Bálint Farkas
Current Organizations / Institutes
The organizations/institutes from the scholar's latest article
Previous Organizations / Institutes
The organizations / institutes from the scholar's other articles in the past ten years
Bálint Farkas has a total of 52 co-authors. Here are the top co-authors by number of shared publications:
- Szilárd György Révész
Hungarian Academy of Sciences
11 shared publications - András Bátkai
Institut für Sekundarbildung und Fachdidaktik, Pädagogische Hochschule Vorarlberg, Feldkirch, Österreich
9 shared publications - Béla Nagy
Univ Szeged, Bolyai Inst, Dept Anal, Aradi Vertanuk Tere 1, H-6720 Szeged, Hungary
8 shared publications - Petra Csomós
Eötvös Loránd University
8 shared publications - Birgit Jacob
School of Mathematics and Natural Sciences University of Wuppertal Gaußstraße 20 42119 Wuppertal Germany
3 shared publications - Merlin Schmitz
IMACM, Bergische Universität Wuppertal, Wuppertal, Germany
3 shared publications
Latest Publications
- Open Access
- 18 November 2025
- Published by Springer Nature in Complex Analysis and Operator Theory
- Vol. 20 (1) , 5
Operator Splitting Based Dynamic Iteration for Linear Infinite-Dimensional Port-Hamiltonian Systems
A dynamic iteration scheme for linear infinite-dimensional port-Hamiltonian systems is proposed. The error of the dynamic iteration is convergent to 0 and subject to a effective decreasing bound. No stability condition is required and the method is in particular applicable to port-Hamiltonian formulations arising from domain decompositions.
- 18 November 2025
- Published by Springer Nature in Complex Analysis and Operator Theory
- Vol. 20 (1) , 5
Open AccessOperator Splitting Based Dynamic Iteration for Linear Infinite-Dimensional Port-Hamiltonian Systems
A dynamic iteration scheme for linear infinite-dimensional port-Hamiltonian systems is proposed. The error of the dynamic iteration is convergent to 0 and subject to a effective decreasing bound. No stability condition is required and the method is in particular applicable to port-Hamiltonian formulations arising from domain decompositions.
- Open Access
Dissipativity-based time domain decomposition for optimal control of hyperbolic PDEs
We propose a time domain decomposition approach to optimal control of partial differential equations (PDEs) based on semigroup theoretic methods. We formulate the optimality system consisting of two coupled forward-backward PDEs, the state and adjoint equation, as a sum of dissipative operators, which enables a Peaceman-Rachford-type fixed-point iteration. The iteration steps may be understood and implemented as solutions of many decoupled, and therefore highly parallelizable, time-distributed optimal control problems. We prove the convergence of the state, the control, and the corresponding adjoint state in function space. Due to the general framework of $C_0$-(semi)groups, the results are particularly well applicable, e.g., to hyperbolic equations, such as beam or wave equations. We illustrate the convergence and efficiency of the proposed method by means of two numerical examples subject to a 2D wave equation and a 3D heat equation.
Open AccessDissipativity-based time domain decomposition for optimal control of hyperbolic PDEs
We propose a time domain decomposition approach to optimal control of partial differential equations (PDEs) based on semigroup theoretic methods. We formulate the optimality system consisting of two coupled forward-backward PDEs, the state and adjoint equation, as a sum of dissipative operators, which enables a Peaceman-Rachford-type fixed-point iteration. The iteration steps may be understood and implemented as solutions of many decoupled, and therefore highly parallelizable, time-distributed optimal control problems. We prove the convergence of the state, the control, and the corresponding adjoint state in function space. Due to the general framework of $C_0$-(semi)groups, the results are particularly well applicable, e.g., to hyperbolic equations, such as beam or wave equations. We illustrate the convergence and efficiency of the proposed method by means of two numerical examples subject to a 2D wave equation and a 3D heat equation.
- 9 December 2024
- Published by Springer Nature in Analysis Mathematica
The semicentennial anniversary of Analysis Mathematica
- 9 December 2024
- Published by Springer Nature in Analysis Mathematica
The semicentennial anniversary of Analysis Mathematica
- Open Access
- 4 March 2023
- Published by Springer Nature in Revista Matemática Complutense
- Vol. 37 (2) , 341-389
A homeomorphism theorem for sums of translates
For a fixed positive integer n consider continuous functions $$K_1,\dots $$ , $$ K_n:[-1,1]\rightarrow {{\mathbb {R}}}\cup \{-\infty \}$$ that are concave and real valued on $$[-1,0)$$ and on (0, 1], and satisfy $$K_j(0)=-\infty $$ . Moreover, let $$J:[0,1]\rightarrow {{\mathbb {R}}}\cup \{-\infty \}$$ be upper bounded and such that $$[0,1]\setminus J^{-1}(\{-\infty \})$$ has at least $$n+1$$ elements, but it is arbitrary otherwise. For $$x_0:=0<x_1<\dots < x_n \le x_{n+1}:=1$$ , so called nodes, and for $$t\in [0,1]$$ consider the sum of translates function $$F(x_1,\ldots ,x_n,t):=J(t)+\sum _{j=1}^n K_j(t-x_j)$$ , and the vector of interval maximum values $$m_j:=m_j(x_1,\ldots ,x_n):=\max _{t\in [x_j,x_{j+1}]}F(x_1,\ldots ,x_n,t)$$ ( $$j=0,1,\ldots ,n$$ ). We describe the structure of the arising interval maxima as the nodes run over the n-dimensional simplex. Applications presented here range from abstract moving node Hermite–Fejér interpolation for generalized algebraic and trigonometric polynomials via Bojanov’s problem to more abstract results of interpolation theoretic flavour.- 4 March 2023
- Published by Springer Nature in Revista Matemática Complutense
- Vol. 37 (2) , 341-389
Open AccessA homeomorphism theorem for sums of translates
For a fixed positive integer n consider continuous functions $$K_1,\dots $$ , $$ K_n:[-1,1]\rightarrow {{\mathbb {R}}}\cup \{-\infty \}$$ that are concave and real valued on $$[-1,0)$$ and on (0, 1], and satisfy $$K_j(0)=-\infty $$ . Moreover, let $$J:[0,1]\rightarrow {{\mathbb {R}}}\cup \{-\infty \}$$ be upper bounded and such that $$[0,1]\setminus J^{-1}(\{-\infty \})$$ has at least $$n+1$$ elements, but it is arbitrary otherwise. For $$x_0:=0<x_1<\dots < x_n \le x_{n+1}:=1$$ , so called nodes, and for $$t\in [0,1]$$ consider the sum of translates function $$F(x_1,\ldots ,x_n,t):=J(t)+\sum _{j=1}^n K_j(t-x_j)$$ , and the vector of interval maximum values $$m_j:=m_j(x_1,\ldots ,x_n):=\max _{t\in [x_j,x_{j+1}]}F(x_1,\ldots ,x_n,t)$$ ( $$j=0,1,\ldots ,n$$ ). We describe the structure of the arising interval maxima as the nodes run over the n-dimensional simplex. Applications presented here range from abstract moving node Hermite–Fejér interpolation for generalized algebraic and trigonometric polynomials via Bojanov’s problem to more abstract results of interpolation theoretic flavour. - Open Access
On the weighted Bojanov-Chebyshev Problem and the sum of translates method of Fenton
Minimax and maximin problems are investigated for a special class of functions on the interval $[0,1]$. These functions are sums of translates of positive multiples of one kernel function and a very general external field function. Due to our very general setting the obtained minimax, equioscillation, and characterization results extend those of Bojanov, Fenton, Hardin, Kendall, Saff and Ambrus, Ball, Erd\'elyi. Moreover, we discover a surprising intertwining phenomenon of interval maxima, which provides new information even in the most classical extremal problem of Chebyshev.Open AccessOn the weighted Bojanov-Chebyshev Problem and the sum of translates method of Fenton
Minimax and maximin problems are investigated for a special class of functions on the interval $[0,1]$. These functions are sums of translates of positive multiples of one kernel function and a very general external field function. Due to our very general setting the obtained minimax, equioscillation, and characterization results extend those of Bojanov, Fenton, Hardin, Kendall, Saff and Ambrus, Ball, Erd\'elyi. Moreover, we discover a surprising intertwining phenomenon of interval maxima, which provides new information even in the most classical extremal problem of Chebyshev. - Open Access
Wiener's lemma along primes and other subsequences
Inspired by subsequential ergodic theorems, we study the validity of Wiener's lemma and the extremal behavior of a measure $\mu$ on the unit circle via the behavior of its Fourier coefficients $\hat\mu(k_n)$ along subsequences $(k_n)$. We focus on arithmetic subsequences such as polynomials, primes and polynomials of primes, and also discuss connections to rigidity sequences, return times sequences and strongly sweeping out sequences as well as measures on $\mathbb{R}$. We also present consequences for orbits of operators and of $C_0$-semigroups on Hilbert and Banach spaces extending the results of Goldstein and Goldstein, Nagy. The results are complemented by some open questions and indication of interesting research directions. After this paper had been published, it was pointed out to us by Emmanuel Lesigne and M\'at\'e Wierdl that there is a gap in the Example on return times sequences along polynomials on page 13. Indeed, to make the argument there work, one needs a Wiener-Wintner type result for polynomial averages with a precise information about the limit, and this is presently out of reach.Open AccessWiener's lemma along primes and other subsequences
Inspired by subsequential ergodic theorems, we study the validity of Wiener's lemma and the extremal behavior of a measure $\mu$ on the unit circle via the behavior of its Fourier coefficients $\hat\mu(k_n)$ along subsequences $(k_n)$. We focus on arithmetic subsequences such as polynomials, primes and polynomials of primes, and also discuss connections to rigidity sequences, return times sequences and strongly sweeping out sequences as well as measures on $\mathbb{R}$. We also present consequences for orbits of operators and of $C_0$-semigroups on Hilbert and Banach spaces extending the results of Goldstein and Goldstein, Nagy. The results are complemented by some open questions and indication of interesting research directions. After this paper had been published, it was pointed out to us by Emmanuel Lesigne and M\'at\'e Wierdl that there is a gap in the Example on return times sequences along polynomials on page 13. Indeed, to make the argument there work, one needs a Wiener-Wintner type result for polynomial averages with a precise information about the limit, and this is presently out of reach. - Open Access
Operator splitting based dynamic iteration for linear infinite-dimensional port-Hamiltonian systems
A dynamic iteration scheme for linear infinite-dimensional port-Hamiltonian systems is proposed. The dynamic iteration is monotone in the sense that the error is decreasing, it does not require any stability condition and is in particular applicable to port-Hamiltonian formulations arising from domain decompositions.
Open AccessOperator splitting based dynamic iteration for linear infinite-dimensional port-Hamiltonian systems
A dynamic iteration scheme for linear infinite-dimensional port-Hamiltonian systems is proposed. The dynamic iteration is monotone in the sense that the error is decreasing, it does not require any stability condition and is in particular applicable to port-Hamiltonian formulations arising from domain decompositions.
- 1 December 2022
- Published by Krasovskii Institute of Mathematics and Mechanics UB RAS in Trudy Instituta Matematiki i Mekhaniki UrO RAN
- Vol. 28 (4) , 262-272
Intertwining of maxima of sum of translates functions with nonsingular kernels
MSC: 26A51, 41A50 DOI: 10.21538/0134-4889-2022-28-4-262-272 In previous papers we investigated so-called sum of translates functions $F(\mathbf x,t):=J(t)+\sum_{j=1}^n \nu_j K(t-x_j)$, where $J:[0,1]\to \underline{\mathbb R}:=\mathbb R\cup\{-\infty\}$ is a "sufficiently nondegenerate" and upper-bounded "field function", and $K:[-1,1]\to \underline{\mathbb R}$ is a fixed- 1 December 2022
- Published by Krasovskii Institute of Mathematics and Mechanics UB RAS in Trudy Instituta Matematiki i Mekhaniki UrO RAN
- Vol. 28 (4) , 262-272
Intertwining of maxima of sum of translates functions with nonsingular kernels
MSC: 26A51, 41A50 DOI: 10.21538/0134-4889-2022-28-4-262-272 In previous papers we investigated so-called sum of translates functions $F(\mathbf x,t):=J(t)+\sum_{j=1}^n \nu_j K(t-x_j)$, where $J:[0,1]\to \underline{\mathbb R}:=\mathbb R\cup\{-\infty\}$ is a "sufficiently nondegenerate" and upper-bounded "field function", and $K:[-1,1]\to \underline{\mathbb R}$ is a fixed- Open Access
Operator splitting for abstract Cauchy problems with dynamical boundary condition
In this work we study operator splitting methods for a certain class of coupled abstract Cauchy problems, where the coupling is such that one of the problems prescribes a "boundary type" extra condition for the other one. The theory of one-sided coupled operator matrices provides an excellent framework to study the well-posedness of such problems. We show that with this machinery even operator splitting methods can be treated conveniently and rather efficiently. We consider three specific examples: the Lie (sequential), the Strang and the weighted splitting, and prove the convergence of these methods along with error bounds under fairly general assumptions.
Open AccessOperator splitting for abstract Cauchy problems with dynamical boundary condition
In this work we study operator splitting methods for a certain class of coupled abstract Cauchy problems, where the coupling is such that one of the problems prescribes a "boundary type" extra condition for the other one. The theory of one-sided coupled operator matrices provides an excellent framework to study the well-posedness of such problems. We show that with this machinery even operator splitting methods can be treated conveniently and rather efficiently. We consider three specific examples: the Lie (sequential), the Strang and the weighted splitting, and prove the convergence of these methods along with error bounds under fairly general assumptions.
- Open Access
- 7 April 2021
- Published by Springer Nature in Nonlinear Differential Equations and Applications NoDEA
- Vol. 28 (3) , 1-46
On a class of stochastic partial differential equations with multiple invariant measures
In this work we investigate the long-time behavior for Markov processes obtained as the unique mild solution to stochastic partial differential equations in a Hilbert space. We analyze the existence and characterization of invariant measures as well as convergence of transition probabilities. While in the existing literature typically uniqueness of invariant measures is studied, we focus on the case where the uniqueness of invariant measures fails to hold. Namely, introducing a generalized dissipativity condition combined with a decomposition of the Hilbert space, we prove the existence of multiple limiting distributions in dependence of the initial state of the process and study the convergence of transition probabilities in the Wasserstein 2-distance. Finally, we apply our results to Lévy driven Ornstein–Uhlenbeck processes, the Heath–Jarrow–Morton–Musiela equation as well as to stochastic partial differential equations with delay.
- 7 April 2021
- Published by Springer Nature in Nonlinear Differential Equations and Applications NoDEA
- Vol. 28 (3) , 1-46
Open AccessOn a class of stochastic partial differential equations with multiple invariant measures
In this work we investigate the long-time behavior for Markov processes obtained as the unique mild solution to stochastic partial differential equations in a Hilbert space. We analyze the existence and characterization of invariant measures as well as convergence of transition probabilities. While in the existing literature typically uniqueness of invariant measures is studied, we focus on the case where the uniqueness of invariant measures fails to hold. Namely, introducing a generalized dissipativity condition combined with a decomposition of the Hilbert space, we prove the existence of multiple limiting distributions in dependence of the initial state of the process and study the convergence of transition probabilities in the Wasserstein 2-distance. Finally, we apply our results to Lévy driven Ornstein–Uhlenbeck processes, the Heath–Jarrow–Morton–Musiela equation as well as to stochastic partial differential equations with delay.