I develop low-rank tensor methods for solving multi-parametric partial differential equations or systems with uncertainties. This task is strongly connected with inverse problems, Bayesian update, data assimilation, optimal design of experiment and optimal control. All these tasks require a huge amount of computational resources, therefore I am also interested in effective parallel algorithms and implementations.
To approximate a random field with as few random variables as possible,
but still retaining the e... more To approximate a random field with as few random variables as possible, but still retaining the essential information, the Karhunen-Lo`eve expansion (KLE) becomes important. Often the random field is characterised by its covariance function. The KLE of a random field requires the solution of eigenvalue problem with the integral operator which has the covariance matrix as its kernel. Usually this eigenvalue problem is solved by a Krylov subspace method with a sparse matrix approximation. We demonstrate the use of the sparse hierarchical matrix (H-matrix) technique with a log-linear computational cost of the matrix-vector product and a log-linear storage requirement.
Parameter identification problems are formulated in a probabilistic language, where the randomnes... more Parameter identification problems are formulated in a probabilistic language, where the randomness reflects the uncertainty about the knowledge of the true values. This setting allows conceptually easily to incorporate new information, e.g. through a measurement, by connecting it to Bayes's theorem. The unknown quantity is modelled as a (may be high-dimensional) random variable. Such a description has two constituents, the measurable function and the measure. One group of methods is identified as updating the measure, the other group changes the measurable function. We connect both groups with the relatively recent methods of functional approximation of stochastic problems, and introduce especially in combination with the second group of methods a new procedure which does not need any sampling, hence works completely deterministically. It also seems to be the fastest and more reliable when compared with other methods. We show by example that it also works for highly nonlinear non-smooth problems with non-Gaussian measures.
Realistic mathematical models of physical processes contain uncertainties. These models are often... more Realistic mathematical models of physical processes contain uncertainties. These models are often described by stochastic differential equations (SDEs) or stochastic partial differential equations (SPDEs) with multiplicative noise, where uncertainties in, e.g. the right-hand side or the coefficients are represented as random fields. To solve a given SPDE numerically one has to discretise the deterministic operator as well as the stochastic fields. The total dimension of the SPDE is the product of the dimensions of the deterministic part and the stochastic part. For approximation of random fields with as few random variables as possible, but still retaining the essential information, the Karhunen-Loève expansion (KLE) becomes important. The KLE of a random field requires the solution of a large eigenvalue problem. Usually it is solved by a Krylov subspace method with a sparse matrix approximation. We demonstrate the use of the low-rank and data sparse hierarchical matrix technique for solving this problem. A log-linear computational cost of the matrix-vector product and a log-linear storage requirement yield to the efficient and fast discretisation of the present random fields.
In parametric equations - stochastic equations are a special case - one may want to approximate t... more In parametric equations - stochastic equations are a special case - one may want to approximate the solution such that it is easy to evaluate its dependence of the parameters. Interpolation in the parameters is an obvious possibility, in this context often labeled as a collocation method. In the frequent situation where one has a "solver" for the equation for a given parameter value - this may be a software component or a program - it is evident that this can independently solve for the parameter values to be interpolated. Such uncoupled methods which allow the use of the original solver are classed as "non-intrusive". By extension, all other methods which produce some kind of coupled system are often - in our view prematurely - classed as "intrusive". We show for simple Galerkin formulations of the parametric problem - which generally produce coupled systems - how one may compute the approximation in a non-intusive way.
In a Bayesian setting, inverse problems and uncertainty quantification (UQ) - the propagation of ... more In a Bayesian setting, inverse problems and uncertainty quantification (UQ) - the propagation of uncertainty through a computational (forward) model - are strongly connected. In the form of conditional expectation the Bayesian update becomes computationally attractive. This is especially the case as together with a functional or spectral approach for the forward UQ there is no need for time-consuming and slowly convergent Monte Carlo sampling. The developed sampling-free non-linear Bayesian update is derived from the variational problem associated with conditional expectation. This formulation in general calls for further discretisation to make the computation possible, and we choose a polynomial approximation. After giving details on the actual computation in the framework of functional or spectral approximations, we demonstrate the workings of the algorithm on a number of examples of increasing complexity. At last, we compare the linear and quadratic Bayesian update on the small but taxing example of the chaotic Lorenz 84 model, where we experiment with the influence of different observation or measurement operators on the update.
The paper deals with the statistical pattern recognition problem for discrete characteristics. We... more The paper deals with the statistical pattern recognition problem for discrete characteristics. We study the behaviour of the minimal empirical error, classifier on the set of arbitrary distributions and the corresponding samples.
ABSTRACT In this work we research the propagation of uncertainties in parameters and airfoil geom... more ABSTRACT In this work we research the propagation of uncertainties in parameters and airfoil geometry to the solution. Typical examples of uncertain parameters are the angle of attack and the Mach number. The discretisation techniques which we used here are the Karhunen-Loève and the polynomial chaos expansions. To integrate high-dimensional integrals in probabilistic space we used Monte Carlo simulations and collocation methods on sparse grids. To reduce storage requirement and computing time, we demonstrate an algorithm for data compression, based on a low-rank approximation of realisations of random fields. This low-rank approximation allows us an efficient postprocessing (e.g. computation of the mean value, variance, etc) with a linear complexity and with drastically reduced memory requirements. Finally, we demonstrate how to compute the Bayesian update for updating a priori probability density function of uncertain parameters. The Bayesian update is also used for incorporation of measurements into the model.
Uncertainty quantification in aerodynamic simulations calls for efficient numerical methods {\col... more Uncertainty quantification in aerodynamic simulations calls for efficient numerical methods {\color{noblue} to reduce computational cost}, especially for the uncertainties caused by random geometry variations which involve a large number of variables. This paper compares five methods, including quasi-Monte Carlo quadrature, polynomial chaos with coefficients determined by sparse quadrature and gradient-enhanced version of kriging, radial basis functions and point collocation polynomial chaos, in their efficiency in estimating statistics of aerodynamic performance upon random perturbation to the airfoil geometry which is parameterized by 9 independent Gaussian variables. The results show that gradient-enhanced surrogate methods achieve better accuracy than direct integration methods with the same computational cost.
To approximate a random field with as few random variables as possible,
but still retaining the e... more To approximate a random field with as few random variables as possible, but still retaining the essential information, the Karhunen-Lo`eve expansion (KLE) becomes important. Often the random field is characterised by its covariance function. The KLE of a random field requires the solution of eigenvalue problem with the integral operator which has the covariance matrix as its kernel. Usually this eigenvalue problem is solved by a Krylov subspace method with a sparse matrix approximation. We demonstrate the use of the sparse hierarchical matrix (H-matrix) technique with a log-linear computational cost of the matrix-vector product and a log-linear storage requirement.
Parameter identification problems are formulated in a probabilistic language, where the randomnes... more Parameter identification problems are formulated in a probabilistic language, where the randomness reflects the uncertainty about the knowledge of the true values. This setting allows conceptually easily to incorporate new information, e.g. through a measurement, by connecting it to Bayes's theorem. The unknown quantity is modelled as a (may be high-dimensional) random variable. Such a description has two constituents, the measurable function and the measure. One group of methods is identified as updating the measure, the other group changes the measurable function. We connect both groups with the relatively recent methods of functional approximation of stochastic problems, and introduce especially in combination with the second group of methods a new procedure which does not need any sampling, hence works completely deterministically. It also seems to be the fastest and more reliable when compared with other methods. We show by example that it also works for highly nonlinear non-smooth problems with non-Gaussian measures.
Realistic mathematical models of physical processes contain uncertainties. These models are often... more Realistic mathematical models of physical processes contain uncertainties. These models are often described by stochastic differential equations (SDEs) or stochastic partial differential equations (SPDEs) with multiplicative noise, where uncertainties in, e.g. the right-hand side or the coefficients are represented as random fields. To solve a given SPDE numerically one has to discretise the deterministic operator as well as the stochastic fields. The total dimension of the SPDE is the product of the dimensions of the deterministic part and the stochastic part. For approximation of random fields with as few random variables as possible, but still retaining the essential information, the Karhunen-Loève expansion (KLE) becomes important. The KLE of a random field requires the solution of a large eigenvalue problem. Usually it is solved by a Krylov subspace method with a sparse matrix approximation. We demonstrate the use of the low-rank and data sparse hierarchical matrix technique for solving this problem. A log-linear computational cost of the matrix-vector product and a log-linear storage requirement yield to the efficient and fast discretisation of the present random fields.
In parametric equations - stochastic equations are a special case - one may want to approximate t... more In parametric equations - stochastic equations are a special case - one may want to approximate the solution such that it is easy to evaluate its dependence of the parameters. Interpolation in the parameters is an obvious possibility, in this context often labeled as a collocation method. In the frequent situation where one has a "solver" for the equation for a given parameter value - this may be a software component or a program - it is evident that this can independently solve for the parameter values to be interpolated. Such uncoupled methods which allow the use of the original solver are classed as "non-intrusive". By extension, all other methods which produce some kind of coupled system are often - in our view prematurely - classed as "intrusive". We show for simple Galerkin formulations of the parametric problem - which generally produce coupled systems - how one may compute the approximation in a non-intusive way.
In a Bayesian setting, inverse problems and uncertainty quantification (UQ) - the propagation of ... more In a Bayesian setting, inverse problems and uncertainty quantification (UQ) - the propagation of uncertainty through a computational (forward) model - are strongly connected. In the form of conditional expectation the Bayesian update becomes computationally attractive. This is especially the case as together with a functional or spectral approach for the forward UQ there is no need for time-consuming and slowly convergent Monte Carlo sampling. The developed sampling-free non-linear Bayesian update is derived from the variational problem associated with conditional expectation. This formulation in general calls for further discretisation to make the computation possible, and we choose a polynomial approximation. After giving details on the actual computation in the framework of functional or spectral approximations, we demonstrate the workings of the algorithm on a number of examples of increasing complexity. At last, we compare the linear and quadratic Bayesian update on the small but taxing example of the chaotic Lorenz 84 model, where we experiment with the influence of different observation or measurement operators on the update.
The paper deals with the statistical pattern recognition problem for discrete characteristics. We... more The paper deals with the statistical pattern recognition problem for discrete characteristics. We study the behaviour of the minimal empirical error, classifier on the set of arbitrary distributions and the corresponding samples.
ABSTRACT In this work we research the propagation of uncertainties in parameters and airfoil geom... more ABSTRACT In this work we research the propagation of uncertainties in parameters and airfoil geometry to the solution. Typical examples of uncertain parameters are the angle of attack and the Mach number. The discretisation techniques which we used here are the Karhunen-Loève and the polynomial chaos expansions. To integrate high-dimensional integrals in probabilistic space we used Monte Carlo simulations and collocation methods on sparse grids. To reduce storage requirement and computing time, we demonstrate an algorithm for data compression, based on a low-rank approximation of realisations of random fields. This low-rank approximation allows us an efficient postprocessing (e.g. computation of the mean value, variance, etc) with a linear complexity and with drastically reduced memory requirements. Finally, we demonstrate how to compute the Bayesian update for updating a priori probability density function of uncertain parameters. The Bayesian update is also used for incorporation of measurements into the model.
Uncertainty quantification in aerodynamic simulations calls for efficient numerical methods {\col... more Uncertainty quantification in aerodynamic simulations calls for efficient numerical methods {\color{noblue} to reduce computational cost}, especially for the uncertainties caused by random geometry variations which involve a large number of variables. This paper compares five methods, including quasi-Monte Carlo quadrature, polynomial chaos with coefficients determined by sparse quadrature and gradient-enhanced version of kriging, radial basis functions and point collocation polynomial chaos, in their efficiency in estimating statistics of aerodynamic performance upon random perturbation to the airfoil geometry which is parameterized by 9 independent Gaussian variables. The results show that gradient-enhanced surrogate methods achieve better accuracy than direct integration methods with the same computational cost.
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Papers by Alexander Litvinenko
but still retaining the essential information, the Karhunen-Lo`eve expansion (KLE) becomes important. Often the random field is characterised by its covariance
function. The KLE of a random field requires the solution of eigenvalue problem with the integral operator which has the covariance matrix as its kernel.
Usually this eigenvalue problem is solved by a Krylov subspace method with a sparse matrix approximation. We demonstrate the use of the sparse hierarchical
matrix (H-matrix) technique with a log-linear computational cost of the matrix-vector product and a log-linear storage requirement.
but still retaining the essential information, the Karhunen-Lo`eve expansion (KLE) becomes important. Often the random field is characterised by its covariance
function. The KLE of a random field requires the solution of eigenvalue problem with the integral operator which has the covariance matrix as its kernel.
Usually this eigenvalue problem is solved by a Krylov subspace method with a sparse matrix approximation. We demonstrate the use of the sparse hierarchical
matrix (H-matrix) technique with a log-linear computational cost of the matrix-vector product and a log-linear storage requirement.