Mathematische Methoden der Unsicherheitsquantifizierung Oliver Ernst Professur Numerische Mathematik Sommersemester 2014
Contents 1 Introduction 1.1 What is Uncertainty Quantification? 1.2 A Case Study: Radioactive Waste Disposal 2 Monte Carlo Methods 2.1 Introduction 2.2 Basic Monte Carlo Simulation 2.3 Improving the Monte Carlo Method 2.4 Multilevel Monte Carlo Estimators 2.5 The Monte Carlo Finite Element Method 3 Random Fields 3.1 Introduction 3.2 Karhunen-Loève Expansion 3.3 Regularity of Random Fields 3.4 Covariance Eigenvalue Decay 4 Stochastic Collocation 4.1 Introduction 4.2 Collocation 4.3 Analytic Parameter Dependence 4.4 Convergence Oliver Ernst (Numerische Mathematik) UQ Sommersemester 2014 6 / 315
Contents 5 Probability Theory 6 Elliptic Boundary Value Problems 7 Collection of Results from Functional Analysis 8 Miscellanea Oliver Ernst (Numerische Mathematik) UQ Sommersemester 2014 296 / 315
Contents 5 Probability Theory 6 Elliptic Boundary Value Problems 7 Collection of Results from Functional Analysis 7.1 Hilbert-Schmidt Operators 8 Miscellanea Oliver Ernst (Numerische Mathematik) UQ Sommersemester 2014 297 / 315
For normed linear spaces X and Y, we denote by L px, Y q the set of all bounded linear operators from X to Y. Definition C.1 Let X and Y be separable Hilbert spaces with norms } } X and } } Y and let tx j u npn denote a CONS of X. A linear operator L : X Ñ Y for which 8ÿ 1{2 }L} HSpX,Y q : }Lx j } 2 Y ă 8 is called a Hilbert-Schmidt operator. We shall write }L} HS if X Y. Proposition C.2 j 1 The mapping } } HSpX,Y q is a norm, called the Hilbert-Schmidt norm, on the space of all Hilbert-Schmidt operators from X to Y, which we denote by HSpX, Y q. In addition, phspx, Y q, } } HSpX,Y q q is Banach space. Oliver Ernst (Numerische Mathematik) UQ Sommersemester 2014 298 / 315
Examples Example C.3 For X Y R n with the Euclidean norm } }, the Hilbert-Schmidt norm of a matrix A P R nˆn coincides with the Frobenius-norm }A} 2 F ř n i,j 1 a2 i,j. Example C.4 Define L P L pl 2 p0, 1qq by pluqpxq ż x 0 upyq dy, u P L 2 p0, 1q, x P p0, 1q. For the CONS tf j pxq? 2 sinpjπxq : j P Nu, we have? 2 plf j qpxq p1 cospjπxqq. jπ L is a Hilbert-Schmidt operator since }Lf j } L 2 p0,1q ď 2? 2 jπ. Oliver Ernst (Numerische Mathematik) UQ Sommersemester 2014 299 / 315
Integral operators Lemma C.5 Let H be a separable Hilbert space. If L P HSpHq, then }L} L phq ď }L} HS. In particular, Hilbert-Schmidt operators are bounded. Definition C.6 For a domain D Ă R d and k P L 2 pd ˆ Dq, the integral operator with kernel function k is defined as the linear operator ż K : u ÞÑ pkuqpx q : kpx, yqupyq dy, x P D. (C.1) Theorem C.7 D An integral operator with kernel function k P L 2 pd ˆ Dq is a Hilbert-Schmidt operator on L 2 pdq. Conversely, any Hilbert-Schmidt operator K on L 2 pdq can be written in the form (C.1) with }K} HS }k} L 2 pdˆdq. Oliver Ernst (Numerische Mathematik) UQ Sommersemester 2014 300 / 315
Compact operators Definition C.8 A set B in a Banach space X is said to be compact if every sequence u n Ă B has a convergent subsequence u nk with limit u P B. Definition C.9 A linear operator L : X Ñ Y, where X and Y are Banach spaces, is said to be compact if the image of any bounded set B Ă X has compact closure in Y, i.e., if LpBq } } Y Theorem C.10 is a compact set in Y for all bounded B Ă X. For k P L 2 pd ˆ Dq the associated integral operator K on L 2 pdq with kernel function k is a compact operator. Oliver Ernst (Numerische Mathematik) UQ Sommersemester 2014 301 / 315
Selfadjoint operators, eigenvalues Definition C.11 An operator L P L phq on a Hilbert space H is said to be selfadjoint if plu, vq pu, Lvq @u, v P H. Proposition C.12 For a domain D Ă R d, if k P L 2 pd ˆ Dq is symmetric, i.e., kpx, yq kpy, x q for all x, y P D, then the integral operator with kernel function k is selfadjoint with respect to the L 2 pdq inner product. Definition C.13 If L P L phq, λ P C is called an eigenvalue of L if there exists nonzero φ P H such that Lφ λφ. The element φ is called an eigenvector or eigenfunction of L. Oliver Ernst (Numerische Mathematik) UQ Sommersemester 2014 302 / 315
Spectral theorem Theorem C.14 (Spectral theorem for selfadjoint compact operators) Let H be a separable Hilbert space and K Ă L phq be selfadjoint and compact. Denote the eigenvalues of K by tλ j u jpn ordered such that λ j`1 ď λ j and denote the associated eigenfunctions by tφ j u. Then (i) All eigenvalues are real and λ j Ñ 0 as j Ñ 8. (ii) The sequence tφ j u can be chosen as a CONS of the range KpHq of K and, (iii) for any u P H, Ku 8ÿ λ k pu, φ j qφ j. j 1 (C.2) Oliver Ernst (Numerische Mathematik) UQ Sommersemester 2014 303 / 315
Nonnegative functions, operators Definition C.15 A function k : D ˆ D Ñ R is nonnegative definite if for any set of points x 1,..., x n P D and numbers a 1,..., a n P R there holds nÿ j,k 1 a j a k kpx j, x k q ě 0. A linear operator L P L phq on a Hilbert space H is nonnegative definite if pu, Luq ě 0 @u P H and positive definite if pu, Luq ą 0 @u P H. Oliver Ernst (Numerische Mathematik) UQ Sommersemester 2014 304 / 315
Nonnegative functions, operators, trace class operators Lemma C.16 For a domain D Ă R d and a nonnegative definite function k P CpD ˆ Dq, the integral operator K on L 2 pdq with kernel function k is nonnegative. Lemma C.17 (Dini) For a bounded domain D let f n P CpDq be such that f n px q ď f n`1 px q for n P N and f n px q Ñ fpx q as n Ñ 8 for all x P D. Then }f f n } 8 Ñ 0 as n Ñ 8. Definition C.18 Let H be a separable Hilbert space. A nonnegative definite operator L P L phq is said to be of trace class if traceplq ă 8, where the trace of L is defined as 8ÿ traceplq : plψ j, ψ j q for any CONS tψ j u jpn of H. j 1 Oliver Ernst (Numerische Mathematik) UQ Sommersemester 2014 305 / 315
Mercer s theorem Theorem C.19 (Mercer) For a bounded domain D, let k P CpD ˆ Dq be a symmetric and nonnegative definite function and let K be the integral operator with kernel function k. There exist eigenfunctions φ j of K with eigenvalues λ j ą 0 such that φ j P CpDq and 8ÿ kpx, yq λ j φ j px qφ j pyq, x, y P D, j 1 where the series converges in CpD ˆ Dq. Furthermore, n sup ˇ ˇkpx, yq ÿ 8ÿ λ j φ j px qφ j pyq ˇ ď sup x,ypd j 1 The operator K is of trace class and ż tracepkq D x PD j n`1 kpx, x q dx. λ j φ j px q 2. (C.3) Oliver Ernst (Numerische Mathematik) UQ Sommersemester 2014 306 / 315