Minimax Rates for Statistical Inverse Problems Under General Source Conditions
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Litao Ding
Abstract
We describe the minimax reconstruction rates in linear ill-posed equations in Hilbert space when smoothness is given in terms of general source sets. The underlying fundamental result, the minimax rate on ellipsoids, is proved similarly to the seminal study by D. L. Donoho, R. C. Liu, and B. MacGibbon [4]. These authors highlighted the special role of the truncated series estimator, and for such estimators the risk can explicitly be given. We provide several examples, indicating results for statistical estimation in ill-posed problems in Hilbert space.
Funding source: National Natural Science Foundation of China
Award Identifier / Grant number: 11331004
Award Identifier / Grant number: 11421110002
Funding statement: The first author is supported by the National Natural Science Foundation of China (Grants No. 11331004 and No. 11421110002) and the Program of Introducing Talents of Discipline to Universities (Grant No. B08018).
References
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© 2018 Walter de Gruyter GmbH, Berlin/Boston
Artikel in diesem Heft
- Frontmatter
- High-Order Semi-Discrete Central-Upwind Schemes with Lax–Wendroff-Type Time Discretizations for Hamilton–Jacobi Equations
- Numerical Modeling of Optical Fibers Using the Finite Element Method and an Exact Non-reflecting Boundary Condition
- Minimax Rates for Statistical Inverse Problems Under General Source Conditions
- Numerical Analysis for the Pure Neumann Control Problem Using the Gradient Discretisation Method
- Combining the DPG Method with Finite Elements
- A Hybrid Method for Solving Inhomogeneous Elliptic PDEs Based on Fokas Method
- Computation of Green’s Function of the Bounded Solutions Problem
- Simplified Generalized Gauss–Newton Method for Nonlinear Ill-Posed Operator Equations in Hilbert Scales
- Symbolic Algorithm of the Functional-Discrete Method for a Sturm–Liouville Problem with a Polynomial Potential
- Assessment of Characteristic Boundary Conditions Based on the Artificial Compressibility Method in Generalized Curvilinear Coordinates for Solution of the Euler Equations
- Approximate Solution of a Singular Integral Equation with a Cauchy Kernel on the Euclidean Plane
- The Dual-Weighted Residual Estimator Realized on Polygonal Meshes
Artikel in diesem Heft
- Frontmatter
- High-Order Semi-Discrete Central-Upwind Schemes with Lax–Wendroff-Type Time Discretizations for Hamilton–Jacobi Equations
- Numerical Modeling of Optical Fibers Using the Finite Element Method and an Exact Non-reflecting Boundary Condition
- Minimax Rates for Statistical Inverse Problems Under General Source Conditions
- Numerical Analysis for the Pure Neumann Control Problem Using the Gradient Discretisation Method
- Combining the DPG Method with Finite Elements
- A Hybrid Method for Solving Inhomogeneous Elliptic PDEs Based on Fokas Method
- Computation of Green’s Function of the Bounded Solutions Problem
- Simplified Generalized Gauss–Newton Method for Nonlinear Ill-Posed Operator Equations in Hilbert Scales
- Symbolic Algorithm of the Functional-Discrete Method for a Sturm–Liouville Problem with a Polynomial Potential
- Assessment of Characteristic Boundary Conditions Based on the Artificial Compressibility Method in Generalized Curvilinear Coordinates for Solution of the Euler Equations
- Approximate Solution of a Singular Integral Equation with a Cauchy Kernel on the Euclidean Plane
- The Dual-Weighted Residual Estimator Realized on Polygonal Meshes
