Chromatographic resolution is an issue of significant importance in preparative chromatography. However, due to the high-dimensional parameter space and nonlinear behavior, baseline separation in multi-component preparative chromatography remains challenging. To overcome these difficulties, we propose a surrogate-assisted optimizer focused on chromatographic resolution that uses a feedforward neural network to emulate PDE-based outlet responses. Based on this surrogate model, we employ the global optimization method of GS-PowerOpt to reshape the nonconvex landscape and robustly identify high-resolution designs. We prove the continuity of the resolution function and establish guarantees for convergence in the smoothed search. Through several case studies, the method consistently identifies designs with well-separated peaks and high resolution. The results offer a practical and theoretically grounded route to effective method development.
Abstract The main theme of this paper is to study Runge approximation for the Helmholtz equation in ℝ 2 {\mathbb{R}^{2}} in an explicit and numerically implementable way, using Fourier series, for both interior and exterior boundary value problems. Compared with existing qualitative Runge approximation results, the main advantage of our approach is its numerical implementability. Our numerical implementation is supported by explicit quantitative estimates.
Abstract We construct examples of distribution-type formal solutions to the Novikov–Veselov equation whose regularization is given by the Moutard transformation and which generalize the delta-type solutions derived by R. Novikov and the author.
Abstract We consider the problem of phase retrieval for continuous and discrete Fourier transforms in multiple dimensions. Using finite difference operators, we give a large class of unexpected examples of non-uniqueness for this problem, including examples with the sparsity condition. A prototype of this construction in the continuous case is given in the work by Novikov and Xu (JFAA, 2026), using linear differential operators. The construction of the present work also yields a large class of non-trivial Pauli partners, i.e., different functions with the same intensities in both configuration and Fourier domains. We also consider the problem of phase retrieval with background information. We show that this problem is not uniquely solvable if the known and unknown parts of the function are not sufficiently disjoint.
Abstract We consider the recovery of a scalar nonlinearity in a semilinear Robin boundary value problem arising, for example, in steady-state heat transfer with nonlinear sources and reactive transport. Given multiple experiments with boundary excitations and full-boundary measurements of the corresponding states, we study the inverse problem of recovering the nonlinearity on the range of observed state values. For the reduced formulation, we establish problem-specific stability, sequential weak-to-strong continuity, and Fréchet differentiability properties of the parameter-to-state map associated with the nonlinear composition r ( u ) {r(u)} . These properties allow us to place the inverse problem within the standard Hilbert-space Tikhonov regularization framework and to record the corresponding existence, stability, and convergence properties of regularized minimizers. We then construct iterative regularization schemes of Gauss–Newton and nonlinear conjugate-gradient type. In addition, we introduce an all-at-once (AAO) Tikhonov formulation that treats the states and the nonlinearity as independent variables and penalizes both data misfit and PDE residuals. This formulation avoids elimination of the states and admits an existence analysis under weaker regularity assumptions on the scalar nonlinearity than those used for the differentiable reduced formulation. We derive the corresponding Gauss–Newton AAO system. Finally, we present two-dimensional numerical experiments for a model problem with synthetic boundary data perturbed by noise, where the unknown nonlinearity is recovered from several boundary excitations and the reduced and AAO reconstruction strategies are compared in terms of reconstruction quality and computational cost.
