Computers and Mathematics with Applications 113 (2022) 214–224 Contents lists available at ScienceDirect Computers and Mathematics with Applications www.elsevier.com/locate/camwa Supermodeling, a convergent data assimilation meta-procedure used in simulation of tumor progression Maciej Paszy ´ nski a , Leszek Siwik a,∗ , Witold Dzwinel a , Keshav Pingali b a AGH University of Sciences and Technology, Faculty of Computer Science, Electronics and Telecommunication, Institute of Computer Science, al. A Mickiewicza 30, 30-059 Krakow, Poland b The University of Texas at Austin, Oden Institute for Computational and Engineering Sciences, United States of America A R T I C L E I N F O A B S T R A C T Keywords: Isogeometric analysis Tumor growth simulation Supermodeling Supermodeling is a modern, model-ensembling paradigm that integrates several self-synchronized imperfect sub-models by controlling a few meta-parameters to generate more accurate predictions of complex systems’ dynamics. Continual synchronization between sub-models is an attractive alternative allowing for accurate trajectory predictions compared to a single model or a classical ensemble of independent models whose decision fusion is based on the majority voting or averaging the outcomes. However, it comes out from numerous observations that the supermodeling procedure’s convergence depends on a few principal factors such as (1) the number of sub-models, (2) their proper selection, and (3) the choice of the convergent optimization procedure, which assimilates the supermodel meta-parameters to data. Herein, we focus on modeling the evolution of the system described by a set of PDEs. We prove that supermodeling is conditionally convergent to a fixed-point attractor regarding only the supermodel meta-parameters. In our proof, we assume constant parametrization of the sub-models. We investigate the formal conditions of the convergence of the supermodeling scheme theoretically. We employ the Banach fixed point theorem for the supermodeling correction operator, updating the synchronization constants’ values iteratively. From the theoretical estimate, we make the following conclusions. The nudging of the supermodel to the ground truth (real data assimilated) should be well balanced because both too small and too large attraction to data cause the supermodel desynchronization. The time-step size can control the convergence of the training procedure, by balancing the Lipshitz continuity constant of the PDE operator. All the sub-models have to be close to the ground-truth along the training trajectory but still sufficiently diverse to explore the phase space better. As an example, we discuss the three-dimensional supermodel of tumor evolution to demonstrate the supermodel’s perfect fit to artificial data generated based on real medical images. 1. Introduction It is well known that the multi-model ensembling approach can be a closer metaphor of an observed phenomenon than a single-model fore- cast [23]. In the multi-model ensemble (MME) approach where the models are not synchronized the MME mean prediction is often more skillful as model errors tend to average out [23], whereas the spread between the model predictions is naturally interpreted as a measure of the uncertainty about the mean [22]. Although MME tends to improve predictions in terms of statistics (i.e., mean and variance), a major draw- back is that averaging uncorrelated trajectories from different models leads to variance reduction and smoothing. It is not intended and is not helpful, thus MME is not designed to produce an improved trajectory * Corresponding author. E-mail addresses: maciej.paszynski@agh.edu.pl (M. Paszy ´ nski), siwik@agh.edu.pl (L. Siwik), dzwinel@agh.edu.pl (W. Dzwinel). that can be seen as a specific forecast [16]. The alternative approach for taking advantage of many trajectories followed by distinctive mod- els and discovering many “basin of attractions” without (premature) loss of the trajectories diversity is combining models dynamically. One of the first naive approaches of this kind has been proposed in [11], while more mature ones can be followed in [25,27]. They introduce connection terms into the model equations that  the state of one model to each other’s states in the ensemble. The computer model’s assimilation to a real phenomenon through a set of observations is a complex inverse problem; thus, its time com- plexity increases exponentially with the number of parameters. This makes data assimilation procedures useless when applied for multiscale https://doi.org/10.1016/j.camwa.2022.03.025 Received 26 February 2021; Received in revised form 26 December 2021; Accepted 12 March 2022 Available online 23 March 2022 0898-1221/ 2022 Elsevier Ltd. All rights reserved.