# Ivan DimovBulgarian Academy of Sciences | BAS · Department of Parallel Algorithms

Ivan Dimov

DSc, PhD

## About

309

Publications

26,208

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Introduction

Currently I am working on High Performance Computing (HPC) on the following field of research: Monte Carlo algorithms, a priori estimates and error analysis; Statistical numerical methods with supper convergent probability error; Parallel algorithms; Mathematical modelling and scientific computation in: - Environmental Mathematics, Gas discharge plasma, Semi-conductor physics, Computational Nano-physics - Wigner Monte Carlo (Signed Particle Formalism) and Financial Mathematics

Additional affiliations

May 2009 - present

**Bulgarian Academy of Sciences, Institute of Information and Communication Technologies**

Position

- Professor, Chair of Scientific Council

March 2005 - July 2009

January 2001 - March 2005

## Publications

Publications (309)

The present study addresses the sensitivity analysis of particle concentration dispersion in the turbulent flow. A stochastic spectral model of turbulence is used to simulate the particle transfer. Sensitivity analysis is performed by estimations of Morris and Sobol indices. This study allows to define the significant and nonsignificant model param...

Nowadays, much of the world has a regional air pollution strategy to limit and decrease the pollution levels across governmental borders and control their impact on human health and ecological systems. Environmental protection is among the leading priorities worldwide. Many challenges in this research area exist since it is a painful subject for so...

High concentration levels of air pollutants may cause damage to plants, animals, and the health of some groups of human beings. Therefore, it is important to investigate different topics related to the high air pollution levels and to find reliable answers to the questions about the possible damages, which might take place when these levels exceed...

Atmospheric chemistry schemes, which are described mathematically by non-linear systems of ordinary differential equations (ODEs), are used in many large-scale air pollution models. These systems of ODEs are badly-scaled, extremely stiff and some components of their solution vectors vary quickly forming very sharp gradients. Therefore, it is necess...

In this paper, various modifications of the Latin Hypercube Sampling algorithm have been used in order to evaluate the sensitivity of an environmental model output results for some dangerous air pollutants with respect to the emission levels and some chemical reaction rates. The environmental security importance is growing rapidly, becoming at pres...

In this work we study advanced stochastic methods for solving a specific multidimensional problems related to computation of European style options in computational finance. Recently, stochastic methods have become a very important tool for high-performance computing of very high-dimensional problems in computational finance. Here, a different kind...

An important issue when large-scale mathematical models are used to support decision makers is their reliability. Sensitivity analysis has a crucial role during the process of validating computational models to ensure their accuracy and reliability. The focus of the present work is to perform global sensitivity analysis of a large-scale mathematica...

In this paper, we present and study highly efficient stochastic methods, including optimal super convergent methods for multidimensional sensitivity analysis of large-scale ecological models and digital twins. The computational efficiency (in terms of relative error and computational time) of the stochastic algorithms for multidimensional numerical...

In previous works (Dimov and Maire in Adv Comput Math 45(3):1499–1519, 2019; Dimov et al. in Appl Math Model 39(15):4494–4510, https://doi.org/10.1016/j.apm.2014.12.018, 2015), we have developed two Monte Carlo algorithms to solve linear systems and Fredholm integral equations of the second kind. These algorithms rely on the computation of a score...

The numerical treatment of an atmospheric chemical scheme, which contains 56 species, is discussed in this paper. This scheme is often used in studies of air pollution levels in different domains, as, for example, in Europe, by large-scale environmental models containing additionally two other important physical processes—transport of pollutants in...

In this work a systematic procedure for multidimensional sensitivity analysis in the area of air pollution modeling by an optimized latin hypercube sampling has been done. The Unified Danish Eulerian Model (UNI-DEM) is used in our investigation, because this is one of the most advanced large-scale mathematical models that describes adequately all p...

A very important problem in the area of neural networks and machine learning is the accurate evaluation of multidimensional integrals. An introduction to the theory of the stochastic approaches with a special choice of optimal generating vectors has been given. A new optimized lattice sequence with a special choice of the optimal generating vector...

This paper is a continuation of the discussion on optimization of the quadrature formulas and their applications in paper [6]. Second-order numerical solutions of Volterra integral equations are constructed using the quadrature formulas obtained in [6]. The numerical results presented in the paper confirm the effectiveness of the methods for numeri...

We compute the Wigner kernel by means of stochastic approaches. In this paper we study an optimized Adaptive Monte Carlo algorithm for evaluation of the Wigner kernel. This is an important problem in quantum mechanics represented by difficult multidimensional integrals. The goal of our work is to present an improved adaptive Monte Carlo algorithm a...

In this work we analyze and compare the performance of an optimal stochastic approach for multidimensional integrals of smooth functions. The purpose of the present study is to compare the optimal Monte Carlo algorithm under consideration with the lattice rules based on the generalized Fibonacci numbers of the corresponding dimension and to discuss...

An optimization version of the van der Corput sequence has been used in our sensitivity studies of the model output results for some air pollutants with respect to the emission levels and some chemical reactions rates. Sensitivity analysis of model outputs to variation or natural uncertainties of model inputs is very significant for improving the r...

In this paper we develop stochastic simulation methods for solving large systems of linear equations, and focus on two issues: (1) construction of global random walk algorithms (GRW), in particular, for solving systems of elliptic equations on a grid, and (2) development of local stochastic algorithms based on transforms to balanced transition matr...

Consider the concentration convection-diffusion process of air-pollution with time-dependent vertical diffusion coefficient modeled by a degenerate parabolic equation. We aim to identify the diffusion coefficient from extra point or integral measurements. For the appropriate solution of the nonlinear inverse problem of determination unknown diffusi...

Sensitivity analysis is a modern promising technique for studying large systems such as ecological systems. The main idea of sensitivity analysis is to evaluate and predict (through computer simulations on large mathematical models) the measure of the sensitivity of the model’s output to the perturbations of some input parameters, and it is a techn...

The carrier transport in nowadays electronic structures is characterized by the nanometer scale and evolution times in the order of a few picoseconds. Under such conditions decoherence effects due to the lattice coexist with quantum coherent dynamics. The hierarchy of models presented in this chapter accounts on different levels of approximation fo...

Homogeneous transport problems feature bulk materials and thus involve only a half of the phase space, the subspace of the carrier momenta. Nevertheless homogeneous simulations provide important semiconductor characteristics such as the dependence of the physical quantities on the time, applied field, carrier energy, temperature, and crystal orient...

The most general computational task of modeling classical transport in semiconductor devices involves the Boltzmann equation accomplished by mixed initial and boundary conditions. The basic step towards the development of for mixed conditions suitable self-consistent algorithms involving statistical weights (event biasing) is the application of the...

The generalized Wigner function of the coupled electron-phonon system depends on both, single electron and phonon degrees of freedom. The averaging over the phonon system following the analysis presented in Chap. 12 gives rise to the reduced Wigner function which depends only on the electron coordinates. A set of three coupled equations is obtained...

The development of a more formal approach begins with the question about the mathematical correspondence between the empirical algorithms and the Boltzmann equation. It has been shown that the Single-Particle or the Ensemble stochastic processes recover the solution of the corresponding stationary or transient Boltzmann equation. The alternative st...

The historical development of microelectronics is characterized by the growing importance of modeling, which gradually embraced all stages of preparation and operation of an integral circuit (IC). Device modeling focuses on the processes determining the electrical properties and behavior of the individual IC elements. The trends towards their minia...

Typical physical scales characterizing quantum electron decoherence are embodied in the femtosecond evolution of optically generated carriers interacting with phonons. A variety of quantum effects can be observed on these scales, such as non- Markovian evolution, giving rise to the Retardation effect, Collision Broadening which is a result of the l...

A algorithms for general stationary inhomogeneous conditions have been considered from the point of view of the Iteration Approach fairly lately, at the beginning of the century. This is related to the higher level of abstraction of the stationary concepts and notions and, in particular, the different ways for evaluation of the mean values which re...

This chapter introduces the Boltzmann and the Wigner transport equations, which will be approached with the numerical theory of the Monte Carlo method in order to devise algorithms for solving them. For this purpose we focus on the heuristic aspects of the physical assumptions about the transport system, which give rise to clearly formulated mathem...

We introduce the basic elements of the semiconductor model, which is in the root of the transport models and provides input parameters for any simulation. These elements are the band structure, the mechanisms for interaction with the lattice, and a set of material parameters characterizing the semiconductor. Details about these concepts can be foun...

The analysis of the response of a carrier system to small changes in an applied electric field is an important task, which provides macroscopic parameters needed for the compact models for circuit simulations. The differential response function depends on the frequency ω of the perturbation and the strength of the applied constant electric field. T...

We summarize the basic elements of the Monte Carlo method for solving integral equations. The method extensively employs concepts and notions of probability theory and statistics. They are marked in italic and defined in the appendix in the case that the reader wants to refresh his insight on the subject during reading this section. Where possible,...

In Chap. 14 we consider the application of the Iteration Approach to the stationary Wigner-Boltzmann equation. Stationary transport problems are characterized by the time independence of the external quantities such as fields and boundary conditions, which ensures the time independence of the corresponding physical averages. The task is first refor...

The transient problem is characterized by an in time developing Wigner state, determined by an initial condition - the Wigner function at a given initial time. Transient quantum algorithms are based on a further development of the concepts of signed particles, introduced for the stationary counterpart Chap. 14. A transition from a single to an ense...

The only requirement for the transition probability is to be admissible to the kernel, that is, to be nonzero where the kernel is not zero. In the case of the Boltzmann equation, however, an arbitrary choice is not efficient, because the kernel is degenerate and contains several delta functions which turn the contribution of a generally constructed...

An important issue when large-scale mathematical models are used to support decision makers is their reliability. Sensitivity analysis of model outputs to variation or natural uncertainties of model inputs is very significant for improving the reliability of these models. A comprehensive experimental study of Monte Carlo algorithm based on adaptive...

Sensitivity analysis is a powerful tool for studying and improving the reliability of mathematical models. Air pollution and meteorological models are in front places among the examples of mathematical models with a lot of natural uncertainties in their input data sets and parameters. In this work some results of the global sensitivity study of the...

In the present paper we construct second order shifted approximations for the first derivative which have exponential and logarithmic generating functions. Applications of the approximations for numerical solution of first order ordinary differential equations and the heat equation are studied in the paper.

Every day we need to solve large problems for which supercomputers are needed. High performance computing (HPC) is a paradigm that allows to efficiently implement large-scale computational tasks on powerful supercomputers unthinkable without optimization. We try to minimize our effort and to maximize the achieved profit.
Many challenging real world...

The book serves as a synergistic link between the development of mathematical models and the emergence of stochastic (Monte Carlo) methods applied for the simulation of current transport in electronic devices. Regarding the models, the historical evolution path, beginning from the classical charge carrier transport models for microelectronics to cu...

Sensitivity studies are nowadays applied to some of the most complicated mathematical models from various intensively developing areas of application. Such a sophisticated model in the area of air pollution modeling is the Danish Eulerian Model, a powerful large scale air pollution model with a long development history. Over the years it was used s...

Sensitivity analysis of model outputs to variation or natural uncertainties of model inputs is very significant for improving the reliability of these models. Several efficient quasi-Monte Carlo algorithms – the van der Corput sequence and lattice rules based on different generating vectors have been used in our sensitivity studies of the model out...

In this paper, we study numerically various approaches, namely an adaptive Monte Carlo algorithm, a particular rank-1 lattice algorithm based on generalized Fibonacci numbers and a Monte Carlo algorithm based on Latin hypercube sampling for computing multidimensional integrals. We compare the performance of the algorithms over three case studies—mu...

We present two approaches for enhancing the accuracy of second order finite difference approximations of two-dimensional semilinear parabolic systems. These are the fourth order compact difference scheme and the fourth order scheme based on Richardson extrapolation. Our interest is concentrated on a system of ten parabolic partial differential equa...

A fundamental problem in Bayesian statistics is the accurate evaluation of multidimensional integrals. A comprehensive experimental study of quasi-Monte Carlo algorithms based on Sobol sequence combined with Matousek linear scrambling and a comparison with adaptive Monte Carlo approach and a lattice rule based on generalized Fibonacci numbers has b...

The large-scale air pollution model UNI-DEM (the Unified Danish Eulerian Model) was used together with several carefully selected climatic scenarios. It was necessary to run the model over a long time-interval (sixteen consecutive years) and to use fine resolution on a very large space domain. This caused great difficulties because it was necessary...

Sensitivity analysis of the results of large and complicated mathematical models is rather tuff and time-consuming task. However, this is quite an important problem as far as their critical applications are concerned. There are many such applications in the area of air pollution modelling. On the other hand, there are lots of natural uncertainties...

In this work we formulate the Wigner equation as an operator equation in a suitable L2 space. This allows us to rigorously express its solution and functionals thereof in terms of von Neumann series and in an already established fashion to represent each term of this series as the contribution of a single signed particle. Then by applying classical...

A new Monte Carlo algorithm for solving systems of Linear Algebraic (LA) equations is presented and studied. The algorithm is based on the “Walk on Equations” Monte Carlo method recently developed by Dimov et al. (Appl Math Model 39:4494–4510, [12]). The algorithm is improved by choosing the appropriate values for the relaxation parameters which le...

The treatment of large-scale air pollution models is not only important for modern society, but also an extremely difficult task. Five important physical and chemical processes: (1) horizontal advection, (2) horizontal diffusion, (3) vertical exchange, (4) emission of different pollutants and (5) dry and wet deposition have to be united and handled...

Air pollution and meteorological models are examples of mathematical models with a lot of natural uncertainties in their input data sets and parameters. Sensitivity analysis is a powerful tool for studying and improving the reliability of such models. In this paper we present the results of a global sensitivity study of the Unified Danish Eulerian...

We present two approaches for enhancing the accuracy of the second-order finite difference approximations of two-dimensional semilinear parabolic systems. These are the fourth-order compact difference scheme and the fourth-order scheme based on Richardson extrapolation. Our interest is concentrated on a system of ten parabolic partial differential...

Richardson Extrapolation is a very general numerical procedure, which can be applied in the solution of many mathematical problems in an attempt to increase the accuracy of the results. It is assumed that this approach is used to handle non-linear systems of ordinary differential equations (ODEs) which arise often in the mathematical description of...

In this paper, we propose and analyse a new unbiased stochastic approach for solving a class of integral equations. We study and compare the proposed unbiased approach against the known biased Monte Carlo method based on evaluation of truncated Liouville-Neumann series. We also compare the proposed algorithm against the deterministic Nystrom method...

Gauge-invariant Wigner theories are formulated in terms of the kinetic momentum, which—being a physical quantity—is conserved after a change of the gauge. These theories rely on a transform of the density matrix, originally introduced by Stratonovich, which generalizes the Weyl transform by involving the vector potential. We thus present an alterna...

The solutions of fractional differential equations (FDEs) have a natural singularity at the initial point. The accuracy of their numerical solutions is lower than the accuracy of the numerical solutions of FDEs whose solutions are differentiable functions. In the present paper we propose a method for improving the accuracy of the numerical solution...

Repeated Richardson Extrapolation can successfully be used in the efforts to improve the efficiency of the numerical treatment of systems of ordinary differential equations (ODEs) mainly by increasing the accuracy of the computed results. It is assumed in this paper that Implicit Runge-Kutta Methods (IRKMs) are used in the numerical solution of sys...

Efficient implementation of the Two-times Repeated Richardson Extrapolation is studied in this paper under the assumption that systems of ordinary differential equations (ODEs) are solved numerically by Explicit Runge-Kutta Methods (ERKMs). The combinations of the Two-times Repeated Richardson Extrapolation with the ERKMs are new numerical methods....