1,434 research outputs found
Sparsity promoting hybrid solvers for hierarchical bayesian inverse problems
No full textThe recovery of sparse generative models from few noisy measurements is an important and challenging problem. Many deterministic algorithms rely on some form of l1-l2 minimization to combine the computational convenience of the l2 penalty and the sparsity promotion of the l1. It was recently shown within the Bayesian framework that sparsity promotion and computational efficiency can be attained with hierarchical models with conditionally Gaussian priors and gamma hyperpriors. The related Gibbs energy function is a convex functional, and its minimizer, which is the maximum a posteriori (MAP) estimate of the posterior, can be computed efficiently with the globally convergent Iterated Alternating Sequential (IAS) algorithm [D. Calvetti, E. Somersalo, and A. Strang, Inverse Problems, 35 (2019), 035003]. Generalization of the hyperpriors for these sparsity promoting hierarchical models to a generalized gamma family either yield globally convex Gibbs energy functionals or can exhibit local convexity for some choices for the hyperparameters [D. Calvetti et al., Inverse Problems, 36 (2020), 025010]. The main problem in computing the MAP solution for greedy hyperpriors that strongly promote sparsity is the presence of local minima. To overcome the premature stopping at a spurious local minimizer, we propose two hybrid algorithms that first exploit the global convergence associated with gamma hyperpriors to arrive in a neighborhood of the unique minimizer and then adopt a generalized gamma hyperprior that promotes sparsity more strongly. The performance of the two algorithms is illustrated with computed examples
Fast simulation of solid tumors thermal ablation treatments with a 3D reaction diffusion model☆
No full textAn efficient computational method for near real-time simulation of thermal ablation of tumors via radio frequencies is proposed. Model
simulations of the temperature field in a 3D portion of tissue containing the tumoral mass for different patterns of source heating can be used
to design the ablation procedure. The availability of a very efficient computational scheme makes it possible to update the predicted outcome of
the procedure in real time. In the algorithms proposed here a discretization in space of the governing equations is followed by an adaptive time
integration based on implicit multistep formulas. A modification of the ode15s MATLAB function which uses Krylov space iterative methods
for the solution of the linear systems arising at each integration step makes it possible to perform the simulations on standard desktop for much
finer grids than using the built-in ode15s. The proposed algorithm can be applied to a wide class of nonlinear parabolic differential equations
Experimental and numerical analysis of a 2-D granular material
No full textPietruszczak and Pande eds., Balkema, Rotterdam
Image inpainting with structural bootstrap priors
No full textIn this article, we consider the following inpainting problem arising in image restoration: part of an image has been removed, and we want to restore the image from the remaining, possibly noisy, portion. We show that if the true image contains no sharp edges, the inpainting can be done rather satisfactorily by means of an isotropic smoothness prior assumption. If, on the other hand, we have information concerning discontinuities in the image, we can input this information into the restoration algorithm using an anisotropic smoothness prior. Based on these observations, we propose an inpainting method based on a bootstrapping procedure that consists of the following steps: first, we smooth out the incomplete image and calculate the gradient field. Since this field is smooth, it can be inpainted satisfactorily. By using the inpainted gradient field, we then construct an anisotropic smoothness prior that pilots the inpainting of the original non-smooth image. The calculations are based on the Bayesian interpretation of the inpainting problem as a statistical inverse problem.
Keywords: Inpainting; Statistical inversion; Structural priors; Boundary condition
Tikhonov regularization and the L-curve for large discrete ill-posed problems
No full textAbstractDiscretization of linear inverse problems generally gives rise to very ill-conditioned linear systems of algebraic equations. Typically, the linear systems obtained have to be regularized to make the computation of a meaningful approximate solution possible. Tikhonov regularization is one of the most popular regularization methods. A regularization parameter specifies the amount of regularization and, in general, an appropriate value of this parameter is not known a priori. We review available iterative methods, and present new ones, for the determination of a suitable value of the regularization parameter by the L-curve criterion and the solution of regularized systems of algebraic equations
Bayes meets krylov: Statistically inspired preconditioners for CGLS
No full textThe solution of linear inverse problems when the unknown parameters outnumber data requires addressing the problem of a nontrivial null space. After restating the problem within the Bayesian framework, a priori information about the unknown can be utilized for determining the null space contribution to the solution. More specifically, if the solution of the associated linear system is computed by the conjugate gradient for least squares (CGLS) method, the additional information can be encoded in the form of a right preconditioner. In this paper we study how the right preconditioner changes the Krylov subspaces where the CGLS iterates live, and we draw a tighter connection between Bayesian inference and Krylov subspace methods. The advantages of a Bayes-meets-Krylov approach to the solution of underdetermined linear inverse problems is illustrated with computed examples
An iterative method with error estimators
No full textAbstractIterative methods for the solution of linear systems of equations produce a sequence of approximate solutions. In many applications it is desirable to be able to compute estimates of the norm of the error in the approximate solutions generated and terminate the iterations when the estimates are sufficiently small. This paper presents a new iterative method based on the Lanczos process for the solution of linear systems of equations with a symmetric matrix. The method is designed to allow the computation of estimates of the Euclidean norm of the error in the computed approximate solutions. These estimates are determined by evaluating certain Gauss, anti-Gauss, or Gauss–Radau quadrature rules
Stochastic modelling of muscle recruitment during activity
No full textMuscle forces can be selected from a space of muscle recruitment strategies that produce stable motion and variable muscle and joint forces. However, current optimization methods provide only a single muscle recruitment strategy. We modelled the spectrum of muscle recruitment strategies while walking. The equilibrium equations at the joints, muscle constraints, static optimization solutions and 15-channel electromyography (EMG) recordings for seven walking cycles were taken from earlier studies. The spectrum of muscle forces was calculated using Bayesian statistics and Markov chain Monte Carlo (MCMC) methods, whereas EMG-driven muscle forces were calculated using EMG-driven modelling. We calculated the differences between the spectrum and EMG-driven muscle force for 1–15 input EMGs, and we identified the muscle strategy that best matched the recorded EMG pattern. The best-fit strategy, static optimization solution and EMG-driven force data were compared using correlation analysis. Possible and plausible muscle forces were defined as within physiological boundaries and within EMG boundaries. Possible muscle and joint forces were calculated by constraining the muscle forces between zero and the peak muscle force. Plausible muscle forces were constrained within six selected EMG boundaries. The spectrum to EMG-driven force difference increased from 40 to 108 N for 1–15 EMG inputs. The best-fit muscle strategy better described the EMG-driven pattern (R2 = 0.94; RMSE = 19 N) than the static optimization solution (R2 = 0.38; RMSE = 61 N). Possible forces for 27 of 34 muscles varied between zero and the peak muscle force, inducing a peak hip force of 11.3 body-weights. Plausible muscle forces closely matched the selected EMG patterns; no effect of the EMG constraint was observed on the remaining muscle force ranges. The model can be used to study alternative muscle recruitment strategies in both physiological and pathophysiological neuromotor conditions
Priorconditioned CGLS-Based Quasi-MAP Estimate, Statistical Stopping Rule, and Ranking of Priors
No full textWe consider linear discrete ill-posed problems within the Bayesian framework, assuming a Gaussian additive noise model and a Gaussian prior whose covariance matrices may be known modulo multiplicative scaling factors. In that context, we propose a new pointwise estimator for the posterior density, the prior conditioned CGLS-based quasi-MAP (qMAP) as a computationally attractive approximation of the classical maximum a posteriori (MAP) estimate, in particular when the effective rank of the matrix A is much smaller than the dimension of the unknown. Exploiting the Bayesian paradigm and the connection between standard normal random variables and the χ2 distribution of their squared Euclidean norms, we propose a new stopping rule, the Maxχ2 rule, to terminate the CGLS iterations in the computation of the qMAP estimate. Moreover, we show that the proposed stopping rule can be used to estimate the possibly unknown scaling factor of the noise covariance, which is tantamount to estimating the noise level. It is shown that the Maxχ2 stopping rule and the proposed noise level estimation are affected by the correlation structure of priorconditioner, i.e., prior-related right preconditioner, but not by its scaling, which in the classical regularization framework would correspond to the Tikhonov regularization parameter. We will also show how the relation between priorconditioners and whitening transformations of the unknown can be used to rank different priorconditioners according to their agreement with the data
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