Statistical Inverse Problems and abstract inverse problems examples - PowerPoint PPT Presentation

Statistical Inverse Problems and Instrumental Variables Thorsten Hohage inverse problems Statistical Inverse Problems and abstract inverse problems examples Instrumental Variables regression problems and statistical inverse problems problem formulation Thorsten Hohage some functional analysis spectral theory for compact operators Institut für Numerische und Angewandte Mathematik spectral theorem for bounded self-adjoint University of Göttingen operators functional calculus regularization RICAM, Linz, 6.9.2008 methods Picard criterion, spectral cutoff other regularization methods definition of regularization methods convergence analysis negative results source conditions choice of

Statistical Inverse outline Problems and inverse problems Instrumental Variables abstract inverse problems Thorsten Hohage examples inverse problems regression problems and statistical inverse problems abstract inverse problems problem formulation examples regression some functional analysis problems and statistical inverse spectral theory for compact operators problems problem formulation spectral theorem for bounded self-adjoint operators some functional functional calculus analysis spectral theory for compact regularization methods operators spectral theorem for Picard criterion, spectral cutoff bounded self-adjoint operators other regularization methods functional calculus regularization definition of regularization methods methods Picard criterion, spectral convergence analysis cutoff other regularization negative results methods definition of regularization source conditions methods convergence convergence in expectation analysis choice of regularization parameters negative results source conditions discrepancy principle choice of

Statistical Inverse Keller’s definition of an inverse problem Problems and Instrumental Variables Thorsten Hohage inverse problems “We call two problems inverses of one another if abstract inverse problems examples the formulation of each involves all or part of the regression problems and solution of the other. Often, for historical statistical inverse problems reasons, one of the two problems has been problem formulation studied extensively for some time, while the some functional analysis other is newer and not so well understood. In spectral theory for compact operators such cases, the former problem is called the spectral theorem for bounded self-adjoint direct problem, while the latter is called the operators functional calculus inverse problem.” regularization methods Picard criterion, spectral cutoff J.B. Keller. Inverse Problems. Am. Math. Mon., 83:107-118 , 1976 other regularization methods definition of regularization methods convergence analysis negative results source conditions choice of

Statistical Inverse Hadamard’s definition of well-posedness Problems and Instrumental Variables Thorsten Hohage inverse problems abstract inverse problems examples Definition regression A problem is called well-posed if problems and statistical inverse problems 1. there exists a solution to the problem (existence), problem formulation some functional 2. there is at most one solution to the problem analysis (uniqueness), spectral theory for compact operators spectral theorem for 3. the solution depends continuously on the data bounded self-adjoint operators (stability). functional calculus regularization methods Otherwise the problem is called ill-posed. Picard criterion, spectral cutoff other regularization methods definition of regularization methods convergence analysis negative results source conditions choice of

Statistical Inverse ill-posedness in terms of operator equations Problems and Instrumental Variables Thorsten Hohage Suppose the inverse problem can be formulated as an inverse problems operator equation abstract inverse problems examples F ( a ) = u regression problems and statistical inverse where x denotes the unknown solution and y the given problems data. problem formulation some functional Then the inverse problem is well-posed in the sense of analysis Hadamard if spectral theory for compact operators spectral theorem for 1. F is surjective (existence) bounded self-adjoint operators functional calculus 2. F in injective (uniqueness) regularization 3. F − 1 is continuous (stability) methods Picard criterion, spectral cutoff Typically, the third condition is violated for inverse other regularization methods definition of regularization problems! methods convergence analysis negative results source conditions choice of

Statistical Inverse first kind integral equations Problems and Instrumental Variables Thorsten Hohage inverse problems abstract inverse problems examples regression problems and statistical inverse Find a function a such that problems � problem formulation some functional k ( x , y ) a ( y ) dy = u ( x ) for all x . analysis spectral theory for compact operators spectral theorem for bounded self-adjoint operators functional calculus regularization methods Picard criterion, spectral cutoff other regularization methods definition of regularization methods convergence analysis negative results source conditions choice of

Statistical Inverse identification of parameters in differential Problems and Instrumental equations Variables Thorsten Hohage inverse problems abstract inverse problems examples Estimate a parameter a in a differential equation regression given noisy measurements of the solution u ! problems and statistical inverse problems The parameter-to-solution operator F : a �→ u is defined problem formulation only implicitely via the differential equation and typically some functional analysis nonlinear even if the differential equation is linear. spectral theory for compact operators spectral theorem for bounded self-adjoint The unknown parameter a might be operators functional calculus ◮ a coefficient in the differential equation, regularization methods ◮ a boundary condition or an initial condition, Picard criterion, spectral cutoff other regularization ◮ a parametrization of the shape of a domain. methods definition of regularization methods convergence analysis negative results source conditions choice of

Statistical Inverse outline Problems and inverse problems Instrumental Variables abstract inverse problems Thorsten Hohage examples inverse problems regression problems and statistical inverse problems abstract inverse problems problem formulation examples regression some functional analysis problems and statistical inverse spectral theory for compact operators problems problem formulation spectral theorem for bounded self-adjoint operators some functional functional calculus analysis spectral theory for compact regularization methods operators spectral theorem for Picard criterion, spectral cutoff bounded self-adjoint operators other regularization methods functional calculus regularization definition of regularization methods methods Picard criterion, spectral convergence analysis cutoff other regularization negative results methods definition of regularization source conditions methods convergence convergence in expectation analysis choice of regularization parameters negative results source conditions discrepancy principle choice of

Statistical Inverse nonparametric regression: random design Problems and Instrumental Variables Estimate a function u in some smoothness class F given Thorsten Hohage i.i.d. random variables ( X i , Y i ) , i = 1 , . . . , n such that inverse problems E ( Y | X = x ) = u ( x ) , abstract inverse problems examples E (( Y − u ( x )) 2 | X = x ) < ∞ . regression problems and statistical inverse problems problem formulation some functional analysis spectral theory for compact operators spectral theorem for bounded self-adjoint operators functional calculus regularization methods Picard criterion, spectral cutoff other regularization methods definition of regularization methods convergence analysis negative results source conditions choice of

Statistical Inverse nonparametric regression: deterministic Problems and Instrumental design Variables Thorsten Hohage inverse problems abstract inverse problems examples regression problems and Estimate a function u in some smoothness class F given statistical inverse problems � � � � problem formulation x ( n ) x ( n ) Y i = u + σ ǫ i , i = 1 , . . . , n some functional i i analysis spectral theory for compact operators where ǫ i are independent random variables satisfying spectral theorem for bounded self-adjoint operators functional calculus E ǫ 2 E ǫ i = 0 , i = 1 . regularization methods Picard criterion, spectral cutoff other regularization methods definition of regularization methods convergence analysis negative results source conditions choice of