学名著译丛『第 1 页』 效率资本市场:理论与实证研究评述* Eugene F. Fama+ 原载《金融学杂志》,1970 年5 月,第25 卷第2 册,第383-417 页 1、引言 资本市场的主要作用是配臵经济体中资本存量的所有权。总的来讲,在理想 市场中,价格提供了准确的资源配臵信号:也就是一个企业能够做出生产——投资决策,以及在假定任何时期的证券价格都―完全反映‖了可得信息的前提下,投资者能够在代表企业经营所有权的证券之间进行选择的市场。一个价格总是能够―完全反映‖可得信息的市场被称为是―有效率的‖。 本文对效率市场模型的理论和实证研究进行了评述。在对有关理论进行研讨 之后,考察了关于证券价格对三个相关信息子集调整的实证研究。首先,讨论了弱形式检验,在该形式下,信息集为历史价格。接下来,考察了半强形式检验,重点放在价格是否对其他公开披露的信息(如公布年度盈利、股票拆细等)做出有效调整。最后,评述了强形式检验,侧重于既定的投资者或群体是否单独拥有与价格形成有关的任何信息1。我们的结论是:除去少数例外情况,效率市场模型成立得相当好。 尽管我们采用从理论到实证研究的方法,从历史观点来看,我们注意到该领 域的实证研究在很大程度上领先于理论的发展。这里给出的理论主要是为了便于判断哪些实证结果从理论观点来看更加具有关联性。然而,我们或多或少地依照历史序列的方式对实证研究本身进行评述。 最后,细心的读者会发现在本文的有些地方没有对相关的研究进行专题讨 论。我对此表示抱歉:该领域的文献浩如烟海,有所取舍是必然的。不过,只要梳理出效率市场研究的主线,并准确地描述该领域当前的进展状况,也就达到了本文的主要目标。 * 本研究项目由国家科学基金提供资助。我对Arthur Laffer,Robert Aliber,Ray Ball,Micheal Jensen,James Lorie,Merton Miller,Charles Nelson,Richard Roll,William Talor 和Ross Watts 提出的意见表示感谢。 + 现就职于Chicago 大学和计量经济学协会。 1 弱形式和强形式检验之间的区别首先由Harry Roberts 提出。 效率资本市场:理论和实证研究评述Eugene F. Fama 金融经济学名著译丛『第 2 页』 2、效率市场理论 2.1 期望回报或“公平游戏”模型 Efficient 1:8μm KTiOPO 4optical parametric oscillator pumped within an Nd:YAG =SrWO 4Raman laser Fen Bai,1,2Qingpu Wang,1,2Zhaojun Liu,1,2,*Xingyu Zhang,1,2Wenjia Sun,1,2 Xuebin Wan,1,2Ping Li,1,2Guofan Jin,1and Huaijin Zhang 3 1 School of Information Science and Engineering,Shandong University,Jinan,Shandong 250100,China 2 Shandong Provincial Key Laboratory of Laser Technology and Application, Shandong University,Jinan,Shandong 250100,China 3 Institute of Crystal Materials,Shandong University,Jinan,Shandong 250100,China *Corresponding author:zhaojunliu@https://www.sodocs.net/doc/096090875.html, Received September 23,2010;revised January 21,2011;accepted January 25,2011; posted February 1,2011(Doc.ID 135511);published March 8, 2011 A 1:8μm optical parametric oscillator (OPO)based on a noncritically phase-matched KTiOPO 4crystal is demon-strated.OPO and stimulated Raman scattering techniques are successfully combined in an acousto-optically Q -switched Nd:YAG =SrWO 4Raman laser.The device efficiently realizes three steps of conversion:from a laser diode wavelength of 808nm to the fundamental wavelength of 1064nm ;next,to the Stokes wavelength of 1180nm ;and finally to the OPO signal wavelength of 1810nm .With an incident diode power of 7:2W and a pulse repetition rate of 15kHz ,an average signal power of 485mW is obtained with a diode-to-signal conversion efficiency of 6.75%.The beam quality factors (M 2)of the signal wave in both horizontal and vertical directions are measured to be 1:7?0:2.The numerical output power results of the system,the thermal lensing,and the stability parameter of the cavity are also discussed.?2011Optical Society of America OCIS codes:190.4410,190.4970,140.3550. An optical parametric oscillator (OPO)is a popular efficient nonlinear frequency conversion technique for generating new laser lines based on a second-order non-linear effect [1–3].Stimulated Raman scattering (SRS)is another well-known method for frequency conversion based on a third-order nonlinear optical process [4–6].OPO in combination with SRS provides a promising way for extending the wavelength range,because both OPO and SRS are flexible and efficient for generating di-verse laser wavelengths.Therefore,combining OPO with SRS is of great significance and is expected to find appli-cations where existing laser systems cannot be effective.We attempt to implement this combination by using Raman radiation to pump an OPO.An OPO can operate only above the oscillating threshold,so the pump radia-tion must be intensive enough to reach the threshold.Moreover,the lower-order laser transverse modes gener-ate the OPO output more efficiently than the higher-order ones [2].Therefore,the pump radiation should have good beam quality for efficient conversion.Solid-state Raman lasers based on the SRS effect satisfy both requirements.First,the Raman laser has been proven to be a good source for generating high-efficiency and high-power Stokes waves [4–6].Second,owing to the beam cleanup effect of SRS [7],the beam quality of the Raman radiation is improved significantly over that of the pump beam.Therefore,it is suggested that one use the Raman radia-tion to generate high-power radiation with good beam quality for pumping an OPO,without any additional beam-shaping elements.To our knowledge,using the Raman laser to pump an OPO is a novel approach.In this Letter,a KTiOPO 4(KTP)intracavity IOPO pumped by an acousto-optically (AO)Q -switched diode-pumped Nd:YAG =SrWO 4Raman laser is demonstrated.A SrWO 4crystal is chosen as the Raman crystal for its high Raman gain and good thermal and mechanical properties [8].A KTP crystal is employed as the OPO crystal and cut for Type II noncritically phase-matched (NCPM)config-uration for the maximum effective nonlinear coefficient and acceptance angle.The 1064nm radiation from the Nd:YAG crystal is Stokes-shifted to a 1180nm laser in the SrWO 4crystal.The 1180nm Raman laser acts as the pumping source for the KTP-OPO,from which the 1810nm signal wave is obtained.At an incident laser diode (LD)power of 7:2W and a pulse repetition rate (PRR)of 15kHz,a signal output power of 485mW is achieved The diode-to-signal conversion efficiency is 6.75%.The beam quality factors (M 2)are 1:7?0:2in both horizontal and vertical directions. The experimental configuration of the KTP-OPO pumped within the Nd:YAG =SrWO 4Raman laser is shown in Fig.1.Both the fundamental and OPO cavities were inside the Raman cavity.A fiber-coupled 808nm LD (25W,NA ?0:22,d core ?600μm)was used as the pump-ing source.The gain medium was an Nd:YAG rod (1:0at :%Nd-doped,?4mm ×10mm).The Raman active Fig.1.(Color online)Optical configuration of the KTP-IOPO pumped by Nd:YAG =SrWO 4Raman laser.LD,laser diode;RM,rear mirror;OC,output coupler;AO,acousto-optical Q -switch;KTP,KTiOPO 4crystal;HR,high reflection;HT,high transmis-sion;AR,antireflection;PR,partial reflection.March 15,2011/Vol.36,No.6/OPTICS LETTERS 813 0146-9592/11/060813-03$15.00/0 ?2011Optical Society of America 确定性:是指自然状态如何出现已知,并替换行动所产生的结果已知。它排除了任何随机事件发生的可能性。 风险:是指那些涉及已知概率或可能性形式出现的随机问题,但排除了未数量化的不确定性问题。即对于未来可能发生的所有事件,以及每一事件发生的概率有准确的认 识。但对于哪一种事件会发生却事先一无所知。 不确定性:是指发生结果尚未不知的所有情形,也即那些决策的结果明显地依赖于不能由决策者控制的事件,并且仅在做出决策后,决策者才知道其决策结果的一类问题。即知道未来世界的可能状态(结果),但对于每一种状态发生的概率不清楚。 自然状态:特定的会影响个体行为的所有外部环境因素。 自然状态的特征:自然状态集合是完全的、相互排斥的(即有且只有一种状态发生) 自然状态的信念(belief):个体会对每一种状态的出现赋予一个主观的判断,即某一特定状态s出现的概率P(s)满足:0≤p(s)≤1,这里的概率p(s)就是一个主观概率,也成为个体对自然的信念。不同个体可能会对自然状态持有不同的信念,但我们通常假定所有的个体的信念相同,这样特定状态出现的概率就是唯一的。 数学期望最大化原则:数学期望收益最大化准则是指使用不确定性下各种可能行为结果的预期值比较各种行动方案优劣。这一准则有其合理性,它可以对各种行为方案进行准确的优劣比较,同时这一准则还是收益最大准则在不确定情形下的推广。 期望效用原则:指出人们在投资决策时不是用“钱的数学期望”来作为决策准则,而是用“道德期望”来行动的。而道德期望并不与得利多少成正比,而与初始财富有关。穷人与富人对于财富增加的边际效用是不一样的。即人们关心的是最终财富的效用,而不是财富的价值量,而且,财富增加所带来的边际效用(货币的边际效用)是递减的。 效用函数的表述和定义:不确定性下的选择问题是其效用最大化的决定不仅对自己行动的选择,也取决于自然状态本身的选择或随机变化。因此不确定下的选择对象被人们称为彩票(Lottery)或未定商品(contingent commodity。 不确定性下的偏好关系表述:个体所有可选择抽奖的集合称为抽奖空间,记为:L=(p,x,y)同样地,假设个体在抽奖空间上存在一个偏好关系,即可以根据自己的标准为所有抽奖排出一个优劣顺序。 公理1: 公理2: 公理3 公平博彩是:指不改变个体当前期望收益的赌局,如一个博彩的随机收益为,其期望收益为,我们就称其为公平博彩。 风险厌恶者:如果经济主体拒绝接受公平博彩,这说明该个体在确定性收益和博彩之间更偏好确定性收益,我们称该主体为风险厌恶者。 风险偏好者:如果一个经济主体在任何时候都愿意接受公平博彩,则称该主体为风险偏好者。定义:u是经济主体的VNM效用函数,W为个体的初始禀赋,如果对于任何满足E(3-)=0,var(3-)〉0的随机变量3-,有u(W)〉E[u(W+3-)],则称个体是(严格)风险厌恶(risk aversion);如果上述不等号方向相反,则称个体是风险偏好(risk loving);如果两边相等,则称个体是风险中性(neutral) 确定性等价值(certainty equivalence):是指经济行为主体对于某一博彩行为的支付意愿。即与某一博彩行为的期望效用所对应的数学期望值(财富价值)。 风险溢价(risk premium):是指风险厌恶者为避免承担风险而愿意放弃的投资收益。或让一个风险厌恶的投资者参与一项博彩所必需获得的风险补偿。 风险溢价与最优资产组合选择定理(绝对风险厌恶系数):如果一个经济主体是严格风险厌Efficient Market Hypothesis(有效市场假说)
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