An adaptive RKHS regularization for Fredholm integral equations

Regularization is a long-standing challenge for ill-posed linear inverse problems, and a prototype is the Fredholm integral equation of the first kind. We introduce a practical RKHS regularization algorithm adaptive to the discrete noisy measurement data and the underlying linear operator. This RKHS arises naturally in a variational approach, and its closure is the function space in which we can identify the true solution. We prove that the RKHS-regularized estimator has a mean-square error converging linearly as the noise scale decreases, with a multiplicative factor smaller than the commonly-used $L^2$-regularized estimator. Furthermore, numerical results demonstrate that the RKHS-regularizer significantly outperforms $L^2$-regularizer when either the noise level decays or when the observation mesh refines.Comment: 18 page

Neurotrophic keratitis after micropulse transscleral diode laser cyclophotocoagulation.

PurposeTo report two cases of neurotrophic keratitis (NK) after micropulse transscleral cyclophotocoagulation (MP-TCP).ObservationsTwo patients with predisposing factors for decreased corneal sensation developed NK 1 month after MP-TCP. Both patients did not heal with initial treatment with topical antibiotic and preservative free artificial tears. One patient required use of a bandage contact lens and the other patient required tarsorrhaphy. Both eyes experienced recurrence of NK.Conclusions and importanceNK can be triggered after MP-TCP in patients with underlying predisposing factors for decreased corneal sensation. This uncommon but vision-threatening complication should be discussed preoperatively with high-risk patients as a possible adverse event after MP-TCP and followed closely postoperatively

MADNESS: A Multiresolution, Adaptive Numerical Environment for Scientific Simulation

MADNESS (multiresolution adaptive numerical environment for scientific simulation) is a high-level software environment for solving integral and differential equations in many dimensions that uses adaptive and fast harmonic analysis methods with guaranteed precision based on multiresolution analysis and separated representations. Underpinning the numerical capabilities is a powerful petascale parallel programming environment that aims to increase both programmer productivity and code scalability. This paper describes the features and capabilities of MADNESS and briefly discusses some current applications in chemistry and several areas of physics