A Wavelet Tour of Option Pricing


 Teresa Webster
 1 years ago
 Views:
Transcription
1 Universität Ulm Fakultät für Mathematik und Wirtschaftswissenschaften A Wavelet Tour of Option Pricing Dissertation zur Erlangung des Doktorgrades Dr. rer. nat. der Fakultät für Mathematik und Wirtschaftswissenschaften der Universität Ulm vorgelegt von Roman Mario Xerxes Rometsch aus Ulm 2
2 Amtierender Dekan: Prof. Dr. Werner Kratz. Gutachter: Prof. Dr. Karsten Urban 2. Gutachter: Prof. Dr. Rüdiger Kiesel Tag der Promotion: 29. Oktober 2
3 A Wavelet Tour of Option Pricing Adaptive Wavelet Methods for Variational Inequalities 2 Mathematics Subject Classification: 35J2, 35J6, 35J86, 35K85, 35Q68, 4A25, 45E5, 49J4, 65K5, 65R2, 65T6, 9G8 c 2 Mario Rometsch Please report errors to:
4 to Sabrina
5 Abstract The efficient numerical solution of elliptic variational inequalities in general and of obstacle problems in particular is of great interest and poses a nontrivial challenge in Numerical Mathematics. Such problems occur for example after the timediscretization of the pricing problem of an American option. The algorithm EVISOLVE, that is presented in this thesis, consists of an adaptive wavelet method that allows for an efficient and reliable solution of such elliptic inequalities. Adaptivity means that the algorithm uses results to adjust itself in order to keep the convergence rate optimal also for nonsmooth solutions and to estimate the error such that a prespecified accuracy can be guaranteed. The convergence and asymptotic optimal complexity of this algorithm is shown. The arithmetic complexity is still suboptimal at the moment. Furthermore, we present Lawa, a C++library, that consists of building blocks for the realization of adaptive wavelet methods. This software has been designed to be applied in research and education. Finally, we show how the pricing problem for American options in various Lévymodels can be solved with uniform waveletgalerkin methods. Zusammenfassung Die effiziente numerische Lösung von elliptischen Variationsungleichungen im Allgemeinen und Hindernisproblemen im Speziellen ist ein wichtiges und nichttriviales Teilgebiet der Numerischen Mathematik und tritt z.b. als Teilproblem bei der quantitativen Bewertung Amerikanischer Optionen auf. Mit dem in dieser Dissertation vorgestellten Algorithmus EVISOLVE, der ein adaptives WaveletVerfahren zur Lösung von elliptischen Variationsungleichungen darstellt, können solche Ungleichungen nun effizient und verlässlich gelöst werden. Adaptiv bedeutet, dass der Algorithmus sich selbst über bereits berechnete Teilresultate anpasst. Dies geschieht mit dem Ziel, die Konvergenzgeschwindigkeit bei nichtglatten Problemen optimal zu halten und Aussagen über die erreichte Genauigkeit zu geben, sodass eine bestimmte Fehlerschranke erreicht wird. Für diesen Algorithmus wird die Konvergenz und die asymptotisch optimale Komplexität gezeigt. Der arithmetische Aufwand ist im Moment noch suboptimal. Weiterhin wird in dieser Arbeit die C++Bibliothek Lawa vorgestellt, die aus Bausteinen zur Realisierung adaptiver WaveletVerfahren besteht und mit Hinblick auf den Einsatz in Forschung und Lehre konzipiert wurde. Schließlich führen wir noch die Bewertung von Amerikanischen Optionen in verschiedenen LévyModellen mit uniformen waveletgalerkin Methoden aus.
6
7 Contents. Introduction 2. Wavelets and Adaptive Methods Biorthogonal Multiresolution Analysis Biorthogonal Spline Multiresolution on the Real Line Properties of Biorthogonal Spline Wavelets Biorthogonal Spline Wavelets on the Interval Adaptive Wavelet Methods for Elliptic Equations Prerequisites Elliptic Operator Equations Uniform Schemes Best Nterm Approximation Besov Regularity for Elliptic Equations in One Dimension Abstract Adaptive Method Prerequisites for an Implementable Adaptive Method Adaptive Wavelet Solver for Elliptic Equations Adaptive Wavelet Solver for Elliptic Equations without Coarsening the Iterands A Simplified Adaptive Wavelet Solver for Elliptic Equations Adaptive Wavelet Methods for Nonlinear Equations Nemytskij Operators and their Properties Recovery Scheme Recoverybased Approximate Matrixvector Product Solving Operator Equations with Lawa Operator Equations Helmholtz Equation Hypersingular Integral Equation Model Problems that come with Lawa Examples for the Helmholtz Problem Examples for the Hypersingular Problem Uniform Schemes Best Nterm Approximation Adaptive Wavelet Methods ELLSOLVE ELLSOLVEWOCOARSENING SADWAVELLSOLVE Semilinear Equations Semilinear Helmholtz Example vii
8 Contents Uniform Methods Adaptive Wavelet Methods Adaptive Wavelet Solution to Variational Inequalities Prerequisites Elliptic Variational Inequalities Parabolic Variational Inequalities Uniform Methods for EVIs and PVIs EVIs PVIs Adaptive Methods for EVIs Besov Regularity for EVIs Abstract Adaptive Method Adaptive Projection Numerical Tests for PROJECT EVISOLVE SADWAVEVISOLVE Adaptive Methods for PVIs Reformulation as an Operator Inequality Formulation as a Biinfinite Inequality in Wavelet Space Towards PVISOLVE Solving Inequalities with Lawa Elliptic Obstacle Problems Secondorder Differential Operator Examples for the Helmholtz Inequality Uniform Schemes On Residual Estimation for Elliptic Obstacle Problems Convergence Rates of Uniform Schemes Best Nterm Approximation EVISOLVE Leaving the Inner Loop with the Exact Residual Leaving the Inner Loop with the Approximate Residual Leaving the Inner Loop with the Simplified Residual SADWAVEVISOLVE Lévy Processes & American Options Options and their Fair Value Lévy Processes The Geometric Brownian Motion The Variance Gamma Process The Normal Inverse Gaussian Process The CGMY Process The Merton Jumpdiffusion The Generalized Hyperbolic Lévy Motion The Kou Model Fully Deterministic Option Pricing viii
9 Contents 7. Setup of the Algebraic System and Singular Quadrature Hadamard FinitePart Integral General Calculation Specific Implementation The CGMY Kernel The Kou Jumpdiffusion The Merton Jumpdiffusion The Normal Inverse Gaussian Model Pricing Options with Lawa European Options BlackScholes Model Merton Jumpdiffusion Model Kou Jumpdiffusion Model CGMY Purejump Model CGMYe Model Normal Inverse Gaussian Model American Options BlackScholes Model Merton Jumpdiffusion Model Kou Jumpdiffusion Model CGMY Purejump Model CGMYe Model Normal Inverse Gaussian Model Further Models and Option Types Conclusion 79.Outlook 8 A. Stochastics 85 B. Function Spaces 87 B.. Sequence Spaces B.2. Lebesgue and Sobolev Spaces B.3. Vectorvalued Lebesgue and Sobolev Spaces B.4. Nonlinear Approximation Spaces C. Singular Quadrature 97 C.. Gaussian Quadrature C.2. Computation of Gaussian Rules C.3. Gaussian FinitePart Quadrature D. Complexity Analysis for Quadratic Programming 2 ix
10
11 List of Figures 2.. Centralized BSplines of order d =,..., Dual BSplines Ñ,,..., Ñ4, Waveletbased image compression Wavelet preconditioning Primal [DKU99] wavelets on the interval The approximation of a function by wavelets Slopes for APPLY Multiscale and local scaling functions representation Slopes for RECOVERY BU and F, d = d = Slopes for RECOVERY BU and F 2, d = d = Slopes for RECOVERY CDD and F, d = d = Slopes for RECOVERY CDD and F 2, d = d = Slopes for RECOVERY CDD and F, d = d = Slopes for RECOVERY CDD and F 2, d = d = Slopes for RECOVERY APPLY CDD and A, d = d = Slopes for RECOVERY APPLY CDD and A, d = d = Slopes for RECOVERY APPLY CDD and A 2, d = d = Plots of the examples for the Helmholtz problem (P) and (P2) Plots of the examples for the Helmholtz problem (P3) and (P4) Plots of the examples for the hypersingular problem (P) and (P2) Convergence rates of uniform schemes for the Helmholtz problem Convergence rates of uniform schemes for the hypersingular problem Convergence rates of best Nterm approximation for the Helmholtz examples Convergence rates of best Nterm approximation for the hypersingular examples ELLSOLVE, Helmholtz problem (P3) ELLSOLVE, Helmholtz problem (P4) ELLSOLVE, hypersingular problem (P2) ELLSOLVEWOCOARSENING, Helmholtz problem (P3) ELLSOLVEWOCOARSENING, Helmholtz problem (P4) ELLSOLVEWOCOARSENING, hypersingular problem (P2) SADWAVELLSOLVE, Helmholtz problem (P3) SADWAVELLSOLVE, Helmholtz problem (P4) SADWAVELLSOLVE, hypersingular problem (P2) Residual reduction rates of the NEWTON solver xi
12 List of Figures 3.8. Convergence rates of uniform schemes for the semilinear Helmholtz problem ELLSOLVEWOCOARSENING, semilinear Helmholtz problem (P3) ELLSOLVEWOCOARSENING, semilinear Helmholtz problem (P4) PROJECT BU, Helmholtz inequality (P) and (P2) PROJECT BU, Helmholtz inequality (P3) and (P4) PROJECT BU, Helmholtz inequality (P5) and (P6) PROJECT CDD, Helmholtz inequality (P) PROJECT CDD, Helmholtz inequality (P2) PROJECT CDD, Helmholtz inequality (P3) PROJECT CDD, Helmholtz inequality (P4) PROJECT CDD, Helmholtz inequality (P5) PROJECT CDD, Helmholtz inequality (P6) Plots of the examples for the Helmholtz inequality problem Plots of the homogenized examples for the Helmholtz inequality problem Error estimation for the projected Richardson iteration, J = Error estimation for the projected Richardson iteration, J = Error estimation for the projected Richardson iteration, J = Convergence rates of uniform schemes for the Helmholtz inequality problem Convergence rates of best Nterm approximation for the Helmholtz inequality problem EVISOLVE for the Helmholtz inequality problem (P4) EVISOLVE for the Helmholtz inequality problem (P5) EVISOLVE for the Helmholtz Inequality problem (P6) EVISOLVE using the approximate residual for the Helmholtz inequality problem (P4) EVISOLVE using the approximate residual for the Helmholtz inequality problem (P5) EVISOLVE using the approximate residual for the Helmholtz inequality problem (P6) EVISOLVE using the simplified residual for the Helmholtz inequality problem (P4) EVISOLVE using the simplified residual for the Helmholtz inequality problem (P5) EVISOLVE using the simplified residual for the Helmholtz inequality problem (P6) SADWAVEVISOLVE for the Helmholtz inequality (P) SADWAVEVISOLVE for the Helmholtz inequality (P2) SADWAVEVISOLVE for the Helmholtz inequality (P3) SADWAVEVISOLVE for the Helmholtz inequality (P4) SADWAVEVISOLVE for the Helmholtz inequality (P5).. 35 xii
13 List of Figures SADWAVEVISOLVE for the Helmholtz inequality (P6) Sample paths of different stock price models Smooth pasting principle in the BlackScholes model Violated smooth pasting principle in the CGMY model European options in the BlackScholes model European options in the Merton jumpdiffusion model European options in the Kou jumpdiffusion model European options in the CGMY model European options in the CGMYe model European options in the Normal Inverse Gaussian model American options in the BlackScholes model American options in the Merton Jumpdiffusion model American options in the Kou Jumpdiffusion model American options in the CGMY model American options in the CGMYe model American options in the Normal Inverse Gaussian model xiii
14
15 List of Algorithms. COARSE[η, v] v η RHS[η, f] f η APPLY[η, A, v] w η GALSOLVE[Λ, A Λ Λ, ṽ Λ, f Λ, η] v Λ ELLSOLVE[ε, A, f] u(ε) ELLSOLVEWOCOARSENING[ε, A, f] u(ε) C(Λ, c) Λ SADWAVELLSOLVE[ε, A, f] u(ε) RECOVERY[η, F, u Λ ] vˆλ PREDICTION[ε, u Λ ] Γ RECONSTRUCTION[Γ, u Λ ] v Γ QUASIINTERPOLATION[v Γ, F] g CHANGEOFBASIS[g] g DECOMPOSITION[u] v PREDICTION CDD[ε, u Λ ] Γ DECOMPOSITION [u] v NEWTON[Λ, A Λ Λ, F, ṽ Λ, f Λ, η] v Λ PRICHARDSON[Λ, A Λ Λ, ũ Λ, f Λ, η] u Λ PPRICHARDSON[Λ, A Λ Λ, ũ Λ, f Λ, η] u Λ PROJECT LSF[η, ˆv] w η PROJECT[η, ˆv] w η EVISOLVE[ε, A, f] u(ε) EXTRAPOLATE[u Λ ] ˆΠ(u Λ ) SADWAVEVISOLVE[ε, A, f] u(ε) xv
16
17 . Introduction... and these wavelets, which science burst upon the world that night, were strange even to the women and men who used them. We start this thesis with a short introduction, where we present the object of analysis, namely the elliptic and the parabolic variational inequality. In order to relate the adaptive methods developed in this thesis later on we give a short overview on the various numerical methods for solving these inequalities. A survey on wavelet methods and their application to computational finance follows together with a comparison of other numerical schemes for the option pricing problem. We close this chapter with an outline concerning the adaptive wavelet library Lawa and a summary of the contributions of this thesis. The Problem To give an idea for the problems analyzed in this thesis, the next figure shows two important applications of variational inequalities, namely the elliptic obstacle problem on the left and the American option pricing problem on the right side Membrane Obstacle V (t, S) Calculated Put value Payoff S t Both problems have in common that they are partially driven by some sort of operator equation, but only in a subdomain. In the other part of the domain the solution rests on the obstacle. It is also known what happens on the boundary of these subdomains, but the important question now is where this boundary is located this is why such problems are also called free boundary problems. The important starting point with existence and uniqueness results for variational inequalities was laid in [LS67]. Since then, a whole field of research has emerged concerning the efficient solution of these problems, from which adaptive methods are nowadays among the most advanced schemes.
18 . Introduction Adaptive Finite Element Methods for Inequalities In this context adaptivity means that schemes use estimates to adapt themselves to the local behaviour of the solution of the problem in order to use a certain budget of computing time in a most efficient way. Adaptive methods for solving variational inequalities in general and obstacle problems in particular are nowadays wellestablished in the form of adaptive finite element methods (AFEM). Error estimators of residual type for elliptic obstacle problems in connection with linear finite elements are derived in [CN, Vee2]. In [Bra5] the author shows that an error estimator based on the Lagrange multiplier is also reliable and efficient but gives rise to a nonconforming contribution. The convergence of a conforming AFEM for obstacle problems again based on the Laplace operator is established in [BCH7] for affine obstacles. However, the presented method attains the same convergence rate as a uniform scheme. Similar findings are presented in [BCH9] for general obstacles where again the adaptive strategy yields only the convergence rate of a uniform scheme while displaying a slightly smaller absolute error. In [BC4] the authors derive error estimators for obstacle problems for the Laplace operator based on averaging techniques. If one is interested in bounding the error in the maximum norm then the estimators from [NSV3] are a good choice as they are reliable and efficient as well, which is demonstrated with the help of examples in 2d and 3d. For parabolic inequalities driven by the BlackScholes operator, the authors show in [MNvPZ7] that their error estimator is reliable and efficient and gives rise to higher convergence rates. The authors also apply the resulting adaptive method to the example of American option pricing. For an extension to inequalities of parabolic and elliptic type that are driven by integrodifferential operators we refer to [NvPZ8]. There, a posteriori error estimates for piecewise linear finite elements are derived and tested at various examples. When it comes to the solution of the discrete systems that arise during an AFEM computation the algorithms from [Hop87] or [Kor94, Kor96] may be used as they are globally convergent and have the same optimal complexity as standard multigrid methods for elliptic equations. Wavelets Compactly supported bases of orthonormal or biorthogonal wavelets are known since [Dau88, CDF92]. Since then, these functions have been used in applications as diverse as for example image procession, audio denoising, data compression and the solution of PDEs. Following the discovery of wavelet bases on the real line, the next step towards wavelet bases on general domains is the construction on the interval [, ] that was done in [DKU99, Pri6, Dij9] to name only a few. Wavelet bases for general domains can be constructed via domain decomposition and have been proposed for example in [CTU99, CTU, BE2, CM, DS99]. It is more or less straightforward how to use wavelets for the solution of operator equations. Later on, still several other advantages like wavelet preconditioning 2
19 [DK92, Jaf92] or waveletbased operator compression were discovered [BCR9, DPS93, Sch98, Ste4, GS6, GS5]. Starting from [CDD], various contributions have been made to adaptive wavelet methods in [CDD2, GHS7, Gan8] for elliptic equations. Proven convergence rates for AFEM schemes are also available as it was shown in [BDD4]. Adaptive methods for nonlinear problems have been analyzed in [DSX, CDD3, CDD4, BU8]. In [DHU, DDU2] optimal adaptive schemes for the solution of saddle point problems like Stokes problem have been developed. There are also specialized wavelet bases and schemes available for the Stokes problem, see [Urb95a, Urb, Urb95b, Urb96, DKU96]. Further applications of adaptive wavelet methods are for example the adaptive optimization of convex functionals in Banach spaces that was presented in [CU5], where this list is far from being complete. Wavelet Methods for Option Pricing Starting from [MvPS4], where waveletgalerkin methods were applied for the first time to the pricing of European options, there has been a tremendous progress in this particular field of research. Since then, many other subproblems from quantitative finance have been analyzed like more sophisticated timestepping for European options [MSW5, MSW6], American option pricing [MNS5], Swing option valuation [WW8], models incorporating stochastic volatility [HMS5, Hil9], variational sensitivity analysis [HSW8], varianceoptimal hedging [Ves9] or multidimensional Lévy models [FRS7, Rei8b, Rei8a, LRZ8, WR8, Win9, RSW9, Rei], where this list does not claim to be complete. See also the survey article [HRSW9]. The simple nature of the domain of problems arising in option pricing, which are essentially cubes, plays in the hands of wavelet methods as there the construction of the bases is straightforward. Finite Differences and MonteCarlo Option Pricing Finite difference methods (FDM) for partial integrodifferential equations (PIDE) arising in option pricing were presented in [CV6]. As these FD methods are not able to treat the singular integral operator directly, the singularity of the Lévy kernel has to be regularized by truncating the jump measure in some small neighbourhood of the origin and adding a corresponding diffusion component to the infinitesimal generator of the process. While this is justified for European options, in [Pow9] it is found that this is not appropriate for exotic contracts like American options or Barrier contracts. In [KL9] the author also mentions that the same effect is true for MonteCarlo methods. Therefore, finite element methods, which do not require regularization, seem to be the natural method of choice for that type of problems. 3
20 . Introduction A Library for Adaptive Wavelet Applications All code that has been developed during the creation of this work is publicly available, see [RS, RSU]. The reason for that is manifold: firstly, we strongly believe in the importance of the underlying code for the validity of the numerical experiments conducted in this thesis that back the theoretical findings. As found in [Hat97], many scientific calculation codes contain serious flaws. For this issue, which has just recently been raised in [Inc] and then extensively discussed in [Sou], the availability of code used in scientific publications is essential for other researchers in that academic field to validate the results of a specific paper. Secondly, the accessibility of work, that has already been accomplished also greatly saves time for people who work in that particular area (the often exerted reinvention of the wheel) or especially who are starting research in that domain. Lastly, every researcher can ask himself the question about the worthiness of numerical results when no one knows how they have been produced. As the algorithms for waveletbased methods are quite involved and this particular field of research is also very young, there are no matured software projects or even commercial packages available. A first attempt to resolve this drawback was done in the IgpmLib as described in [BBJ + 2]. However, this library has some slight design flaws, which mainly originate from the poor compiler compliance to the C++standard back then and the fact that the library was in the first place not intended for adaptive methods. Therefore, back in 25 Alexander Stippler came to the decision that only a complete rewrite from scratch would solve this problem and he began developing the library Lawa, which is an abbreviation of Library for Adaptive Wavelet Applications within his Ph.D.Project [Sti]. The author of this thesis joined this project in 27, which is publicly available at under the terms of the GNU General Public License [Fre9, Version 2] or [Fre7, Version 3]. Of course, the library is still under development and thus far from being complete nor free from defects and so this thesis can only represent a snapshot of the methods currently available in Lawa. However, in making it a public project the authors hope to encourage other researcher on working with that foundation. Contributions The contributions of this thesis are the design of adaptive wavelet algorithms for the solution of variational inequalities and the development of a modern replacement of the IgpmLib in view of a library for adaptive wavelet applications that is free as in free speech, portable and usable in research and education. Starting in Chapter 2, we provide a short introduction to wavelet and multiscale methods with a focus on the adaptive wavelet algorithms from [CDD, CDD2, GHS7, BK6]. We also present one of the first implementations of the RECOVERY scheme, which approximatively evaluates nonlinear 4
21 functions of wavelet expansions, as developed in [BU8, CDD4] including some numerical tests. Then in Chapter 3, we introduce two different model problems for an operator equation that come with Lawa and which can be solved by the adaptive algorithms from the second chapter. Building on a slightly different variant of the RECOVERY scheme, we design in Chapter 4 an adaptive projection algorithm and based on that a first adaptive wavelet algorithm EVI SOLVE for solving elliptic variational inequalities, that is fully computable with proven convergence. As of today, there are no previous adaptive wavelet methods for variational inequalities known to the author. Besides analysing the Besov regularity of the elliptic variational inequality, a second simplified adaptive wavelet algorithm SADWAVEVISOLVE is presented. Finally, a Galerkin solver for the resulting finitedimensional variational inequalities is derived and a heuristic residualbased error bound is proposed, which is also tested at some examples. We also present some building blocks towards an adaptive wavelet scheme for parabolic inequalities. In Chapter 5 the two adaptive algorithms are applied to a set of both smooth and singular elliptic test examples. When we come to the quantitative finance part of this thesis in Chapter 6, the gap between the American option pricing problem and parabolic variational inequalities is bridged by recalling major theorems from [MvPS4, MNS5]. We also present various different Lévy models that are commonly used. In Chapter 7 we apply the concept of the Hadamard finite part integral to the setup of the algebraic system stemming from the discretized option pricing problem. Lastly, we show in Chapter 8 how uniform schemes may be applied to the pricing problem and plot the value functions of European and American options in the various models from Chapter 6. We summarise the findings of this thesis in Chapter 9 and provide in Chapter some starting points for further research in this field that are now imminent. Miscellaneous Throughout this thesis, a b will always mean a Cb where the constant C R = { x R x } does not depend on any other quantity in the statement unless explicitly mentioned. Conversely, b a is short for b Ca and finally a b stands for a b and b a. We will always denote functions in italics u(x) while boldfaced characters indicate (infinite)dimensional vectors u l 2 or matrices A : l 2 l 2. For a vector u l 2, we denote by u Λ its restriction to the index set Λ while we use the expression u Λ in order to point out its support Λ. The characters V and H always denote real Hilbert spaces, where as usual V and H stand for their topological duals. Typical examples used in this thesis are V = H (Ω) and H = L 2(Ω). 5
22
23 2. Wavelets and Adaptive Methods All this time the Guard was looking at her, first through a telescope, then through a microscope, and then through an operaglass. (Lewis Carroll) The first chapter now introduces the main tool that we are going to use in the course of this thesis for solving operator equations and inequalities. For this we will loosely follow the book [Urb9] and the survey article [Dah97], where also many proofs of the various statements may be found. The interested reader who wants to gain additional insight in the history of wavelets is also referred to those. Afterwards the adaptive wavelet methods for the solution of operator equations from [CDD, CDD2, GHS7, BK6] are described as they are implemented in Lawa and because they form the starting point for our development of the adaptive methods for operator inequalities later on. At the end of this chapter we have a look at the RECOVERY scheme that allows to evaluate nonlinear operators in an efficient way using a wavelet basis. There we will also present some experiments concerning the quantitative behaviour of the methods from [BU8] and [CDD4]. 2.. Biorthogonal Multiresolution Analysis We begin with a basic concept. Definition 2. (Biorthogonal Multiresolution Analysis) For the Hilbert space L 2 (Ω) from Definition B.7, Ω R, with inner product (, ) = (, ) ;Ω and norm = ;Ω we are considering two sequences of closed, nested subspaces {S j } j N and { S j } j N such that their union is dense in L 2 (Ω): ( ) S j S j+, clos L2 (Ω) S j = L 2 (Ω), S j S j+, clos L2 (Ω) j= ( j= S j ) = L 2 (Ω). We assume that the spaces S j and S j are generated by the basis functions ϕ j,k and ϕ j,k, respectively, which means that S j = S(Φ j ) = span(φ j ), with Φ j = { ϕj,k k I j }, where Ij Z denotes the index set of the basis functions on level j. If then for j N it holds that {, k = l, (ϕ j,k, ϕ j,l ) =, otherwise, ϕ j,k S j, ϕ j,l S j, (2.) 7
AN INTRODUCTION TO NUMERICAL METHODS AND ANALYSIS
AN INTRODUCTION TO NUMERICAL METHODS AND ANALYSIS Revised Edition James Epperson Mathematical Reviews BICENTENNIAL 0, 1 8 0 7 z ewiley wu 2007 r71 BICENTENNIAL WILEYINTERSCIENCE A John Wiley & Sons, Inc.,
More informationMathematics Course 111: Algebra I Part IV: Vector Spaces
Mathematics Course 111: Algebra I Part IV: Vector Spaces D. R. Wilkins Academic Year 19967 9 Vector Spaces A vector space over some field K is an algebraic structure consisting of a set V on which are
More informationMATHEMATICAL METHODS OF STATISTICS
MATHEMATICAL METHODS OF STATISTICS By HARALD CRAMER TROFESSOK IN THE UNIVERSITY OF STOCKHOLM Princeton PRINCETON UNIVERSITY PRESS 1946 TABLE OF CONTENTS. First Part. MATHEMATICAL INTRODUCTION. CHAPTERS
More informationLecture 10. Finite difference and finite element methods. Option pricing Sensitivity analysis Numerical examples
Finite difference and finite element methods Lecture 10 Sensitivities and Greeks Key task in financial engineering: fast and accurate calculation of sensitivities of market models with respect to model
More informationInner Product Spaces
Math 571 Inner Product Spaces 1. Preliminaries An inner product space is a vector space V along with a function, called an inner product which associates each pair of vectors u, v with a scalar u, v, and
More informationHøgskolen i Narvik Sivilingeniørutdanningen STE6237 ELEMENTMETODER. Oppgaver
Høgskolen i Narvik Sivilingeniørutdanningen STE637 ELEMENTMETODER Oppgaver Klasse: 4.ID, 4.IT Ekstern Professor: Gregory A. Chechkin email: chechkin@mech.math.msu.su Narvik 6 PART I Task. Consider twopoint
More informationSolution of Linear Systems
Chapter 3 Solution of Linear Systems In this chapter we study algorithms for possibly the most commonly occurring problem in scientific computing, the solution of linear systems of equations. We start
More informationMoving Least Squares Approximation
Chapter 7 Moving Least Squares Approimation An alternative to radial basis function interpolation and approimation is the socalled moving least squares method. As we will see below, in this method the
More informationContinued Fractions and the Euclidean Algorithm
Continued Fractions and the Euclidean Algorithm Lecture notes prepared for MATH 326, Spring 997 Department of Mathematics and Statistics University at Albany William F Hammond Table of Contents Introduction
More informationIntroduction to the Finite Element Method
Introduction to the Finite Element Method 09.06.2009 Outline Motivation Partial Differential Equations (PDEs) Finite Difference Method (FDM) Finite Element Method (FEM) References Motivation Figure: cross
More informationNotes on Factoring. MA 206 Kurt Bryan
The General Approach Notes on Factoring MA 26 Kurt Bryan Suppose I hand you n, a 2 digit integer and tell you that n is composite, with smallest prime factor around 5 digits. Finding a nontrivial factor
More informationMathematical Finance
Mathematical Finance Option Pricing under the RiskNeutral Measure Cory Barnes Department of Mathematics University of Washington June 11, 2013 Outline 1 Probability Background 2 Black Scholes for European
More informationModel order reduction via Proper Orthogonal Decomposition
Model order reduction via Proper Orthogonal Decomposition Reduced Basis Summer School 2015 Martin Gubisch University of Konstanz September 17, 2015 Martin Gubisch (University of Konstanz) Model order reduction
More informationFactorization Theorems
Chapter 7 Factorization Theorems This chapter highlights a few of the many factorization theorems for matrices While some factorization results are relatively direct, others are iterative While some factorization
More information1 VECTOR SPACES AND SUBSPACES
1 VECTOR SPACES AND SUBSPACES What is a vector? Many are familiar with the concept of a vector as: Something which has magnitude and direction. an ordered pair or triple. a description for quantities such
More informationNonlinear Iterative Partial Least Squares Method
Numerical Methods for Determining Principal Component Analysis Abstract Factors Béchu, S., RichardPlouet, M., Fernandez, V., Walton, J., and Fairley, N. (2016) Developments in numerical treatments for
More information1 Sets and Set Notation.
LINEAR ALGEBRA MATH 27.6 SPRING 23 (COHEN) LECTURE NOTES Sets and Set Notation. Definition (Naive Definition of a Set). A set is any collection of objects, called the elements of that set. We will most
More informationPoint Lattices in Computer Graphics and Visualization how signal processing may help computer graphics
Point Lattices in Computer Graphics and Visualization how signal processing may help computer graphics Dimitri Van De Ville Ecole Polytechnique Fédérale de Lausanne Biomedical Imaging Group dimitri.vandeville@epfl.ch
More informationx1 x 2 x 3 y 1 y 2 y 3 x 1 y 2 x 2 y 1 0.
Cross product 1 Chapter 7 Cross product We are getting ready to study integration in several variables. Until now we have been doing only differential calculus. One outcome of this study will be our ability
More informationApplied Linear Algebra I Review page 1
Applied Linear Algebra Review 1 I. Determinants A. Definition of a determinant 1. Using sum a. Permutations i. Sign of a permutation ii. Cycle 2. Uniqueness of the determinant function in terms of properties
More informationNumerical Analysis An Introduction
Walter Gautschi Numerical Analysis An Introduction 1997 Birkhauser Boston Basel Berlin CONTENTS PREFACE xi CHAPTER 0. PROLOGUE 1 0.1. Overview 1 0.2. Numerical analysis software 3 0.3. Textbooks and monographs
More information3. Interpolation. Closing the Gaps of Discretization... Beyond Polynomials
3. Interpolation Closing the Gaps of Discretization... Beyond Polynomials Closing the Gaps of Discretization... Beyond Polynomials, December 19, 2012 1 3.3. Polynomial Splines Idea of Polynomial Splines
More informationCHAPTER 1 Splines and Bsplines an Introduction
CHAPTER 1 Splines and Bsplines an Introduction In this first chapter, we consider the following fundamental problem: Given a set of points in the plane, determine a smooth curve that approximates the
More informationCONTROLLABILITY. Chapter 2. 2.1 Reachable Set and Controllability. Suppose we have a linear system described by the state equation
Chapter 2 CONTROLLABILITY 2 Reachable Set and Controllability Suppose we have a linear system described by the state equation ẋ Ax + Bu (2) x() x Consider the following problem For a given vector x in
More informationPiecewise Cubic Splines
280 CHAP. 5 CURVE FITTING Piecewise Cubic Splines The fitting of a polynomial curve to a set of data points has applications in CAD (computerassisted design), CAM (computerassisted manufacturing), and
More informationMean value theorem, Taylors Theorem, Maxima and Minima.
MA 001 Preparatory Mathematics I. Complex numbers as ordered pairs. Argand s diagram. Triangle inequality. De Moivre s Theorem. Algebra: Quadratic equations and expressions. Permutations and Combinations.
More information4: SINGLEPERIOD MARKET MODELS
4: SINGLEPERIOD MARKET MODELS Ben Goldys and Marek Rutkowski School of Mathematics and Statistics University of Sydney Semester 2, 2015 B. Goldys and M. Rutkowski (USydney) Slides 4: SinglePeriod Market
More informationDomain Decomposition Methods. Partial Differential Equations
Domain Decomposition Methods for Partial Differential Equations ALFIO QUARTERONI Professor ofnumericalanalysis, Politecnico di Milano, Italy, and Ecole Polytechnique Federale de Lausanne, Switzerland ALBERTO
More informationLINEAR ALGEBRA W W L CHEN
LINEAR ALGEBRA W W L CHEN c W W L Chen, 1997, 2008 This chapter is available free to all individuals, on understanding that it is not to be used for financial gain, and may be downloaded and/or photocopied,
More informationScientific Computing: An Introductory Survey
Scientific Computing: An Introductory Survey Chapter 10 Boundary Value Problems for Ordinary Differential Equations Prof. Michael T. Heath Department of Computer Science University of Illinois at UrbanaChampaign
More informationWavelet analysis. Wavelet requirements. Example signals. Stationary signal 2 Hz + 10 Hz + 20Hz. Zero mean, oscillatory (wave) Fast decay (let)
Wavelet analysis In the case of Fourier series, the orthonormal basis is generated by integral dilation of a single function e jx Every 2πperiodic squareintegrable function is generated by a superposition
More informationThe Characteristic Polynomial
Physics 116A Winter 2011 The Characteristic Polynomial 1 Coefficients of the characteristic polynomial Consider the eigenvalue problem for an n n matrix A, A v = λ v, v 0 (1) The solution to this problem
More informationA new FeynmanKacformula for option pricing in Lévy models
A new FeynmanKacformula for option pricing in Lévy models Kathrin Glau Department of Mathematical Stochastics, Universtity of Freiburg (Joint work with E. Eberlein) 6th World Congress of the Bachelier
More informationNumerical Analysis Lecture Notes
Numerical Analysis Lecture Notes Peter J. Olver 5. Inner Products and Norms The norm of a vector is a measure of its size. Besides the familiar Euclidean norm based on the dot product, there are a number
More informationOverview of Math Standards
Algebra 2 Welcome to math curriculum design maps for Manhattan Ogden USD 383, striving to produce learners who are: Effective Communicators who clearly express ideas and effectively communicate with diverse
More informationMASSACHUSETTS INSTITUTE OF TECHNOLOGY 6.436J/15.085J Fall 2008 Lecture 5 9/17/2008 RANDOM VARIABLES
MASSACHUSETTS INSTITUTE OF TECHNOLOGY 6.436J/15.085J Fall 2008 Lecture 5 9/17/2008 RANDOM VARIABLES Contents 1. Random variables and measurable functions 2. Cumulative distribution functions 3. Discrete
More informationBANACH AND HILBERT SPACE REVIEW
BANACH AND HILBET SPACE EVIEW CHISTOPHE HEIL These notes will briefly review some basic concepts related to the theory of Banach and Hilbert spaces. We are not trying to give a complete development, but
More informationThe van Hoeij Algorithm for Factoring Polynomials
The van Hoeij Algorithm for Factoring Polynomials Jürgen Klüners Abstract In this survey we report about a new algorithm for factoring polynomials due to Mark van Hoeij. The main idea is that the combinatorial
More informationIllPosed Problems in Probability and Stability of Random Sums. Lev B. Klebanov, Tomasz J. Kozubowski, and Svetlozar T. Rachev
IllPosed Problems in Probability and Stability of Random Sums By Lev B. Klebanov, Tomasz J. Kozubowski, and Svetlozar T. Rachev Preface This is the first of two volumes concerned with the illposed problems
More informationDecember 4, 2013 MATH 171 BASIC LINEAR ALGEBRA B. KITCHENS
December 4, 2013 MATH 171 BASIC LINEAR ALGEBRA B KITCHENS The equation 1 Lines in twodimensional space (1) 2x y = 3 describes a line in twodimensional space The coefficients of x and y in the equation
More informationLecture Notes to Accompany. Scientific Computing An Introductory Survey. by Michael T. Heath. Chapter 10
Lecture Notes to Accompany Scientific Computing An Introductory Survey Second Edition by Michael T. Heath Chapter 10 Boundary Value Problems for Ordinary Differential Equations Copyright c 2001. Reproduction
More informationCommunication on the Grassmann Manifold: A Geometric Approach to the Noncoherent MultipleAntenna Channel
IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 48, NO. 2, FEBRUARY 2002 359 Communication on the Grassmann Manifold: A Geometric Approach to the Noncoherent MultipleAntenna Channel Lizhong Zheng, Student
More information24. The Branch and Bound Method
24. The Branch and Bound Method It has serious practical consequences if it is known that a combinatorial problem is NPcomplete. Then one can conclude according to the present state of science that no
More informationInner Product Spaces and Orthogonality
Inner Product Spaces and Orthogonality week 34 Fall 2006 Dot product of R n The inner product or dot product of R n is a function, defined by u, v a b + a 2 b 2 + + a n b n for u a, a 2,, a n T, v b,
More informationMaster of Mathematical Finance: Course Descriptions
Master of Mathematical Finance: Course Descriptions CS 522 Data Mining Computer Science This course provides continued exploration of data mining algorithms. More sophisticated algorithms such as support
More informationFRACTIONAL INTEGRALS AND DERIVATIVES. Theory and Applications
FRACTIONAL INTEGRALS AND DERIVATIVES Theory and Applications Stefan G. Samko Rostov State University, Russia Anatoly A. Kilbas Belorussian State University, Minsk, Belarus Oleg I. Marichev Belorussian
More informationStephane Crepey. Financial Modeling. A Backward Stochastic Differential Equations Perspective. 4y Springer
Stephane Crepey Financial Modeling A Backward Stochastic Differential Equations Perspective 4y Springer Part I An Introductory Course in Stochastic Processes 1 Some Classes of DiscreteTime Stochastic
More informationFACTORING POLYNOMIALS IN THE RING OF FORMAL POWER SERIES OVER Z
FACTORING POLYNOMIALS IN THE RING OF FORMAL POWER SERIES OVER Z DANIEL BIRMAJER, JUAN B GIL, AND MICHAEL WEINER Abstract We consider polynomials with integer coefficients and discuss their factorization
More informationMetric Spaces. Chapter 7. 7.1. Metrics
Chapter 7 Metric Spaces A metric space is a set X that has a notion of the distance d(x, y) between every pair of points x, y X. The purpose of this chapter is to introduce metric spaces and give some
More informationLinear Codes. Chapter 3. 3.1 Basics
Chapter 3 Linear Codes In order to define codes that we can encode and decode efficiently, we add more structure to the codespace. We shall be mainly interested in linear codes. A linear code of length
More information1. Introduction. O. MALI, A. MUZALEVSKIY, and D. PAULY
Russ. J. Numer. Anal. Math. Modelling, Vol. 28, No. 6, pp. 577 596 (2013) DOI 10.1515/ rnam20130032 c de Gruyter 2013 Conforming and nonconforming functional a posteriori error estimates for elliptic
More informationThe Dirichlet Unit Theorem
Chapter 6 The Dirichlet Unit Theorem As usual, we will be working in the ring B of algebraic integers of a number field L. Two factorizations of an element of B are regarded as essentially the same if
More informationNumerical methods for American options
Lecture 9 Numerical methods for American options Lecture Notes by Andrzej Palczewski Computational Finance p. 1 American options The holder of an American option has the right to exercise it at any moment
More informationPOLYNOMIAL HISTOPOLATION, SUPERCONVERGENT DEGREES OF FREEDOM, AND PSEUDOSPECTRAL DISCRETE HODGE OPERATORS
POLYNOMIAL HISTOPOLATION, SUPERCONVERGENT DEGREES OF FREEDOM, AND PSEUDOSPECTRAL DISCRETE HODGE OPERATORS N. ROBIDOUX Abstract. We show that, given a histogram with n bins possibly noncontiguous or consisting
More informationVector and Matrix Norms
Chapter 1 Vector and Matrix Norms 11 Vector Spaces Let F be a field (such as the real numbers, R, or complex numbers, C) with elements called scalars A Vector Space, V, over the field F is a nonempty
More informationLinear Algebra Done Wrong. Sergei Treil. Department of Mathematics, Brown University
Linear Algebra Done Wrong Sergei Treil Department of Mathematics, Brown University Copyright c Sergei Treil, 2004, 2009, 2011, 2014 Preface The title of the book sounds a bit mysterious. Why should anyone
More informationClassification of Cartan matrices
Chapter 7 Classification of Cartan matrices In this chapter we describe a classification of generalised Cartan matrices This classification can be compared as the rough classification of varieties in terms
More informationClass Meeting # 1: Introduction to PDEs
MATH 18.152 COURSE NOTES  CLASS MEETING # 1 18.152 Introduction to PDEs, Fall 2011 Professor: Jared Speck Class Meeting # 1: Introduction to PDEs 1. What is a PDE? We will be studying functions u = u(x
More informationSolving polynomial least squares problems via semidefinite programming relaxations
Solving polynomial least squares problems via semidefinite programming relaxations Sunyoung Kim and Masakazu Kojima August 2007, revised in November, 2007 Abstract. A polynomial optimization problem whose
More information1 Norms and Vector Spaces
008.10.07.01 1 Norms and Vector Spaces Suppose we have a complex vector space V. A norm is a function f : V R which satisfies (i) f(x) 0 for all x V (ii) f(x + y) f(x) + f(y) for all x,y V (iii) f(λx)
More information3 Orthogonal Vectors and Matrices
3 Orthogonal Vectors and Matrices The linear algebra portion of this course focuses on three matrix factorizations: QR factorization, singular valued decomposition (SVD), and LU factorization The first
More informationNOTES ON LINEAR TRANSFORMATIONS
NOTES ON LINEAR TRANSFORMATIONS Definition 1. Let V and W be vector spaces. A function T : V W is a linear transformation from V to W if the following two properties hold. i T v + v = T v + T v for all
More informationInner product. Definition of inner product
Math 20F Linear Algebra Lecture 25 1 Inner product Review: Definition of inner product. Slide 1 Norm and distance. Orthogonal vectors. Orthogonal complement. Orthogonal basis. Definition of inner product
More informationNotes on Determinant
ENGG2012B Advanced Engineering Mathematics Notes on Determinant Lecturer: Kenneth Shum Lecture 918/02/2013 The determinant of a system of linear equations determines whether the solution is unique, without
More informationTOPOLOGY: THE JOURNEY INTO SEPARATION AXIOMS
TOPOLOGY: THE JOURNEY INTO SEPARATION AXIOMS VIPUL NAIK Abstract. In this journey, we are going to explore the so called separation axioms in greater detail. We shall try to understand how these axioms
More informationProbability and Statistics
CHAPTER 2: RANDOM VARIABLES AND ASSOCIATED FUNCTIONS 2b  0 Probability and Statistics Kristel Van Steen, PhD 2 Montefiore Institute  Systems and Modeling GIGA  Bioinformatics ULg kristel.vansteen@ulg.ac.be
More informationMathematics (MAT) MAT 061 Basic Euclidean Geometry 3 Hours. MAT 051 PreAlgebra 4 Hours
MAT 051 PreAlgebra Mathematics (MAT) MAT 051 is designed as a review of the basic operations of arithmetic and an introduction to algebra. The student must earn a grade of C or in order to enroll in MAT
More informationExact shapereconstruction by onestep linearization in electrical impedance tomography
Exact shapereconstruction by onestep linearization in electrical impedance tomography Bastian von Harrach harrach@math.unimainz.de Institut für Mathematik, Joh. GutenbergUniversität Mainz, Germany
More informationHow High a Degree is High Enough for High Order Finite Elements?
This space is reserved for the Procedia header, do not use it How High a Degree is High Enough for High Order Finite Elements? William F. National Institute of Standards and Technology, Gaithersburg, Maryland,
More informationTHE DIMENSION OF A VECTOR SPACE
THE DIMENSION OF A VECTOR SPACE KEITH CONRAD This handout is a supplementary discussion leading up to the definition of dimension and some of its basic properties. Let V be a vector space over a field
More informationComputer Graphics. Geometric Modeling. Page 1. Copyright Gotsman, Elber, Barequet, Karni, Sheffer Computer Science  Technion. An Example.
An Example 2 3 4 Outline Objective: Develop methods and algorithms to mathematically model shape of real world objects Categories: WireFrame Representation Object is represented as as a set of points
More informationAu = = = 3u. Aw = = = 2w. so the action of A on u and w is very easy to picture: it simply amounts to a stretching by 3 and 2, respectively.
Chapter 7 Eigenvalues and Eigenvectors In this last chapter of our exploration of Linear Algebra we will revisit eigenvalues and eigenvectors of matrices, concepts that were already introduced in Geometry
More information3. INNER PRODUCT SPACES
. INNER PRODUCT SPACES.. Definition So far we have studied abstract vector spaces. These are a generalisation of the geometric spaces R and R. But these have more structure than just that of a vector space.
More informationFurther Study on Strong Lagrangian Duality Property for Invex Programs via Penalty Functions 1
Further Study on Strong Lagrangian Duality Property for Invex Programs via Penalty Functions 1 J. Zhang Institute of Applied Mathematics, Chongqing University of Posts and Telecommunications, Chongqing
More informationFourthOrder Compact Schemes of a Heat Conduction Problem with Neumann Boundary Conditions
FourthOrder Compact Schemes of a Heat Conduction Problem with Neumann Boundary Conditions Jennifer Zhao, 1 Weizhong Dai, Tianchan Niu 1 Department of Mathematics and Statistics, University of MichiganDearborn,
More informationESSENTIAL COMPUTATIONAL FLUID DYNAMICS
ESSENTIAL COMPUTATIONAL FLUID DYNAMICS Oleg Zikanov WILEY JOHN WILEY & SONS, INC. CONTENTS PREFACE xv 1 What Is CFD? 1 1.1. Introduction / 1 1.2. Brief History of CFD / 4 1.3. Outline of the Book / 6 References
More informationChapter 5. Banach Spaces
9 Chapter 5 Banach Spaces Many linear equations may be formulated in terms of a suitable linear operator acting on a Banach space. In this chapter, we study Banach spaces and linear operators acting on
More informationA Novel Fourier Transform Bspline Method for Option Pricing*
A Novel Fourier Transform Bspline Method for Option Pricing* Paper available from SSRN: http://ssrn.com/abstract=2269370 Gareth G. Haslip, FIA PhD Cass Business School, City University London October
More information1 Solving LPs: The Simplex Algorithm of George Dantzig
Solving LPs: The Simplex Algorithm of George Dantzig. Simplex Pivoting: Dictionary Format We illustrate a general solution procedure, called the simplex algorithm, by implementing it on a very simple example.
More informationNEW YORK STATE TEACHER CERTIFICATION EXAMINATIONS
NEW YORK STATE TEACHER CERTIFICATION EXAMINATIONS TEST DESIGN AND FRAMEWORK September 2014 Authorized for Distribution by the New York State Education Department This test design and framework document
More informationTHE DYING FIBONACCI TREE. 1. Introduction. Consider a tree with two types of nodes, say A and B, and the following properties:
THE DYING FIBONACCI TREE BERNHARD GITTENBERGER 1. Introduction Consider a tree with two types of nodes, say A and B, and the following properties: 1. Let the root be of type A.. Each node of type A produces
More informationApplications to Data Smoothing and Image Processing I
Applications to Data Smoothing and Image Processing I MA 348 Kurt Bryan Signals and Images Let t denote time and consider a signal a(t) on some time interval, say t. We ll assume that the signal a(t) is
More informationTESLA Report 200303
TESLA Report 233 A multigrid based 3D spacecharge routine in the tracking code GPT Gisela Pöplau, Ursula van Rienen, Marieke de Loos and Bas van der Geer Institute of General Electrical Engineering,
More informationEXIT TIME PROBLEMS AND ESCAPE FROM A POTENTIAL WELL
EXIT TIME PROBLEMS AND ESCAPE FROM A POTENTIAL WELL Exit Time problems and Escape from a Potential Well Escape From a Potential Well There are many systems in physics, chemistry and biology that exist
More informationCSE 494 CSE/CBS 598 (Fall 2007): Numerical Linear Algebra for Data Exploration Clustering Instructor: Jieping Ye
CSE 494 CSE/CBS 598 Fall 2007: Numerical Linear Algebra for Data Exploration Clustering Instructor: Jieping Ye 1 Introduction One important method for data compression and classification is to organize
More informationFUNCTIONAL ANALYSIS LECTURE NOTES: QUOTIENT SPACES
FUNCTIONAL ANALYSIS LECTURE NOTES: QUOTIENT SPACES CHRISTOPHER HEIL 1. Cosets and the Quotient Space Any vector space is an abelian group under the operation of vector addition. So, if you are have studied
More informationOn the Adaptive Tensor Product Wavelet Galerkin Method with Applications in Finance
Fakultät für Mathematik und Wirtschaftswissenschaften On the Adaptive Tensor Product Wavelet Galerkin Method with Applications in Finance Dissertation zur Erlangung des Doktorgrades Dr. rer. nat. der Fakultät
More informationApplied Computational Economics and Finance
Applied Computational Economics and Finance Mario J. Miranda and Paul L. Fackler The MIT Press Cambridge, Massachusetts London, England Preface xv 1 Introduction 1 1.1 Some Apparently Simple Questions
More informationA gentle introduction to the Finite Element Method. Francisco Javier Sayas
A gentle introduction to the Finite Element Method Francisco Javier Sayas 2008 An introduction If you haven t been hiding under a stone during your studies of engineering, mathematics or physics, it is
More informationNotes 11: List Decoding Folded ReedSolomon Codes
Introduction to Coding Theory CMU: Spring 2010 Notes 11: List Decoding Folded ReedSolomon Codes April 2010 Lecturer: Venkatesan Guruswami Scribe: Venkatesan Guruswami At the end of the previous notes,
More informationModern Optimization Methods for Big Data Problems MATH11146 The University of Edinburgh
Modern Optimization Methods for Big Data Problems MATH11146 The University of Edinburgh Peter Richtárik Week 3 Randomized Coordinate Descent With Arbitrary Sampling January 27, 2016 1 / 30 The Problem
More informationIdeal Class Group and Units
Chapter 4 Ideal Class Group and Units We are now interested in understanding two aspects of ring of integers of number fields: how principal they are (that is, what is the proportion of principal ideals
More information2.3 Convex Constrained Optimization Problems
42 CHAPTER 2. FUNDAMENTAL CONCEPTS IN CONVEX OPTIMIZATION Theorem 15 Let f : R n R and h : R R. Consider g(x) = h(f(x)) for all x R n. The function g is convex if either of the following two conditions
More informationStochastic Inventory Control
Chapter 3 Stochastic Inventory Control 1 In this chapter, we consider in much greater details certain dynamic inventory control problems of the type already encountered in section 1.3. In addition to the
More informationRNCodings: New Insights and Some Applications
RNCodings: New Insights and Some Applications Abstract During any composite computation there is a constant need for rounding intermediate results before they can participate in further processing. Recently
More informationValuation of American Options
Valuation of American Options Among the seminal contributions to the mathematics of finance is the paper F. Black and M. Scholes, The pricing of options and corporate liabilities, Journal of Political
More informationDETERMINANTS. b 2. x 2
DETERMINANTS 1 Systems of two equations in two unknowns A system of two equations in two unknowns has the form a 11 x 1 + a 12 x 2 = b 1 a 21 x 1 + a 22 x 2 = b 2 This can be written more concisely in
More informationMATH 304 Linear Algebra Lecture 20: Inner product spaces. Orthogonal sets.
MATH 304 Linear Algebra Lecture 20: Inner product spaces. Orthogonal sets. Norm The notion of norm generalizes the notion of length of a vector in R n. Definition. Let V be a vector space. A function α
More informationComputational Optical Imaging  Optique Numerique.  Deconvolution 
Computational Optical Imaging  Optique Numerique  Deconvolution  Winter 2014 Ivo Ihrke Deconvolution Ivo Ihrke Outline Deconvolution Theory example 1D deconvolution Fourier method Algebraic method
More informationISOMETRIES OF R n KEITH CONRAD
ISOMETRIES OF R n KEITH CONRAD 1. Introduction An isometry of R n is a function h: R n R n that preserves the distance between vectors: h(v) h(w) = v w for all v and w in R n, where (x 1,..., x n ) = x
More information