Search Results (3)

A new high-order hybrid method integrating neural networks and corrected finite differences is developed for solving elliptic equations with irregular interfaces and discontinuous solutions. Standard fourth-order finite difference discretization becomes invalid near such interfaces due to the discontinuities and requires corrections based on Cartesian derivative jumps. In traditional numerical methods, such as the augmented matched interface and boundary (AMIB) method, these derivative jumps can be reconstructed via additional approximations and are solved together with the unknown solution in an iterative procedure. Nontrivial developments have been carried out in the AMIB method in treating sharply curved interfaces, which, however, may not work for interfaces with geometric singularities. In this work, machine learning techniques are utilized to directly predict these Cartesian derivative jumps without involving the unknown solution. To this end, physics-informed neural networks (PINNs) are trained to satisfy the jump conditions for both closed and open interfaces with possible geometric singularities. The predicted Cartesian derivative jumps can then be integrated in the corrected finite differences. The resulting discrete Laplacian can be efficiently solved by fast Poisson solvers, such as fast Fourier transform (FFT) and geometric multigrid methods, over a rectangular domain with Dirichlet boundary conditions. This hybrid method is both easy to implement and efficient. Numerical experiments in two and three dimensions demonstrate that the method achieves fourth-order accuracy for the solution and its derivatives. Full article

The objective of this paper was to present a new inverse problem statement and numerical method for the Volterra integral equations with piecewise continuous kernels. For such Volterra integral equations of the first kind, it is assumed that kernel discontinuity curves are the desired ones, but the rest of the information is known. The resulting integral equation is nonlinear with respect to discontinuity curves which correspond to integration bounds. A direct method of discretization with a posteriori verification of calculations is proposed. The family of quadrature rules is employed for approximation purposes. It is shown that the arithmetic complexity of the proposed numerical method is $\mathcal{O}\left(N^3\right)$. The method has first-order convergence. A generalization of the method is also proposed for the case of an arbitrary number of discontinuity curves. The illustrative examples are included to demonstrate the efficiency and accuracy of proposed solver. Full article

Factors associated with olaparib toxicity remain unknown in ovarian cancer patients. The large inter-individual variability in olaparib pharmacokinetics could contribute to the onset of early significant adverse events (SAE). We aimed to retrospectively analyze the pharmacokinetic/pharmacodynamic relationship for toxicity in ovarian cancer patients from “real life” data. The clinical endpoint was the onset of SAE (grade III/IV toxicity or dose reduction/discontinuation). Plasma olaparib concentration was assayed using liquid chromatography at any time over the dosing interval. Trough concentrations (CminPred) were estimated using a population pharmacokinetic model. The association between toxicity and clinical characteristics or CminPred was assessed by logistic regression and non-parametric statistical tests. Twenty-seven patients were included, among whom 13 (48%) experienced SAE during the first six months of treatment. Olaparib CminPred was the only covariate significantly associated with increased risk of SAE onset (odds ratio = 1.31, 95%CI = [1.10; 1.57], for each additional 1000 ng/mL). The ROC curve identified a threshold of CminPred = 2500 ng/mL for prediction of SAE onset (sensitivity/specificity 0.62 and 1.00, respectively). This study highlights a significant association between olaparib plasma exposure and SAE onset and identified the threshold of 2500 ng/mL trough concentration as potentially useful to guide dose adjustment in ovarian cancer patients. Full article