Convolution Equations Research Articles

Convolution equation and operators on the euclidean motion group

Let $G = \mathbb{R}^2\rtimes SO(2)$ be the Euclidean motion group, let g be the Lie algebra of G and let U(g) be the universal enveloping algebra of g. Then U(g) is an infinite dimensional, linear associative and non-commutative algebra consisting of invariant differential operators on G. The Dirac measure on G is represented by $\delta_G$, while the convolution product of functions or measures on G is represented by $\ast$. Among other notable results, it is demonstrated that for each u in U(g), there is a distribution E on G such that the convolution equation $u\ast E = \delta_G$ is solved by method of convolution. Further more, it is established that the (convolution) operator $A^\prime : C^\infty_c(G)\rightarrow C^\infty(G),$, which is defined as $A^\prime f = f\ast T^n\delta(t)$ extends to a bounded linear operator on $L^2(G)$, for $f\in C^\infty_c(G)$, the space of infinitely differentiable functions on G with compact support. Furthermore, we demonstrate that the left convolution operator LT denoted as $L_Tf = T\ast f$ commutes with left translation, for $T\in D^\prime(G)$.

Model reference adaptive controller with interpretable fuzzy rules for linear MIMO systems

This paper presents a method for fuzzy model reference adaptive control system design for a partially known multi-input multi-output (MIMO) linear dynamic plant. Differential or convolutional equations can describe the plant to be controlled, and the fuzzy controller is assumed to be from the MIMO sector-bounded nonstationary nonlinearities class. The task is to obtain robust adaptive stabilization. Some absolute stability theorems and integral evaluations of the state and control vectors are applied in the MIMO fuzzy adaptive controller design procedure for the first time. A method for automatically obtaining a near-optimal nonlinear dynamic reference model is described. A practical step-by-step procedure of the MIMO adaptive fuzzy controller design is proposed, resulting in highly interpretable MIMO fuzzy control rules based on triangular fuzzy sets and strong triangular fuzzy partition. An example of the fuzzy controller design according to the proposed procedure is given.

Eliminating the Second-Order Time Dependence from the Time Dependent Schrödinger Equation Using Recursive Fourier Transforms

A strategy is developed for writing the time-dependent Schrödinger Equation (TDSE), and more generally the Dyson Series, as a convolution equation using recursive Fourier transforms, thereby decoupling the second-order integral from the first without using the time ordering operator. The energy distribution is calculated for a number of standard perturbation theory examples at first- and second-order. Possible applications include characterization of photonic spectra for bosonic sampling and four-wave mixing in quantum computation and Bardeen tunneling amplitude in quantum mechanics.

Matrix equation representation of the convolution equation and its unique solvability

Abstract We consider the convolution equation F * X = B F* X=B , where F ∈ R 3 × 3 F\in {{\mathbb{R}}}^{3\times 3} and B ∈ R m × n B\in {{\mathbb{R}}}^{m\times n} are given and X ∈ R m × n X\in {{\mathbb{R}}}^{m\times n} is to be determined. The convolution equation can be regarded as a linear system with a coefficient matrix of special structure. This fact has led to many studies including efficient numerical algorithms for solving the convolution equation. In this study, we show that the convolution equation can be represented as a generalized Sylvester equation. Furthermore, for some realistic examples arising from image processing, we show that the generalized Sylvester equation can be reduced to a simpler form, and we analyze the unique solvability of the convolution equation.

An embedding model for temporal knowledge graphs with long and irregular intervals

As the basis of various downstream applications such as automated question answering and knowledge reasoning, Temporal Knowledge Graph (TKG) embedding learning has attracted many interests. Recent works have explored several methods to learn dynamic embedding for TKGs and led to success in many applications, however, they are difficult to be optimized. In order to accurately catch semantic information propagation of entity-to-entity and entity-to-relation, as well as track dynamics of entities and relations in continuous time, we proposed a novel TKG embedding model ODETKGE. Instead of using recurrent neural network to model dynamics of semantic representations in continuous time, we leverage neural ordinary differential equations to model the dynamic evolution, which enable us to capture long-range dependencies between embeddings at different timestamps, track the dynamics of TKGs with irregular intervals and remain computationally efficient. Furthermore, to effectively model semantics of knowledge graph structure, we combine entity-relation collaboratively updated graph convolutional networks and neural ordinary differential equations. Additionally, to acquire important semantics such as creation and disappearance of relations, we utilize a graph transition layer. Comparing with RNN based models, the proposed ODETKGE can use fewer parameters to achieve deeper layers, and make the transformation of representations between two timestamps smoother. Extensive experiments have been executed on real-world datasets and the results verify that ODETKGE is superior than the state-of-the-art models in link forecasting, long-interval link forecasting, irregular-interval link forecasting, ablation study and training time.

Finite Element Analysis for Linear Viscoelastic Materials Considering Time-Dependent Poisson’s Ratio: Variable Stiffness Method

For linear viscoelastic materials, this paper proposes a finite element analysis method based on an integral constitutive relationship that can simultaneously consider the relaxation behavior of the elastic modulus and the creep Poisson’s ratio. Firstly, the generalized Maxwell model is employed to depict the relaxation characteristics of the elastic modulus, while the generalized Kelvin model is used to represent the creep Poisson’s ratio. Subsequently, the element relaxation stiffness matrix is established, thereby forming a convolutional finite element equation. Furthermore, the recursive calculation of the convolutional integral is derived, and the calculation steps of the finite element for viscoelasticity considering the time-dependent nature of both the elastic modulus and Poisson’s ratio are established. Finally, the accuracy of the proposed algorithm is verified through two numerical examples with linear viscoelastic material. The results indicate that the proposed variable stiffness method for the finite element analysis of linear viscoelastic materials can simultaneously consider the changes in the elastic modulus and Poisson’s ratio over time, thereby establishing a new path and idea for the more accurate simulation of viscoelastic materials’ mechanical properties. Compared with the initial strain method for linear viscoelastic materials, the variable stiffness method proposed in this paper effectively avoids the assumption of constant stress during the micro time interval, thus significantly enhancing the finite element calculation accuracy of linear viscoelastic materials. The proposed method establishes a simulation algorithm that matches existing commercial software with viscoelastic material experiments by considering the elastic modulus and Poisson’s ratio as material parameters.

Convolution, deconvolution, the unit hydrograph and flood routing

Convolution equations are used to relate the input and the output of a system such as rainfall and runoff, or inflow and outflow of a river reach. There have been numerous reports of unsatisfactory results from the deconvolution necessary to calculate the connecting transfer function. The cause is that the equations are ill-conditioned, and it is shown here that the fundamental theoretical solution is that of wild oscillations such as has often been found computationally. A spectral method is proposed for numerical solution, where, instead of individual point values, the transfer function is expressed as series of given continuous functions, where the problem is to determine the coefficients of those functions. The resulting equations have been found to be well-conditioned, and solutions obtained were smooth, bounded, and enabled a certain amount of physical interpretation of the transfer function. The method has been applied to several problems, including typical rainfall–runoff ones and flood routing and wave propagation problems, with quite satisfactory results. Another problem for deconvolution is found to be the traditional use of truncated equations. A remedy is only to use later output data points where convolution with input data does not reach back beyond the initial one. For the routing of larger flood events, the linear methods employed were found to be not so accurate. However as they are a first approximation that requires no knowledge of stream geometry or resistance, and as either discharge or water level hydrographs can be used, they may be useful.

Dynamics of one-dimensional maps and Gurtin–Maccamy's population model. Part I. Asymptotically constant solutions

UDC 517.9 Motivated by the recent work by Ma and Magal [Proc. Amer. Math. Soc. (2021); https://doi.org/10.1090/proc/15629] on the global stability property of the Gurtin–MacCamy's population model, we consider a family of scalar nonlinear convolution equations with unimodal nonlinearities. In particular, we relate the Ivanov and Sharkovsky analysis of singularly perturbed delay differential equations in [https://doi.org/10.1007/978-3-642-61243-5_5] with the asymptotic behavior of solutions of the Gurtin–MacCamy's system. According the classification proposed in [https://doi.org/10.1007/978-3-642-61243-5_5], we can distinguish three fundamental kinds of continuous solutions of our equations, namely, solutions of the asymptotically constant type, relaxation type and turbulent type. We present various conditions assuring that all solutions belong to the first of these three classes. In the setting of unimodal convolution equations, these conditions suggest a generalized version of the famous Wright's conjecture.

On the combination of Lebesgue and Riemann integrals in theory of convolution equations

On the combination of Lebesgue and Riemann integrals in theory of convolution equations

Oscillatory Solution of a Convolutional Volterra Integral Equation

Oscillatory solutions play a pivotal role in understanding functional differential and integral equations, offering insights into the behaviour of these equations' solutions, and assisting in understanding their growth, stability, and convergence properties. This study establishes the oscillatory solution of a convolutional Volterra integral equation using mathematical proofs. Theorems for oscillatory solutions are proposed and proven based on well-defined assumptions, along with an illustrated example. The proofs presented herein reveal that the convolutional Volterra integral equation can exhibit oscillatory or non-oscillatory behavior, contingent upon the characteristics of the function within the integral.

Ob asimptotike spektra integral'nogo operatora s logarifmicheskim yadrom spetsial'nogo vida

We study the asymptotic behavior of the spectrum of an integral operator similarto an integral operator with a logarithmic kernel depending on the sum of arguments. Bya simple change of variables, the corresponding equation is reduced to an integral equation ofconvolution type defined on a finite interval (as is well known, such equations in the general casecannot be solved by quadratures). Next, using the Fourier transform, the equation is reduced toa conjugation problem for analytic functions and then to an infinite system of linear algebraicequations, the isolation of the main terms in which allows deriving a relation that determinesthe spectrum of the original problem.

Solution space characterisation of perturbed linear Volterra integrodifferential convolution equations: The [formula omitted] case

In this paper we characterise the Lp stability of the following perturbed linear Volterra integrodifferential convolution equation, x′(t)=∫[0,t]ν(ds)x(t−s)+f(t),t≥0,where ν is a finite signed Borel measure. Additionally we provide a framework which points to necessary and sufficient conditions on the forcing function that ensures the solution lies in a particular function space. We highlight how such a result is of interest when studying perturbed Stochastic Functional Differential Equations (SFDEs).

A fast time domain solver for the equilibrium Dyson equation

We consider the numerical solution of the real-time equilibrium Dyson equation, which is used in calculations of the dynamical properties of quantum many-body systems. We show that this equation can be written as a system of coupled, nonlinear, convolutional Volterra integro-differential equations, for which the kernel depends self-consistently on the solution. As is typical in the numerical solution of Volterra-type equations, the computational bottleneck is the quadratic-scaling cost of history integration. However, the structure of the nonlinear Volterra integral operator precludes the use of standard fast algorithms. We propose a quasilinear-scaling FFT-based algorithm which respects the structure of the nonlinear integral operator. The resulting method can reach large propagation times and is thus well-suited to explore quantum many-body phenomena at low energy scales. We demonstrate the solver with two standard model systems: the Bethe graph and the Sachdev-Ye-Kitaev model.

Reconstruction of transient acoustic field using sparse real-time near-field acoustic holography

A sparse real-time near-field acoustic holography method is proposed to precisely and stably reconstruct a transient sound field. By a time-domain impulse response function, a time domain convolution equation between the pressure time-wavenumber spectra on the hologram and reconstruction planes is first established. Then, a smoothed ℓ0-norm optimization algorithm is applied to solve the serious ill-conditioned problem of the inverse process to get the pressure time-wavenumber spectrum of the reconstruction plane. The key part of solving this problem is to approximate replace the discontinuous ℓ0-norm by using a suitable continuous Gaussian function family, and a steepest ascent algorithm is introduced to minimize the continuous function for obtaining the optimal solution. Finally, the pressure time-wavenumber spectra of the reconstruction plane for all wavenumbers at all times are solved, and the corresponding time-dependent pressures are gained by the two-dimensional space inverse Fourier transform. A numerical simulation is conducted to observe the reconstruction capacity of the proposed method. The simulation effect prove that the proposed method can accurately reconstruct the transient sound field, and its reconstruction results are compared to those of the real-time near-field acoustic holography (RTNAH) with Tikhonov regularization and YALL1 modal for verifying the superiority of the proposed method. The YALL1 module utilizes an alternating direction algorithm for solving the ℓ1 minimization problems. Several parameters including the noise, position of the holograph plane and two initialization values are discussed in the simulation. The comparison and discussion results show that the reconstruction effect of the proposed method is much better than those of RTNAH with Tikhonov regularization and YALL1 modal under different parameter configurations. An experiment with a percussive steel plate is implemented for further validating the effectivity of the proposed method.

Interpolation Problem with Knots on a Line for Solutions of a Multidimensional Convolution Equation

Interpolation Problem with Knots on a Line for Solutions of a Multidimensional Convolution Equation

Restoration of a blurred photographic image of a moving object obtained at the resolution limit

Objectives. When processing images of the Earth’s surface obtained from satellites, the problem of restoring a blurry image of a moving object is of great practical importance. The aim of this work is to study the possibility of improving the quality of restoration of blurry images obtained at the limit of the resolution of the camera.Methods. Digital signal processing methods informed by the theory of incorrect and ill-conditioned problems were used.Results. The proposed method for restoring a blurred photographic image of a moving object differs from traditional approaches in that the discrete convolution equation, to which the problem of restoring a blurred image is reduced, is obtained by approximating the corresponding integral equation based on the Kotelnikov interpolation series rather than on the traditional basis of the quadrature formula. In the work, formulas are obtained for calculating the kernel of the convolution obtained using the Kotelnikov interpolation series. The discrete convolution inversion problem, which belongs to the class of ill-posed problems, requires regularization. Results of traditional approaches to restoring blurred images using the quadrature formula with Tikhonov regularization and the proposed method based on the Kotelnikov interpolation series are compared. Although the quality of the blurred image restoration is almost the same in both cases, in the quadrature formula the blur value is expressed as an integer number of pixels, while, when using the Kotelnikov series, this value can also be specified in fractions of a pixel.Conclusions. The expediency of discretizing the convolution describing the image distortion of the blur type on the basis of the Kotelnikov interpolation series when processing a blurred image obtained at the limit of the resolution of the camera is demonstrated. In this case, the amount of blur can be expressed in fractions of a pixel. This situation typically arises when processing satellite photography of the Earth’s surface.

Convolution equations with variable time nonlocal coefficients

We consider a one-dimensional nonlocal equation of the form −M((a∗(g∘u))(t))u′′(t)=λf(t,u(t)),t∈(0,1)subject to boundary data. Using classical topological fixed point theory together with a nonstandard order cone, existence of at least one positive solution is established. The novelty lies in the fact that the nonlocal element, (a∗(g∘u))(t), is time varying.

Stability of fixed points in generalized fractional maps of the orders $$0< \alpha <1$$

Caputo fractional (with power-law kernels) and fractional (delta) difference maps belong to a more widely defined class of generalized fractional maps, which are discrete convolutions with some power-law-like functions. The conditions of the asymptotic stability of the fixed points for maps of the orders $0< \alpha <1$ that are derived in this paper are narrower than the conditions of stability for the discrete convolution equations in general and wider than the well-known conditions of stability for the fractional difference maps. The derived stability conditions for the fractional standard and logistic maps coincide with the results previously observed in numerical simulations. In nonlinear maps, one of the derived limits of the fixed-point stability coincides with the fixed-point - asymptotically period two bifurcation point.

Linear topological invariants for kernels of convolution and differential operators

We establish the condition (Ω) for smooth kernels of various types of convolution and differential operators. By the (DN)-(Ω) splitting theorem of Vogt and Wagner, this implies that these operators are surjective on the corresponding spaces of vector-valued smooth functions with values in a product of Montel (DF)-spaces whose strong duals satisfy the condition (DN), e.g., the space D′(Y) of distributions over an open set Y⊆Rn or the space S′(Rn) of tempered distributions. Most notably, we show that:(i)EP(X)={f∈E(X)|P(D)f=0} satisfies (Ω) for any differential operator P(D) and any open convex set X⊆Rd.(ii)Let P∈C[ξ1,ξ2] and X⊆R2 open be such that P(D):E(X)→E(X) is surjective. Then, EP(X) satisfies (Ω).(iii)Let μ∈E′(Rd) be such that E(Rd)→E(Rd),f↦μ⁎f is surjective. Then, {f∈E(Rd)|μ⁎f=0} satisfies (Ω). The central result in this paper is that the space of smooth zero solutions of a general convolution equation satisfies the condition (Ω) if and only if the space of distributional zero solutions of the equation satisfies the condition (PΩ). The above and related statements then follow from known results concerning (PΩ) for distributional kernels of convolution and differential operators [3,15,16].

Dynamic Productivity Prediction Method of Shale Condensate Gas Reservoir Based on Convolution Equation

The dynamic productivity prediction of shale condensate gas reservoirs is of great significance to the optimization of stimulation measures. Therefore, in this study, a dynamic productivity prediction method for shale condensate gas reservoirs based on a convolution equation is proposed. The method has been used to predict the dynamic production of 10 multi-stage fractured horizontal wells in the Duvernay shale condensate gas reservoir. The results show that flow-rate deconvolution algorithms can greatly improve the fitting effect of the Blasingame production decline curve when applied to the analysis of unstable production of shale gas condensate reservoirs. Compared with the production decline analysis method in commercial software HIS Harmony RTA, the productivity prediction method based on a convolution equation of shale condensate gas reservoirs has better fitting affect and higher accuracy of recoverable reserves prediction. Compared with the actual production, the error of production predicted by the convolution equation is generally within 10%. This means it is a fast and accurate method. This study enriches the productivity prediction methods of shale condensate gas reservoirs and has important practical significance for the productivity prediction and stimulation optimization of shale condensate gas reservoirs.