# Showing papers in "Lobachevskii Journal of Mathematics in 2021"

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TL;DR: In this article, the existence and uniqueness of solution of an analogue of the Gellerstedt problem with nonlocal assumptions on the boundary and integral gluing conditions for the parabolic-hyperbolic type equation with nonlinear terms and Gerasimov-Caputo operator of differentiation was studied.

Abstract: This work is devoted to study the existence and uniqueness of solution of an analogue of the Gellerstedt problem with nonlocal assumptions on the boundary and integral gluing conditions for the parabolic-hyperbolic type equation with nonlinear terms and Gerasimov–Caputo operator of differentiation. Using the method of integral energy, the uniqueness of solution have been proved. Existence of solution was proved by the method of successive approximations of factorial law for Volterra type nonlinear integral equations.

34 citations

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TL;DR: In this paper, a boundary value problem with the Poincare-Tricomi condition for a degenerate partial differential equation of elliptic-hyperbolic type of the second kind was studied.

Abstract: In this paper we study a boundary value problem with the
Poincare–Tricomi condition for a degenerate partial differential
equation of elliptic-hyperbolic type of the second kind. In the
hyperbolic part of a degenerate mixed differential equation of the
second kind the line of degeneracy is a characteristic. For this
type of differential equations a class of generalized solutions is
introduced in the characteristic triangle. Using the properties of
generalized solutions, the modified Cauchy and Dirichlet problems
are studied. The solutions of these problems are found in the
convenient form for further investigations. A new method has been
developed for a differential equation of mixed type of the second
kind, based on energy integrals. Using this method, the uniqueness
of the considering problem is proved. The existence of a solution
of the considering problem reduces to investigation of a singular
integral equation and the unique solvability of this problem is
proved by the Carleman–Vekua regularization method.

27 citations

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TL;DR: In this paper, the existence and uniqueness of the inverse boundary value problem for a mixed type partial differential equation with Hilfer operator with spectral parameter in a positive rectangular domain and a negative rectangular domain was studied.

Abstract: In this paper, we consider an inverse boundary value problem for a
mixed type partial differential equation with Hilfer operator of
fractional integro-differentiation in a positive rectangular
domain and with spectral parameter in a negative rectangular
domain. The differential equation depends from another positive
parameter in mixed derivatives. With respect to first variable
this equation is a fractional-order nonhomogeneous differential
equation in the positive part of the considering segment, and with
respect to second variable is a second-order differential equation
with spectral parameter in the negative part of this segment.
Using the Fourier series method, the solutions of direct and
inverse boundary value problems are constructed in the form of a
Fourier series. Theorems on the existence and uniqueness of the
problem are proved for regular values of the spectral parameter.
It is proved the stability of the solution with respect to
redefinition functions, and with respect to parameter given in
mixed derivatives. For irregular values of the spectral parameter,
an infinite number of solutions in the form of a Fourier series
are constructed.

23 citations

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TL;DR: In this paper, the Schrodinger operator (energy operator) is constructed as a self-adjoint extension of the symmetric Laplace operator, and an essential spectrum is described and the condition for the existence of the eigenvalue of the Schodoringer operator is studied.

Abstract: We consider a one-dimensional two-particle quantum system interacted by two identical point interactions situated symmetrically with respect to the origin at the points $$\pm x_{0}$$
. The corresponding Schrodinger operator (energy operator) is constructed as a self-adjoint extension of the symmetric Laplace operator. An essential spectrum is described and the condition for the existence of the eigenvalue of the Schrodinger operator is studied. The main results of the work are based on the study of the operator extension spectrum of the operator $$h_{\mu}.$$

22 citations

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TL;DR: In this paper, two-particle Schrodinger operators with fixed quasi-momentum of particles pair were studied and a partition of the Bose-Hubbard Hamiltonian parameter plane into several connected components was established.

Abstract: We study two-particle Schrodinger operators $${H}_{\lambda\mu}(k)$$
, with the fixed quasi-momentum of particles pair $$k\in\mathbb{T}$$
, on $$L^{2,o}(\mathbb{T},\eta)$$
. These operators are associated to the Bose–Hubbard Hamiltonian $$\widehat{\mathbb{H}}_{\lambda\mu}$$
of a system of two identical quantum-mechanical particles (fermions) interacting via zero-range potential $$\mu\in\mathbb{R}$$
on one site and potential $$\lambda\in\mathbb{R}$$
on neighboring sites. We establish a partition of the $$\lambda-\mu$$
parameter-plane into several connected components where the Schrodinger operator $$H_{\lambda\mu}(k)$$
can have only a definite (constant) number of eigenvalues. The eigenvalues may locate and below the bottom of the essential spectrum and above its top.

14 citations

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12 citations

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TL;DR: The nonlinear two-point boundary value problem with parameter for systems of ordinary differential equations is investigated by the parametrization method in this article, which consists in partitioning the interval apart, the introduction of additional parameters and reducing the original problem to the equivalent multi-point problem with parameters.

Abstract: The nonlinear two-point boundary value problem with parameter for systems of ordinary differential equations is investigated by the parametrization method. The method consists in partition of the interval apart, the introduction of additional parameters and reducing the original problem to the equivalent multi-point boundary value problem with parameters. The algorithms for finding a solution to the equivalent boundary value problem are constructed. The conditions for convergence of the algorithms that ensure the solvability of nonlinear two-point boundary value problem with parameter are obtained. Definition of the isolated solution to nonlinear two-point boundary value problem with parameter, which is a modification of the well-known definition for the nonlinear two-point boundary value problem without parameter, is introduced.

12 citations

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TL;DR: It is proved that for anymu, the discrete spectrum of the family of discrete Schrodinger-type operators is a singleton and the asymptotics of e(\mu) are found as $$mu\searrow 0$$ and $$\mu
earrow 0.$$

Abstract: We consider the family $$\widehat{\mathbf{h}}_{\mu}:=\widehat{\varDelta}\widehat{\varDelta}-\mu\widehat{\delta}_{x0},$$
$$\mu\in\mathbb{R},$$
of discrete Schrodinger-type operators in one-dimensional lattice $$\mathbb{Z}$$
, where $$\widehat{\varDelta}$$
is the discrete Laplacian and $$\widehat{\delta}_{x0}$$
is the Dirac’s delta potential concentrated at $$0.$$
We prove that for any $$\mu
eq 0$$
the discrete spectrum of $$\widehat{\mathbf{h}}_{\mu}$$
is a singleton $$\{e(\mu)\},$$
and $$e(\mu)<0$$
for $$\mu>0$$
and $$e(\mu)>4$$
for $$\mu<0.$$
Moreover, we study the properties of $$e(\mu)$$
as a function of $$\mu,$$
in particular, we find the asymptotics of $$e(\mu)$$
as $$\mu\searrow 0$$
and $$\mu
earrow 0.$$

12 citations

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TL;DR: In this article, the lateral order on vector lattices is investigated, and it is shown that with every element of a complete vector lattice, a continuous orthogonally additive projection is associated with a lateral-to-order continuous projection.

Abstract: In this article we explore orthogonally additive (nonlinear) operators in vector lattices. First we investigate the lateral order on vector lattices and show that with every element $$e$$
of a $$C$$
-complete vector lattice $$E$$
is associated a lateral-to-order continuous orthogonally additive projection $$\mathfrak{p}_{e}\colon E\to\mathcal{F}_{e}$$
. Then we prove that for an order bounded positive $$AM$$
-compact orthogonally additive operator $$S\colon E\to F$$
defined on a $$C$$
-complete vector lattice $$E$$
and taking values in a Dedekind complete vector lattice $$F$$
all elements of the order interval $$[0,S]$$
are $$AM$$
-compact operators as well.

10 citations

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TL;DR: In this paper, the authors considered the unique solvability of a boundary value problem for a third-order partial integro-differential equation with a degenerate kernel and multiple characteristics.

Abstract: In this paper, we consider the questions of the unique solvability of a boundary value problem for a third-order partial integro-differential equation with a degenerate kernel and multiple characteristics. An explicit solution of the boundary value problem is constructed. In this case, a combination of three methods was used: the method for constructing Green’s function, the method of Fourier series and the Fredholm method for the degenerate kernel.

10 citations

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TL;DR: The concept of Pythagorean fuzzy set and intuitionistic fuzzy set was introduced in this article, where the authors defined the concept of $$q$$q$$¯¯ -rung orthopair fuzzy topological space and studied some properties of this concept.

Abstract: The concept of $$q$$
-rung orthopair fuzzy set is the extension of the concept of both intuitionistic fuzzy set and Pythagorean fuzzy set. The aim of this paper is to define the concept of $$q$$
-rung orthopair fuzzy topological space and to study some properties of this concept. For this purpose, firstly the basic definition of $$q$$
-rung orthopair fuzzy topology is introduced and an example is constructed and then the concept of $$q$$
-rung orthopair fuzzy continuity is studied and it is shown that a $$q$$
-rung orthopair fuzzy topology can be established on the domain or range of a function.

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TL;DR: In this paper, the impact of porous media disorder and its coupling with flow rates, favorable and unfavorable viscosity ratios, as well as surface tensions on the dynamics of interfaces development during two-phase drainage flow was investigated.

Abstract: This paper systematically investigates the impact of porous media
disorder and its coupling with flow rates, favorable and
unfavorable viscosity ratios, as well as surface tensions on the
dynamics of interfaces development during two-phase drainage flow.
A special attention is paid to establishing relationship between
the dynamics of fluid–fluid and fluid–solid interfacial lengths,
the pore selectivity and the displacement efficiency using imaging
of fluids distribution in porous media. As samples of study, we
used artificially generated models of porous media with different
disorder parameters and with two-types of pore-channels systems
"— hexagonal and square. In our methodology, the disorder
defines the range of grain size distribution and is applied to
control the pore size range. For two-phase flow simulation, the
lattice Boltzmann equations and the color-gradient model are
applied. It was established the linear relationships between
fluid–fluid and fluid–solid interfacial length and saturation of
the invaded fluid. During numerical simulations at different
disorders, the lack of disorder effect on the fluid–fluid
interface dynamics and negative disorder impact on the
fluid–solid interface dynamics was found. When varying the flow
parameters, it was identified that the increase in the
fluid–fluid interface dynamics is accompanied by a decrease in
the fluid–solid interface dynamics. For all displacement
mechanisms considered in this paper, except capillary fingering,
an inverse relationship between pore selectivity and pore number,
involved in displacement, was detected. We found a shift of pore
selectivity towards higher values with increasing disorder which
negatively impacts on the displacement efficiency. In capillary
fingering regime, a strong tendency to minimize fluid–fluid
interfacial length with surface tension explains the lack of
relationship between pore selectivity and pore number which leads
to bad predictable displacement efficiency in this regime.

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TL;DR: In this paper, an inverse problem for determining the order of time-fractional derivative in a nonhomogeneous subdiffusion equation with an arbitrary elliptic differential operator with constant coefficients in a 3-dimensional torus is considered.

Abstract: An inverse problem for determining the order of time-fractional derivative in a nonhomogeneous subdiffusion equation with an arbitrary elliptic differential operator with constant coefficients in $$N$$
-dimensional torus is considered. Using the classical Fourier method it is proved, that the value of the solution at a fixed time instant as the observation data recovers uniquely the order of fractional derivative. Generalization to an arbitrary $$N$$
-dimensional domain and to elliptic operators with variable coefficients is considered.

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TL;DR: In this article, the family of boundary value problems for a system of integro-differential equations of mixed type is considered, and conditions for the unique solvability of the considered problem are obtained in terms of the solveability to family of Cauchy problems and the hybrid system.

Abstract: The family of boundary value problems for a system of integro-differential equations of mixed type is considered. First, considered problem is reduced to the family of boundary value problems for the Fredholm integro-differential equations with an unknown function and the integral relation. Further, based on the introduction of an additional parameter as a value of the solution at the beginning line of the domain, the problem is reduced to an equivalent problem containing the family of Cauchy problems for the system of Fredholm integro-differential equations with a parameter and the unknown function, and the hybrid system of functional and integral equations for the parameter and the unknown function, respectively. Conditions for the unique solvability of the considered problem are obtained in the terms of the solvability to family of Cauchy problems and the hybrid system.

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TL;DR: In this article, the authors deal with various generalized solutions to one-dimensional pressureless gas dynamics along with the entropy solutions other solutions are considered that include the combination of concentration and decay processes.

Abstract: The paper deals with various generalized solutions to one-dimensional pressureless gas dynamics. Along with the entropy solutions other solutions are considered that include the combination of concentration and decay processes. This research is motivated by multidimensional case where the concentration mechanism seems to be not enough for the construction of rigorous theory. Thus the present paper describes in one-dimensional setting a possible behavior that could appear in multidimensional case.

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TL;DR: In this paper, a modification of the Ramsey model that describes the consumer behavior of the households is presented, and the impact of the large amount of households can be modelled by a mean field term, which leads to a Kolmogorov-Fokker-Planck equation, evolving forward in time to describe the evolution of the probability density function.

Abstract: We present a modification of the Ramsey model that describes the consumer behavior of the households. We assume that the salary of the households is a stochastic process, defined by the stochastic differential equation (SDE). The impact of the large amount of the households can be modelled by a mean field term. This leads to a Kolmogorov–Fokker–Planck equation, evolving forward in time that describes the evolution of the probability density function of the households. Considering a Hamilton–Jacobi–Bellman equation, evolving backwards in time that describes the optimal strategy of the households behavior, we obtain a Mean Field Game problem. We present a self-similar solution of the Hamilton–Jacobi–Bellman equation and introduce the numerical solution of the Kolmogorov–Fokker–Planck equation.

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TL;DR: In this article, an explicit solution of the -dimensional anomalous diffusion equation in the infinite domain with non-zero initial condition and vanishing condition at infinity was derived from the parabolic integro-differential equation with memory in which the kernel is the Mittag-Liffler function.

Abstract: This paper intends on obtaining the explicit solution of $$n$$
-dimensional anomalous diffusion equation in the infinite domain with non-zero initial condition and vanishing condition at infinity. It is shown that this equation can be derived from the parabolic integro-differential equation with memory in which the kernel is $$t^{-\alpha}E_{1-\alpha,1-\alpha}(-t^{1-\alpha})$$
, $$\alpha\in(0,1),$$
where $$E_{\alpha,\beta}$$
is the Mittag-Liffler function. Based on Laplace and Fourier transforms the properties of the Fox H-function and convolution theorem, explicit solution for anomalous diffusion equation is obtained.

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TL;DR: In this paper, the approaches and methods to create a semantic library within the mathematics subject domain are described and a step-by-step description of the general ontology within the subject domain is given.

Abstract: The paper describes the approaches and methods to create a semantic library within the ‘‘Mathematics’’ subject domain. The theoretical background of the research involves the approach based on ontologies used when creating semantic libraries. The paper suggests a step-by-step description of the general ontology within the subject domain. The results obtained are well-grounded and contribute to distinct data integration.

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TL;DR: A tensor-train decomposition of a tensor approximating the integrand is constructed and used to evaluate a multivariate quadrature formula in this article, where the authors show how to deal with singularities and conduct theoretical analysis of the integration accuracy.

Abstract: Numerical integration is a classical problem emerging in many fields of science. Multivariate integration cannot be approached with classical methods due to the exponential growth of the number of quadrature nodes. We propose a method to overcome this problem. Tensor-train decomposition of a tensor approximating the integrand is constructed and used to evaluate a multivariate quadrature formula. We show how to deal with singularities in the integration domain and conduct theoretical analysis of the integration accuracy. The reference open-source implementation is provided.

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TL;DR: In this article, the authors combine the grid-characteristic method on structured computational grids with the discontinuous Galerkin method, to describe the contact boundary between the subdomains of the integration domain.

Abstract: An advantage of the discontinuous Galerkin method is the ability to describe complex contact surfaces by using unstructured computational grids. However, the use of the discontinuous Galerkin method requires significant computational resources, including at the preprocessing stage. The grid-characteristic method on structured regular computational grids saves computational resources, but problems arise when taking into account complex inhomogeneities, including surface topography. Therefore, it is of interest to combine the grid-characteristic method on structured computational grids with the discontinuous Galerkin method, to which this paper is devoted. The work describes in detail the Galerkin method and the description of the contact boundary between the subdomains of the integration domain in which the grid-characteristic method and the discontinuous Galerkin method are used. The corresponding calculation algorithm is discussed. Examples are given on the calculation of the propagation of seismic waves from the hypocenter of an earthquake to the Earth’s surface, taking into account the surface topography.

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TL;DR: In this article, a spatial non-steady problem of the influence of a moving heat flow source (laser heating) on the surface of half-space using superposition principle and transient functions method is presented, and a numerical analytical algorithm based on discretization in time and space coordinates is developed.

Abstract: The solution of a spatial non-steady problem of the influence of a moving heat flow source (laser heating) on the surface of half-space using superposition principle and transient functions method is presented, and a numerical analytical algorithm based on discretization in time and space coordinates is developed and realized.

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TL;DR: In this paper, the optimal control problem with concentrated parameters for a degenerate differential equation with the Caputo operator and with coefficients from the Lebesgue space is studied, and a new version of the method of increments is applied and the concept of a conjugate equation with an integral form is essentially used.

Abstract: In this paper, on the base of Pontryagin maximum principle, the optimal control problem with concentrated parameters for a degenerate differential equation with the Caputo operator and with coefficients from the Lebesgue space is studied. The efficiency indicator of the considered optimal control problem has an integral form of fractional order. A new version of the method of increments is applied and the concept of a conjugate equation with an integral form is essentially used.

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TL;DR: In this paper, the existence and uniqueness of solution of an analogue of the Gellerstedt type problem with integral gluing condition for a loaded mixed parabolic-hyperbolic type equation have been investigated.

Abstract: In this work, the existence and the uniqueness of solution of an analogue of the Gellerstedt type problem with integral gluing condition for a loaded mixed parabolic-hyperbolic type equation have been investigated. Considering mixed type equation has a nonlinear loaded parts. Using the method of integral energy the uniqueness of solution has been proved. The existence of solution is proved, using the well-known successive approximations method.

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TL;DR: In this article, an initial-boundary value problem for a time-fractional subdiffusion equation with an arbitrary order elliptic differential operator is considered, and the existence of the classical solution of the posed problem is proved by the classical Fourier method.

Abstract: An initial-boundary value problem for a time-fractional subdiffusion equation with an arbitrary order elliptic differential operator is considered. Uniqueness and existence of the classical solution of the posed problem are proved by the classical Fourier method. Sufficient conditions for the initial function and for the right-hand side of the equation are indicated, under which the corresponding Fourier series converge absolutely and uniformly. In the case of an initial-boundary value problem on $$N$$
-dimensional torus, one can easily see that these conditions are not only sufficient, but also necessary.

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TL;DR: In this paper, the authors studied zero sets of entire functions in the Schwartz algebra, defined as the Fourier-Laplace transform image of the space of all distributions compactly supported on the real line.

Abstract: In this paper we study zero sets of entire functions in the Schwartz algebra. This algebra is defined as the Fourier–Laplace transform image of the space of all distributions compactly supported on the real line. We obtain the conditions under which given complex sequence forms zero set of some invertible in the sense of Ehrenpreis element of the Schwartz algebra (slowly decreasing function).

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TL;DR: In this paper, the existence of optimal control of linear and nonlinear stochastic systems with quadratic criterion qualities is proved by using dynamic programming, where the control is included with a small parameter.

Abstract: Problems of optimal control of linear and nonlinear stochastic systems with quadratic criterion qualities are studied. For such problems the existence of optimal control in feedback control form is proved by method of dynamic programming. The work consists of two parts. The first part deals with the linear problem. In the first part, the existence of a solution to the Cauchy problem for the generalized Riccati equation is proved by a method based on the idea of the Bellman linearization scheme. The proof consists in the direct application of existence theorems for ordinary differential equations to the generalized Riccati equation. The main part of the article is its second part, which concerns the study of a nonlinear problem. Meaningful result is obtained only when the control is included with a small parameter in the stochastic part. Existence for small $$\varepsilon$$
of the solution of the Bellman equation corresponding to the nonlinear problem are proved using the abstract implicit function theorem.

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TL;DR: In this paper, the authors considered the problem of solarity of sets in dual spaces and gave an answer to the unique farthest point problem for sets with constant acyclic values.

Abstract: Two max- and min-approximation problems on solarity of sets in dual spaces are considered. It is shown that if the metric projection onto a set $$M\subset X^{*}$$
is $$w^{*}$$
-upper semicontinuous and has nonempty $$w^{*}$$
-closed acyclic values, then $$M$$
is a sun. In particular, a Chebyshev set with $$w^{*}$$
-continuous metric projection is a sun. In the max-approximation setting, a set with $$w^{*}$$
-upper-semicontinuous $$\max$$
-projection with nonempty $$w^{*}$$
-closed acyclic values is shown to be local $$\max$$
-sun. As a result, it follows that that a uniquely remotal set with $$w^{*}$$
-continuous $$\max$$
-projection operator is a singleton, which gives an answer to the well-known unique farthest point problem in dual spaces for sets with $$w^{*}$$
-continuous farthest-point mapping.

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TL;DR: In this paper, a linear boundary value problem for essentially loaded differential equations with additional parameters at the loading points is investigated, where the method of parameterization is used for solving the problem.

Abstract: A linear boundary value problem for essentially loaded differential equations is investigated. Using the properties of essentially loaded differential equation and assuming the invertibility of the matrix compiled through the coefficients at the values of the derivative of the desired function at load points, we reduce the considering problem to a two-point boundary value problem for loaded differential equations. The method of parameterization is used for solving the problem. The linear boundary value problem for loaded differential equations by introducing additional parameters at the loading points is reduced to equivalent boundary value problem with parameters. The equivalent boundary value problem with parameters consists of the Cauchy problem for the system of ordinary differential equations with parameters, boundary condition and continuity conditions. The solution of the Cauchy problem for the system of ordinary differential equations with parameters is constructed using the fundamental matrix of differential equation. The system of linear algebraic equations with respect to the parameters are composed by substituting the values of the corresponding points in the built solutions to the boundary condition and the continuity condition. Numerical method for finding solution of the problem is suggested, which based on the solving the constructed system and the Bulirsch–Stoer method for solving Cauchy problem on the subintervals.

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TL;DR: The ExtraNoise project as mentioned in this paper investigates the influence of system noise on HPC application performance and finds that although system noise heavily influences execution time and energy consumption, it does not change the computational effort a program performs.

Abstract: Many contemporary HPC systems expose their jobs to substantial amounts of interference, leading to significant run-to-run variation. For example, application runtimes on Theta, a Cray XC40 system at Argonne National Laboratory, vary by up to 70
$$\%$$
, caused by a mix of node-level and system-level effects, including network and file-system congestion in the presence of concurrently running jobs. This makes performance measurements generally irreproducible, heavily complicating performance analysis and modeling. On noisy systems, performance analysts usually have to repeat performance measurements several times and then apply statistics to capture trends. First, this is expensive and, second, extracting trends from a limited series of experiments is far from trivial, as the noise can follow quite irregular patterns. Attempts to learn from performance data how a program would perform under different execution configurations experience serious perturbation, resulting in models that reflect noise rather than intrinsic application behavior. On the other hand, although noise heavily influences execution time and energy consumption, it does not change the computational effort a program performs. Effort metrics that count how many operations a machine executes on behalf of a program, such as floating-point operations, the exchange of MPI messages, or file reads and writes, remain largely unaffected and—rare non-determinism set aside—reproducible. This paper addresses initial stage of an ExtraNoise project, which is aimed at revealing and tackling key questions of system noise influence on HPC applications.