The method of Tikhonov regularization is commonly used to obtain regularized solutions of ill-posed linear inverse problems. We use its natural connection to optimal Bayes estimators to determine optimal experimental designs that can be used with Tikhonov regularization; they are designed to control a measure of total relative efficiency. We present an iterative/semidefinite programming hybrid method to explore the configuration space efficiently. Two examples from geophysics are used to illustrate the type of applications to which the methodology can be applied.
CurtisA (1999b) Optimal design of focused experiments and surveys. Geophysical Journal International,139, 205−15.
10.
CurtisA (2004) Theory of model-based geophysical survey and experimental design Part A - Linear Problems. The Leading Edge, 23, 997−1004.
11.
DraperNRHunterWG (1967) The use of prior distributions in the design of experiments for parameter estimation in non-linear situations. Biometrika, 54, 147−53.
12.
FedorovVV (1972) Theory of optimal experiments. Academic Press.
13.
FedorovVVHacklP (1997) Model-oriented design of experiments. Springer.
14.
GuestTCurtisA (2009) Iteratively constructive sequential design of experiments and surveys with nonlinear parameter-data relationships. Journal of Geophysical Research,114, B0430.7.
15.
GuestTCurtisA (2010) Optimal trace selection for AVA processing of shale-sand reservoirs. Geophysics, 75, C37−47.
16.
HaberEHoreshLTenorioL (2008) Numerical methods for experimental design of large-scale linear ill-posed inverse problems. Inverse Problems, 24, 055012.
17.
HaberEHoreshLTenorioL (2010) Numerical methods the design of large-scale nonlinear discrete ill-posed inverse problems. Inverse Problems,26, 025002.
18.
HaberEMagnantZLuceroCTenorioL (2012) Numerical methods for A-optimal designs with a sparsity constraint for ill-posed inverse problems. Computational Optimization and Applications,52, 293−314.
19.
HansenPC (1998) Rank-deficient and discrete ill-posed problems: Numerical aspects of linear inversion. SIAM.
20.
HuanHMarzoukY (2013) Simulation-based optimal Bayesian experimental design for nonlinear systems. Journal of Computational Physics,32, 288−317.
21.
KaipioJSomersaloE (2004) Statistical and computational inverse problems. Springer.
22.
KieferJWolfowitzJ (1960) Equivalence of two extremal problems. Canadian Journal of Mathematics,14, 849−79.
23.
LehmannELCasellaG (1998) Theory of point estimation (2nd edn). Springer.
NguyenNMillerAJ (1992) A review of some exchange algorithms for constructing discrete D-optimal designs. Computational Statistics and Data Analysis, 14, 489−98.
O’SullivanF (1986) A statistical perspective on ill-posed inverse problems. Statistical Science, 1, 502−27.
28.
PukelsheimF (2006) Optimal design of experiments.SIAM.
29.
ReayJR (1965) Generalizations of a theorem of carathéodory. American Mathematical Society, memoir No. 54.
30.
RockafellarRT (1970) Convex analysis. Princeton University Press.
31.
SteinbergDMHunterWG (1984) Experimental design: review and comment. Technometrics, 26, 71−97.
32.
TohKCTütüncilRHToddMJ (2002) SDPT3. A Matlab software for semidefinite-quadratic-linear programming. www.math.nus.edu.sg/~mattohkc/sdpt3.html
33.
VogelCR (2002) Computational methods for inverse problems. SIAM.
34.
WynnHP (1970) The sequential generation of D-optimum experimental designs. The Annals of Mathematical Statistics., 41, 1655−64.
35.
WynnHP (1972) Results in the theory and construction of D-optimum experimental designs. Journal of the Royal Statistical Society. Series B (Methodological), 34, 133−47.
36.
WynnHP (1984) Jack Kiefer’s contributions to experimental design. The Annals of Statisics, 12, 416−23.
